Invariant Markov processes under Lie group actions
Ming Liao
February 26, 2017
Contents
Introduction
1
1 Invariant Markov processes under actions of topological groups
1.1 Invariant Markov processes under group actions . . . . . . . . . . .
1.2 Lévy processes in topological groups . . . . . . . . . . . . . . . . . .
1.3 Lévy process in topological homogeneous spaces . . . . . . . . . . .
1.4 Inhomogeneous Lévy processes . . . . . . . . . . . . . . . . . . . . .
1.5 Markov processes under a non-transitive action . . . . . . . . . . .
1.6 Angular part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Lévy processes in Lie groups
2.1 Generators of Lévy processes in Lie groups
2.2 Lévy measure . . . . . . . . . . . . . . . .
2.3 Stochastic integral equations . . . . . . . .
2.4 Lévy processes in a matrix group . . . . .
2.5 Proof of Theorem 2.9, part 1 . . . . . . . .
2.6 Proof of Theorem 2.9, part 2 . . . . . . . .
3 Lévy processes in homogeneous spaces
3.1 Generators on homogeneous spaces . . .
3.2 From Lie groups to homogeneous spaces
3.3 Convolution semigroups . . . . . . . . .
3.4 Riemannian Brownian motion . . . . . .
4 Lévy processes in compact Lie groups
4.1 Fourier analysis on compact Lie groups
4.2 Lévy processes in compact Lie groups .
4.3 Symmetric Lévy processes . . . . . . .
4.4 bi-invariant Lévy processes . . . . . . .
4.5 Examples . . . . . . . . . . . . . . . .
i
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5
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27
31
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39
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45
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58
67
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71
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97
98
100
108
113
119
5 Spherical transform and Lévy -Khinchin formula
125
5.1
Spherical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2
Lévy -Khinchin formula on homogeneous spaces . . . . . . . . . . . . . . . . 128
5.3
Symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.4
Examples of symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.5
Compact symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.6
Noncompact and Euclidean types . . . . . . . . . . . . . . . . . . . . . . . . 149
6 Inhomogeneous Lévy processes in Lie groups
157
6.1
Additive processes in Euclidean spaces . . . . . . . . . . . . . . . . . . . . . 157
6.2
Measure functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.3
Martingale representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.4
Finite variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.5
A martingale property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.6
A transformation rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.7
Representation in the case of finite variation . . . . . . . . . . . . . . . . . . 187
6.8
Uniqueness of triple and quadruple . . . . . . . . . . . . . . . . . . . . . . . 189
6.9
Representation in the reduced form . . . . . . . . . . . . . . . . . . . . . . . 193
6.10 Some simple properties and proof of Theorem 2.1 . . . . . . . . . . . . . . . 197
6.11 Adding jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
7 Proofs of main results
209
7.1
Matrix-valued measure functions
. . . . . . . . . . . . . . . . . . . . . . . . 209
7.2
Proofs of (A) and (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
7.3
Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
7.4
Uniqueness in distribution under finite variation . . . . . . . . . . . . . . . . 230
7.5
Proof of Theorem 6.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
8 Inhomogenous Lévy processes in homogeneous spaces
245
8.1
Martingale representation on homogeneous spaces . . . . . . . . . . . . . . . 245
8.2
Finite variation and irreducibility . . . . . . . . . . . . . . . . . . . . . . . . 251
8.3
Additional properties and proof of Theorem 3.3 . . . . . . . . . . . . . . . . 257
8.4
Proof of Theorems 8.4 and 8.6 . . . . . . . . . . . . . . . . . . . . . . . . . . 261
9 Decomposition of Markov processes
267
9.1
Decomposition into radial and angular parts . . . . . . . . . . . . . . . . . . 267
9.2
A skew-product decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 274
ii
Appendices
A.1 Lie groups . . . . . . . . . . . . . . . . . . . .
A.2 Action of Lie groups and homogeneous spaces
A.3 Stochastic processes . . . . . . . . . . . . . . .
A.4 Stochastic integrals . . . . . . . . . . . . . . .
A.5 Stochastic differential equations . . . . . . . .
A.6 Poisson random measures . . . . . . . . . . .
A.7 Some miscellaneous results . . . . . . . . . . .
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281
281
284
285
289
293
296
299
Bibliography
305
Index
312
iii
iv
Introduction
The invariance of probability distributions under various transformations plays an important
role in the probability theory. In many examples, the transformations form a Lie group. In
the classical theory, the translation invariant Markov processes in a Euclidean space Rn
can be identified with Lévy processes, which are processes xt that have independent and
stationary increments in sense that for any s < t, xt − xs is independent of the process up to
time s and its distribution depends only on t − s. Lévy processes in Rn have been extensively
studied, but they still generate enormous interests, see Applebaum [3], Bertoin [8] and Sato
[82] for some of modern books on this subject. By the celebrated Lévy -Khinchin formula,
a Lévy process may be represented by a triple of a drift vector, a covariance matrix and a
Lévy measure, in the sense that its distribution is determined by the triple, and to any such
triple, there is a Lévy process, unique in distribution. The purpose of this work is to study
more general invariant Markov processes under the actions of Lie groups, centered around
some useful ways to represent such processes in the spirit of the classical Lévy -Khinchin
representation.
Invariant Markov processes under Lie group actions may be considered at three different
levels of generality. First, we may consider Markov processes in a Lie group G that are invariant under the left (or right) translations. Such processes are direct extensions of classical
Lévy processes in Rn , and can be identified with processes in G that have independent and
−1
stationary increments of the form x−1
s xt (or xt xs ) for s < t. At the second level, we may
consider a Markov process xt in a manifold X that is invariant under the transitive action of
a Lie group G on X. In this case, X may be identified with a homogeneous space G/K, and
xt with a Markov process in G/K invariant under the natural G-action. Hunt [41] (1956)
obtained a formula for the generator of an invariant Markov process in G or G/K (when K
is compact), which allows us to represent such a process in distribution by a triple of a drift
vector, a covariance matrix and a Lévy measure, just as for a Lévy process in Rn .
We will call a Markov process in G invariant under left translations, or a Markov process
in G/K invariant under the natural G-action, a Lévy process. The one-dimensional distributions of a Lévy process form a (weakly) continuous convolution semigroup of probability
measures on G or on G/K, and to study the distribution of the Lévy process is the same
1
as to study the associated convolution semigroup. We will develop some basic properties
of convolution semigroups on G and on G/K. We will also obtain a Lévy -Khinchin type
formula for convolution semigroups on compact Lie groups and symmetric spaces.
A homogeneous space G/K does not possess a natural product structure. To study
invariant Markov processes, and the convolution of probability measures, on G/K, we have
developed a theory of a kind of product “in distribution” on G/K that allows us to carry
over almost all the results on the Lie group G to the homogeneous space G/K. For example,
it is easy to show that Lévy processes in G/K, which are defined as G-invariant Markov
processes, can be characterized as processes that have independent and stationary increments
in a suitable sense, just as their counterparts in G.
At the third level of generality, we may consider a Markov process xt invariant under
the non-transitive action of a Lie group G. In this case, under some suitable assumptions,
xt may be decomposed into a radial part and an angular part. The radial part can be an
arbitrary Markov process in a subspace that is transversal to G-orbits, whereas under the
conditional distribution given the radial part, the angular part is a (time) inhomogeneous
Markov process in a standard G-orbit that is invariant under the G-action.
This leads us to consider inhomogeneous Markov processes that are invariant under a Lie
group action. It is easy to show that inhomogeneous Markov processes xt in a Lie group G
invariant under left translations may be identified with processes that have independent, but
not necessarily stationary, increments x−1
s xt for s < t. Such processes will be called inhomogeneous Lévy processes in G. For stochastically continuous inhomogeneous Lévy processes
in Lie groups, Feinsilver [25] (1978) obtained a martingale property, generalizing a result
of Stroock-Varadhan [84] (1973) for continuous processes. By this martingale property, an
inhomogeneous Lévy process is determined in distribution by a time dependent triple of a
deterministic path, a covariance matrix function and a Lévy measure function. A different
form of martingale representation on more general locally compact groups, in terms of the
abstract Fourier analysis, is obtained in Heyer-Pap [38].
A large portion of this book will be devoted to an extension of this result. We will establish a representation of an inhomogeneous Lévy process by a time dependent triple through
a martingale property, not only on Lie groups, but also on homogeneous spaces, and we will
not assume the process is stochastically continuous. Such a representation is useful because
in principle, the triple can be explicitly constructed and from which the process can be generated, whereas other types of representations, such as convolution semigroups or transition
semigroups, although important in theoretical development, usually cannot be written down
explicitly. To include processes that may not be stochastically continuous is also useful. For
example, in the decomposition of a Markov process xt invariant under a non-transitive action
as mentioned earlier, the conditioned angular part, as an inhomogeneous Lévy process, may
2
not be stochastically continuous even if xt is so. We note that a representation for inhomogeneous Lévy process in Rn by a time dependent triple via Fourier transform, without
assuming stochastic continuity, is obtained in Jacod-Shiryaev [45].
As our purpose in this book is limited to the representation of invariant Markov processes, many aspects of Lévy processes in Lie groups and homogeneous spaces are not discussed
here. For example, Lévy processes in noncompact semisimgple Lie groups and symmetric
spaces exhibit interesting path limiting properties which are not present for their counterparts in Euclidean spaces. Limiting properties of products of random matrices and random
walks in semisimple Lie groups, which may be regarded as discrete time Lévy processes, were
studied by Furstenberg-Kesten [26] (1960), Furstenberg [27] (1963), Tutubalin [87] (1965),
Virtser [89] (1970), Guivarc’h-Raugi [31] (1985), and Raugi [78] (1997). The study of Limiting properties of Brownian motion in semisimple Lie groups and symmetric spaces can be
traced to Dynkin [18] (1961), Orihara [70] (1970) and Malliavin-Malliavin [67] (1972), and
continued in Norris-Rogers-Williams [69] (1986), Taylor [85, 86] (1988, 1991), Babillot [7]
(1991) and Liao [54, 60] (1998,2012). Author’s monograph [55] (2004) provides an account
of limiting and dynamical properties of Lévy processes in noncompact semisimple Lie groups
and symmetric spaces.
The convolution of probability measures on Lie groups and more general locally compact
groups has been studied extensively. Our discussion of this subject is limited to the basic
properties, and the results on Lie groups and associated homogeneous spaces, as in [56, 58,
63]. For a comprehensive treatment, the reader is referred to Heyer’s 1977 classic treatise [35],
which is still highly relevant today. Some of more recent developments can be found in Siebert
[83], Heyer-Pap [37], Pap [72], Dani-McCrudden [15, 16], Hazod-Siebert [32] and Saloff-Coste
[81]. A comprehensive analysis of convolution of probability measures on compact Lie groups
can be found in Applebaum’s recent book [5].
We now provide a brief description for the content of each chapter. In Chapter 1, the
basic definitions and properties of invariant Markov processes are developed under the general setting of actions of locally compact topological groups, including Lévy processes in
topological groups and topological homogeneous spaces, inhomogeneous Lévy processes, and
invariant Markov processes under non-transitive actions. Lévy processes in Lie groups, and
in homogeneous spaces of Lie groups, are discussed in more details in Chapters 2 and 3,
centered around Hunt’s generator formula.
In Chapter 4, we study the distributions of Lévy processes in compact Lie groups based
on Peter-Weyl Theorem, and in Chapter 5, we do the same on symmetric spaces using
spherical transform. Both methods are natural extensions of classical Fourier analysis on
Euclidean spaces. The main results are a Lévy -Khinchin type formula, the smoothness of
distribution densities, and the exponential convergence of the distribution to the normalized
3
Haar measure in the compact cases.
Lévy processes in Lie groups and homogeneous spaces are treated in author’s 2004 monograph [55], so Chapters 2, 3 and 4 have some overlap with Chapters 1, 2 and 4 in [55], but the
present work contains many improvements and additions, including some useful properties
of convolution semigroups on Lie groups and homogeneous spaces.
The representation of inhomogeneous Lévy processes in Lie groups by time-dependent
triples, via a martingale property, as described earlier, is presented in Chapter 6 together
with several different forms of this representation and other related results. The main part
of the proof is given in Chapter 7. The results for inhomogeneous Lévy processes in a
homogeneous space G/K are essentially parallel to those on a Lie group G, and are described
in Chapter 8. These results are applied in Chapter 9 to studya decomposition of an invariant
Markov process under a non-transitive Lie group action, as mentioned earlier. We will obtain
a skew-product decomposition for a continuous invariant Markov process, similar to the well
known skew-product of Brownian motion in Rn .
Some facts of Lie groups and stochastic analysis used in this work are reviewed in the
appendices at the end of the book.
4
Chapter 1
Invariant Markov processes under
actions of topological groups
Although the main purpose of this work is to study invariant Markov processes under actions
of Lie groups, in this first chapter, we will develop the basic definitions and properties under
the more general action of a topological group which is assumed to be locally compact. A
special case is a Markov process in a topological group G invariant under the action of G on
itself by left translations. Such a process may be characterized by independent and stationary
increments, and will be called a Lévy process in G, see section §1.2. A more general case is a
Markov process in a topological space X under the transitive action of a topological group G.
Such a process may be regarded as a process in a topological homogeneous space G/K of G.
We will develop some basic properties under the assumption that K is compact. Because such
a process may be characterized by independent and stationary increments in a certain sense,
so will be called a Lévy process in G/K, see §1.3. These results may be extended to time
inhomogeneous invariant Markov processes, called inhomogeneous Lévy processes, see §1.4.
In all these cases, the distribution of the process is characterized by a convolution semigroup
of probability measures on G or on G/K. It is shown in §1.5 and §1.6 that an invariant
Markov process under a non-transitive action, under some suitable assumptions, may be
decomposed into a radial part and an angular part. The radial part can be an arbitrary
Markov process in a subspace that is transversal to the orbits of the group action, and given
the radial process, the angular part is an inhomogeneous Lévy process in the standard orbit.
The reader is referred to Appendix A.3 for the basic facts of Markov processes.
5
1.1
Invariant Markov processes under group actions
We now state some general definitions and conventions to be used in this work, most of
which are also commonly used in the literature. On a topological space X, let B(X) be its
Borel σ-algebra, and let Bb (X) and B+ (X) be respectively the spaces of B(X)-measurable
functions that are bounded and nonnegative. A measure µ on X is always assumed to be
defined on B(X) unless when explicitly stated otherwise. For a measurable function f , the
∫
integral f dµ may be written as µ(f ). If µ is a measure on a measurable space X and if F
is a measurable map from X to another measurable space Y , then F µ denotes the measure
on Y given by F µ(f ) = µ(f ◦ F ) for any measurable function f ≥ 0 on Y . This may also be
written as F µ(B) = µ(F −1 (B)) for measurable B ⊂ X.
A topological group G is a group and a topological space such that both the product
map G × G → G, (g, h) 7→ gh, and the inverse map G → G, g 7→ g −1 , are continuous. A
continuous action of G on a topological space X is a continuous map G × X → X given by
(g, x) 7→ gx such that g(hx) = (gh)x and ex = x for g, h ∈ G and x ∈ X, where e is the
identity element of G. In the sequel, the action of a topological group G on a topological
space X is always assumed to be continuous unless when explicitly stated otherwise. The
action is called linear if X is a linear space and the map x 7→ gx is linear for each g ∈ G.
The group action defined above is also called a left action. We may also consider a right
action when the action map is written as (x, g) 7→ xg and satisfies (xg)h = x(gh). In the
sequel, all actions are assumed to be left actions unless when explicitly stated otherwise.
A function f or a measure µ on X is called invariant under a measurable map g: X → X,
or g-invariant for short, if f ◦g = f or gµ = µ. An operator T on X with domain D(T ) being
a set of functions on X is map from D(T ) to a possibly different set of functions on X. It is
called g-invariant if ∀f ∈ D(T ), f ◦ g ∈ D(T ) and T (f ◦ g) = (T f ) ◦ g. In particular, a kernel
K from X to itself, which is a family of measures K(x, ·) on X as defined in Appendix A.3,
may be regarded as an operator on X with domain D(K) = B+ (X), then it is g-invariant if
(Kf ) ◦ g = K(f ◦ g) for f ∈ B+ (X). This is equivalent to K(g(x), B) = K(x, g −1 (B)) for
x ∈ X and B ∈ B(X).
A function f , a measure µ, an operator T or a kernel K on X is called invariant under
the action of a group G, or G-invariant for short, if it is g-invariant for any g ∈ G.
For g ∈ G, let lg , rg and cg be respectively the left translation, the right translation and
the conjugation map: G → G, defined by lg x = gx, rg x = xg and cg x = gxg −1 for x ∈ G.
The group G acts on itself by left translation and also by conjugation, whereas the right
translation is a right action of G on itself. A function f or a measure µ on G is called left
invariant if it is invariant under the action of G on itself by left translations. that is, if
f ◦ lg = f or lg µ = µ for g ∈ G. Similarly, an operator T on G is called left invariant if
for f ∈ D(T ), f ◦ lg ∈ D(T ) and (T f ) ◦ lg = T (f ◦ lg ) for g ∈ G. If this holds only for g
6
contained in a subgroup K of G, then f , µ or T is called K-left invariant. The (K-) right
invariant and (K-) conjugate invariant functions, measures and operators on G are defined
similarly. When they are both left and right (resp K-left and K-right) invariant, then they
are called bi-invariant (resp. K-bi-invariant).
Throughout this chapter, let X be a topological space and let G be a topological group
that acts continuously on X, both are equipped with lcscH (locally compact and second
countable Hausdorff) topologies. We will let C(X), Cb (X), Cc (X) and C0 (X) denote respectively the spaces of continuous functions, bounded continuous functions, continuous functions
with compact supports and continuous functions convergent to 0 at infinity (under the onepoint compactification topology) on X. Note that C0 (X) may also be characterised as the
space of continuous functions f on X such that for any ε > 0, there is a compact K ⊂ X
with |f | < ε on K c (the complement of K in X).
A Markov process xt in X, t ∈ R+ = [0, ∞), usually means a family of processes, one for
each starting point x ∈ X, associated to a transition semigroup Pt and satisfying the simple
Markov property. Recall from Appendix A.3 that this means: for t > s ≥ 0 and f ∈ Bb (X),
E[f (xt ) | Fs ] = Pt−s f (xs )
P almost surely,
(1.1)
where {Ft } is the natural filtration of process xt . If (1.1) holds under a larger filtration {Ft },
a possibly stronger requirement, then xt is called a Markov process associated to {Ft } or
an {Ft }-Markov process. The symbol Px is used to denote the distribution of the process
starting at x on the canonical path space and Ex is the associated expectation. Occasionally,
a Markov process means a single process with a given initial distribution and this should be
clear from context. We will allow a Markov process xt to have a possibly finite life time ζ
as described in Appendix A.3.
We may also consider an inhomogeneous Markov process xt in X, which is a family of
processes, one for each pair of starting time s ∈ R+ and starting point x ∈ X, associated to a two-parameter transition semigroup Ps,t , 0 ≤ s ≤ t, and satisfying the following
inhomogeneous Markov property:
E[f (xt ) | Fs ] = Ps,t f (xs ),
P -almost surely.
(1.2)
where {Ft } is the natural filtration of process xt (see Appendix A.3). If the above holds for
a larger filtration {Ft }, then the inhomogeneous Markov process xt is said to be associated
to {Ft }.
A Markov process xt in X is called invariant under a measurable map g: X → X, or
g-invariant for short, if its transition semigroup Pt , as an operator for each t, is g-invariant,
that is, if
Pt (g(x), B) = Pt (x, g −1 (B))
(1.3)
7
for any t ∈ R+ , x ∈ X and B ∈ B(X). Note that (1.3) is equivalent to
Egx [f (xt )] = Ex [f (gxt )]
(1.4)
for any t ∈ R+ , x ∈ X and f ∈ Bb (X).
Proposition 1.1 A Markov process xt in X is g-invariant if and only if for any x ∈ X, the
process gxt with x0 = x has the same distribution as the process xt with x0 = gx.
Proof It is clear that the same distribution implies the g-invariance (1.4). Now assume (1.4).
Let {Ft } be the natural filtration of process xt . Then for 0 < t1 < t2 and f1 , f2 ∈ Bb (X), by
the simple Markov property,
Egx [f1 (xt1 )f2 (xt2 )] = Egx {f1 (xt1 )Egx [f2 (xt2 ) | Ft1 ]} = Egx [f1 (xt1 )Pt2 −t2 f2 (xt1 )]
= Ex [f1 (gxt1 )Pt2 −t1 f (gxt1 )] = Ex [f1 (gxt1 )Pt2 −t1 (f ◦ g)f (xt1 )]
= Ex {f1 (gxt1 )Ex [f2 (gxt2 ) | Ft1 ]} = Ex [f1 (gxt1 )f2 (gxt2 )].
Inductively, it can be shown that for t1 < t2 < · · · < tn ,
Egx [f1 (xt1 )f2 (xt2 ) · · · fn (xtn )] = Ex [f1 (gxt1 )f2 (gxt2 ) · · · fn (gxtn )].
This proves the same distribution. 2
The g-invariance for an inhomogeneous Markov process xt in X is defined in a similar
fashion. The process xt is called invariant under a measurable map g: X → X, or g-invariant,
if its transition semigroup Ps,t is g-invariant, that is, if
Ps,t (gx, B) = Ps,t (x, g −1 (B)).
∀t ≥ s ≥ 0, x ∈ X and B ∈ B(X),
(1.5)
The following proposition may be proved as Proposition 1.1.
Proposition 1.2 An inhomogeneous Markov process xt is g-invariant if and only if for all
s ∈ R+ and x ∈ X, the process gxt , t ≥ s, with xs = x has the same distribution as process
xt , t ≥ s, with xs = gx.
A Markov process xt in X is called invariant under the action of a group G, or G-invariant,
if its transition semigroup Pt is g-invariant for any g ∈ G. Similarly, an inhomogeneous
Markov process xt in X is called G-invariant if its transition semigroup Ps,t is G-invariant.
8
1.2
Lévy processes in topological groups
A Markov process xt in a topological group G invariant under the action of G on itself by
left translation is called left invariant. Its transition semigroup Pt is left invariant in the
sense that
Pt (f ◦ lg )(x) = Pt f (gx)
(1.6)
for t ∈ R+ , x, g ∈ G and f ∈ Bb (G).
A process is said to have rcll paths, or called a rcll process, if almost all its paths are
right continuous with left limits in its state space. In literature, a rcll process is often called
càdlàg (French “continue à droite, limite à gauche”). Most processes in this work will be
assumed to be rcll. We will see in Remarks 1.5, 1.15, 1.21 and 1.28 that many of these
processes in fact have rcll versions if they are continuous in distribution in some sense.
Let xt be a rcll left invariant Markov process in G. Recall e is the identity element of
G. Then Pt f (x) = Pt (f ◦ lx )(e) = Ee [f (xxt )]. By the rcll paths, one sees that Pt is a Feller
transition semigroup (see Appendix A.3 for definition of Feller processes), and hence xt is a
Feller process.
Let xt be a process in G with an infinite life time, and let {Ftx } be its natural filtration.
x
It is said to have independent increments if for s < t, x−1
s xt is independent of Fs . It is
said to have stationary increments if for s < t, the distribution of x−1
s xt depends only on
−1
t − s, that is, if x−1
s xt and x0 xt−s have the same distribution. A rcll process xt is called a
Lévy process in G if it has independent and stationary increments.
The classical example is a Lévy process xt in the d-dim Euclidean space Rd when Rd is
regarded as an additive group. In this case, the increment x−1
s xt is written as xt − xs .
For a Lévy process xt in G, let
xet = x−1
0 xt .
(1.7)
It is clear that process xet is also a Lévy process in G starting at e and is independent of x0 .
Let xt be a Lévy process in G. For t ∈ R+ , x ∈ G and f ∈ Bb (G), put
Pt f (x) = E[f (xxet )].
(1.8)
It is easy to see that this defines a transition semigroup Pt on G, which is conservative, that
is, Pt 1 = 1, and is left invariant.
For t > s,
e
E[f (xt ) | Fs ] = E[f (xs x−1
s xt ) | Fs ] = E[f (xxt−s )] |x=xs = Pt−s f (xs ).
This shows that the Lévy process xt is a left invariant Markov process.
9
Now let xt be a rcll left invariant Markov process in G with an infinite life time. Then
for t > s and f ∈ Bb (G),
−1
E[f (x−1
)(xs ) = Pt−s f (e).
s xt ) | Fs ] = E[f (x xt ) | Fs ] |x=xs = Pt−s (f ◦ lx−1
s
This shows that xt has independent and stationary increments, and hence is a Lévy process.
We have proved that the class of Lévy processes in G coincides with the class of left invariant
rcll Markov processes in G with infinite life times.
Let xt be a left invariant rcll Markov process in G with a possibly finite life time.
Then Pt 1(x) = Pt (1 ◦ lx )(e) = Pt 1(e), which is right continuous in t due to the right
continuity of paths. By the semigroup property and the left invariance of Pt , Ps+t 1(e) =
∫
Ps (e, dx)Pt 1(x) = Ps 1(e)Pt 1(e). It follows that Pt 1(x) = e−λt for all x ∈ G and for some
fixed λ ≥ 0. Let P̂t = eλt Pt . It is easy to see that P̂t is a conservative and left invariant Feller
transition semigroup. The Feller process x̂t associated to P̂t is a left invariant rcll Markov
process with an infinite life time, and hence is a Lévy process. Let τ be an exponential
random variable of rate λ (τ = ∞ if λ = 0) that is independent of process x̂t , and let x′t be
the process x̂t killed at time τ , that is, x′t = x̂t for t < τ and x′t = ∆ for t ≥ τ , where ∆ is the
point at infinity (see Appendix A.3). It is easy to show that x′t is a rcll Markov process with
transition semigroup Pt and hence is identical in distribution to process xt . To summarize,
we have proved the following result.
Theorem 1.3 A left invariant rcll Markov process in G with an infinite life time is a
Lévy process. Conversely, a Lévy process xt in G is a left invariant rcll Markov process
with an infinite life time and its transition semigroup is given by (1.8).
In general, a left invariant rcll Markov process xt in G with a possibly finite life time is
a Feller process and is identical in distribution to a Lévy process x̂t killed at an independent
exponential time of rate λ ≥ 0. The transition semigroup Pt of xt and P̂t of x̂t are related
as Pt = e−λt P̂t .
Remark 1.4 Let xt be a process in G with an infinite life time. The above proof shows
that xt is a left invariant Markov process if and only if it has independent and stationary
increments.
Remark 1.5 Let xt be a left invariant Markov process in G without assuming an infinite life
∫
time, and let Pt be its transition semigroup. Then Pt f (x) = Pt (f ◦ lx )(e) = G f (xy)µt (dy),
where µt = Pt (e, ·). If µt → δe (the unit point mass at e) weakly as t → 0, then Pt is Feller
and hence xt has a rcll version. Therefore, if xt also has an infinite life time, then after a
modification on a null set for each t ≥ 0, xt becomes a Lévy process in G.
10
Given a filtration {Ft } on the underlying probability space (Ω, F, P ). A Lévy process
xt is called associated to {Ft }, or an {Ft }-Lévy process, if it is adapted to {Ft } and for
any s < t, x−1
s xt is independent of Fs . A Lévy process is clearly associated to its natural
filtration. By the proof of Theorem 1.3, it is easy to see that a Lévy process is associated to
a filtration if and only if it is associated to the same filtration as a Markov process.
It is easy to see that if xt is a Lévy process associated to a filtration {Ft } and if s > 0 is
e
fixed, then x′t = x−1
s xs+t is a Lévy process identical in distribution to the process xt and is
independent of Fs . The following theorem says that s may be replaced by a stopping time.
Theorem 1.6 Let xt be a Lévy process associated to a filtration {Ft } in G. If τ is an {Ft }stopping time with P (τ < ∞) > 0, then under the conditional probability P (· | τ < ∞), the
process x′t = x−1
τ xτ +t is a Lévy process in G that is independent of Fτ . Moreover, the process
′
xt under P (· | τ < ∞) has the same distribution as the process xet under P .
Proof First assume τ takes only discrete values. Fix 0 < t1 < t2 < · · · < tk , ϕ ∈ Cb (Gk )
and ξ ∈ (Fτ )+ . Because ξ1[τ =t] ∈ (Ft )+ , where 1A denotes the indicator of set A, we have
−1
E[ϕ(x−1
τ xτ +t1 , · · · , xτ xτ +tk )ξ | τ < ∞]
=
∑
−1
E[ϕ(x−1
t xt+t1 , · · · , xt xt+tk )ξ1[τ =t] ]/P (τ < ∞)
t<∞
=
∑
−1
E[ϕ(x−1
t xt+t1 , · · · , xt xt+tk )]E(ξ1[τ =t] )/P (τ < ∞)
t<∞
−1
= E[ϕ(x−1
0 xt1 , · · · , x0 xtk )]E(ξ | τ < ∞)
(1.9)
−1
−1
−1
Setting ξ = 1 yields E[ϕ(x−1
τ xτ +t1 , · · · , xτ xτ +tk ) | τ < ∞] = E[ϕ(x0 xt1 , · · · , x0 xtk )].
Therefore, for a general ξ ∈ (Fτ0 )+ , the expression in (1.9) is equal to
−1
E[ϕ(x−1
τ xτ +t1 , · · · , xτ xτ +tk ) | τ < ∞] E(ξ | τ < ∞).
This proves the desired result for a discrete stopping time τ .
For a general stopping time τ , let τn = (k + 1)2−n on the set [k · 2−n ≤ τ < (k + 1)2−n ]
for k = 0, 1, 2, . . .. Then τn are discrete stopping times and τn ↓ τ as n ↑ ∞. The result for
τ follows from the discrete case and the right continuity of xt . 2
The convolution of two measures µ and ν on G is the measure µ ∗ ν on G determined by
µ ∗ ν(f ) =
∫
f (xy)µ(dx)ν(dy)
(1.10)
for f ∈ B+ (G), and it is a probability measure if so are µ and ν. It is easy to check that the
convolution is associative, that is, (µ ∗ ν) ∗ γ = µ ∗ (ν ∗ γ), and hence, it is meaningful write
a convolution product µ1 ∗ µ2 ∗ · · · ∗ µn or an n-fold convolution power µ∗n = µ ∗ µ ∗ · · · ∗ µ.
11
A family of probability measures µt on G, t ≥ 0, is called a convolution semigroup on G
if µs+t = µs ∗µt . Although convolution semigroup may be defined for more general measures,
in this work, a convolution semigroup will always mean a family of probability measures. It
is called continuous if µt → µ0 weakly as t → 0. By [35, Theorem 1.5.7], if µt is continuous,
then µt → µs weakly as t → s for all s > 0. An alternative proof will be given shortly.
Let xt be a Lévy process in G. It is easy to see that Pt (e, ·), the distributions µt of
e
xt , t ≥ 0, form a continuous convolution semigroup of probability measures on G (the
continuity follows from the right continuity of paths) with µ0 = δe , which will be called
the convolution semigroup associated to the Lévy process gt . On the other hand, if µt is a
continuous convolution semigroup on G with µ0 = δe , then
Pt f (x) = µt (f ◦ lx ),
(1.11)
for f ∈ C0 (G), defines a left invariant conservative Feller transition semigroup Pt on G. There
is a Feller process xt associated to Pt , which is left invariant, and hence is a Lévy process
with µt as the associated convolution semigroup.
Because xt is a Feller process, it is stochastically continuous, that is, xt = xt− almost
surely for all t > 0 (see for example ([46, Proposition 25.20]). It follows that µt is in fact
continuous in t under the weak convergence, as mentioned before.
To summarize, we have the following result.
Theorem 1.7 For a Lévy process xt in G, the distributions µt of xet = x−1
0 xt form a continuous convolution semigroup on G with µ0 = δe . conversely, if µt is such a convolution
semigroup on G, then there is a Lévy process xt in G with µt as distribution of xet .
Using the Markov property, is easy to show that the finite dimensional distributions of a
Lévy process xt in G with associated convolution semigroup µt are given by
∫
E[f (xt1 , xt2 . . . , xtn )] =
f (x0 x1 , x0 x1 x2 , . . . , x0 x1 x2 · · · xn )
µ0 (dx0 )µt1 (dx1 )µt2 −t1 (dx2 ) · · · µtn −tn−1 (dxn )
(1.12)
for f ∈ Bb (X n ) and 0 ≤ t1 < t2 < · · · < tn , where µ0 is the initial distribution.
Remark 1.8 In the definition of Lévy processes, if the increments x−1
s xt for s < t are
−1
replaced by xt xs , this will lead to a different definition of Lévy processes, which coincide with
Markov processes in G that are invariant under right translations. In [55], the Lévy processes
defined earlier and defined here are called respectively left and right Lévy processes. Because
the group G is in general non-commutative, left and right Lévy processes are two different
classes of processes, but they are in natural duality under the map xt 7→ x−1
t . All the
preceding results established for left Lévy processes hold also for right Lévy processes with
12
−1
e
suitable changes. For example, xet = x−1
0 xt should be changed to xt = xt x0 and (1.11) holds
with lx replaced by rx . Note that a continuous convolution semigroup µt on G with µ0 = δe
can be used to generate either a left Lévy process by (1.11) or a right Lévy process by the
counter part of (1.11) with lx replaced by rx .
In this work, we will exclusively consider left Lévy processes, unless when explicitly stated
otherwise, and we will omit the adjective “left” in its name.
1.3
Lévy process in topological homogeneous spaces
Let X be a topological space and let G be a topological group that acts transitively on X,
both are equipped with lcscH topologies. Fix a point o in X. The isotropy subgroup K of
G at o, defined by K = {g ∈ G; go = o}, is a closed subgroup of G.
For any closed subgroup K of G, the space G/K of left cosets gK, g ∈ K, is called a
homogeneous space of G. It is equipped with the quotient topology, under which the natural
projection
π : G → G/K, g 7→ gK,
is continuous and open. Under this topology, the natural action of G on G/K, given by
xK 7→ gxK for g ∈ G, is continuous. Moreover, when K is the isotropy subgroup at o ∈ X
as defined above, then under the map: gK 7→ go, G/K is homeomorphic to X, and the
G-action on X is just the natural action of G on G/K (see Theorem 3.2 in [33, chapter II]).
We may identify X and G/K together with the associated G-actions in this way. Under
this identification, the natural projection π: G → G/K is just the map: G → X given by
g 7→ go, where o = eK.
Note that if G is a topological group with a lcscH topology, and if K is a compact
subgroup, then the homogeneous space X = G/K, under the quotient topology, is a lcscH
space.
In the rest of this section, we will assume K is compact. We will also assume the homogeneous space G/K has a continuous local section in the sense that there are a neighborhood
U of o in X and a continuous map ϕ: U → G such that π ◦ ϕ = idU (the identity map
on U ). Because the isotropy subgroup at a different point go, g ∈ G, is gKg −1 , it can be
shown that these assumptions are independent of the choice for the point o ∈ X. Under
these assumptions, it can be shown that the map: (x, k) 7→ ϕ(x)k is a homeomorphism from
U × K onto a neighborhood of e of G, see Appendix A.7 for more details.
It is well known and also easy to see that a continuous local section, even a smooth
local section, exists if G is a Lie group. By Theorems 4.6 and 4.7, and Corollary 4.8 in
[50, Chapter I], the group G of isometries on a connected Riemannian manifold X has a
Lie group structure and acts smoothly on X, and the isotropy subgroup K of G at any
13
o ∈ X is compact. In this case, if G acts transitively on X, then X = G/K satisfies all the
assumptions stated here.
A measurable map S: X → G is called a section map on X if π ◦ S = idX . By the
existence of a continuous local section, one can always construct a section map on X (see
Appendix A.7). In general, it may not be continuous on X, but for any x ∈ X, there is a
section map that is continuous on a neighborhood of x (even smooth there when G is a Lie
group).
A measure on a lcscH space is called a Radon measure if it has a finite charge on any
compact set. Note that the usual definition of Radon measures on Hausdorff spaces includes
a regularity condition, which is automatically satisfied on a lcscH space. It is well known
that on a topological group G equipped with a lcscH topology, there is a nontrivial left (resp.
right) invariant Radon measure µ, called a left (resp. right) Haar measure on G, which is
unique up to a multiplicative constant. In general, left Haar measures do not agree with right
Haar measures, but when they do, the group G is called unimodular. In this case, we will
simply say Haar measures. It is easy to show (by applying right translations and the inverse
map to a left Haar measure) that a compact group K is unimodular and there is a unique
left invariant probability measure on K, which is also invariant under right translations and
the inverse map on K. This measure is called the normalized Haar measure on the compact
group K, and is denoted as ρK or simply dk in computation.
The convolution between two measures µ and ν on X is the measure µ ∗ ν defined by
µ ∗ ν(f ) =
∫
∫
f (S(x)ky)dkµ(dx)ν(dy),
X×X
(1.13)
K
for any f ∈ B+ (X). This definition does not depend on the choice for the section map S
(because if S ′ is another section map, then S ′ (x) = S(x)kx for some kx ∈ K), and reduces to
the convolution on G when K = {e}. Because for any g ∈ G and x ∈ X, S(gx) = gS(x)k for
some k = k(g, x) ∈ K, it is easy to show that the convolution on X, as on G, is associative,
that is, (µ ∗ ν) ∗ γ = µ ∗ (ν ∗ γ), so the n-fold convolution µ1 ∗ µ2 ∗ · · · ∗ µn is well defined.
Recall a measure µ on X is called K-invariant if kµ = µ for any k ∈ K. If ν is a
K-invariant measure on X, then µ ∗ ν can be written a little more concisely as
µ ∗ ν(f ) =
∫
f (S(x)y)µ(dx)ν(dy)
(1.14)
X×X
for f ∈ B+ (X), which does not depend on the choice for the section map S. Moreover, if µ
is also K-invariant, then so is µ ∗ ν.
A convolution semigroups on X and its continuity are defined in the same way as on G
given in §1.2. Thus, a family of probability measures µt on X, t ∈ R+ , is called a convolution
semigroup on X if µs+t = µs ∗ µt , and it is called continuous if µt → µ0 weakly as t → 0.
Recall that if µt is a continuous convolution semigroup on G, then µt → µs weakly as t → s
14
for any s > 0. The same holds for a continuous convolution semigroup µt on X, as an easy
consequence of Proposition 1.12 later.
The following result provides a basic relation between invariant measures on G and on
X = G/K. Recall a measure µ on G is called K-left or K-right invariant if lk µ = µ or
rk µ = µ for k ∈ K, and K-bi-invariant if it is both K-left and K-right invariant.
Proposition 1.9 The map
µ 7→ ν = πµ
is a bijection from the set of K-right invariant measures µ on G onto the set of measures ν
on X. It is also a bijection from the set of K-bi-invariant measures µ on G onto the set of
K-invariant measures ν on X. Moreover, if ν is a measure on X, then the unique K-right
invariant measure µ on G satisfying ν = πµ is given by
∫
∀f ∈ B+ (G),
∫
µ(f ) =
f (S(x)k)dk ν(dx),
X
(1.15)
K
where S is any section map on X. Furthermore, the map µ → πµ preserves the convolution
in the sense that for any measures µ1 and µ2 on G,
π(µ1 ∗ µ2 ) = (πµ1 ) ∗ (πµ2 ),
(1.16)
provided one of the following three conditions holds: µ1 is K-right invariant, or µ2 is K-left
invariant, or µ2 is K-conjugate invariant.
Proof For g ∈ G, S ◦ π(g) = gk for some k ∈ K. If µ is a K-right invariant measure on G
with πµ = ν, then for f ∈ B+ (G),
∫
µ(f ) =
K
µ(f ◦ rk )dk =
∫
K
µ(f ◦ rk ◦ S ◦ π)dk =
∫
K
ν(f ◦ rk ◦ S)dk.
This shows that µ satisfies (1.15). Conversely, using the K-right invariance of dk, it is easy
to show that µ given by (1.15) is K-right invariant with πµ = ν. It is also clear that µ is
K-bi-invariant if and only if ν is K-invariant. For two measures µ1 and µ2 on G, satisfying
one of the three conditions stated above, let ν1 = πµ1 and ν2 = πµ2 . Then for f ∈ B+ (X),
π(µ1 ∗ µ2 )(f ) =
∫
=
∫
=
∫
=
∫
∫
f (π(g1 g2 ))µ1 (dg1 )µ2 (dg2 ) =
∫
f (g1 kπ(g2 ))µ1 (dg1 )ρK (dk)µ2 (dg2 ) =
f (π(g1 kg2 ))µ1 (dg1 )ρK (dk)µ2 (dg2 )
f (gky)µ1 (dg)ρK (dk)ν2 (dy)
f (S(π(g))k ′ ky)µ1 (dg)ρK (dk)ν2 (dy) (for some k ′ ∈ K)
f (S(x)ky)ν1 (dx)ρK (dk) ν2 (dy) = ν1 ∗ ν2 (f ). 2
15
Proposition 1.10 Let ρG be a left Haar measure on G.
(a) ρG is K-right invariant.
(b) ρX = πρG is a G-invariant measure on X = G/K, and any G-invariant measure on X
is cρX for some constant c ≥ 0. In particular, if G is compact and if ρG is the normalized
Haar measure on G, then ρX is the unique G-invariant probability measure on X.
Proof: For k ∈ K, rk ρG is left invariant and so rk ρG = λ(k)ρG for some λ(k) > 0. For
f ∈ Cc (G), ρG (f ◦ rk ) = λ(k)ρG (f ). This shows k 7→ λ(k) is continuous. Then k 7→ λ(k −1 ) is
a continuous group homomorphism from K into the multiplicative group (0, ∞). Its range
as a compact subgroup is necessarily {1}, and hence rk ρG = ρG . This proves (a). It is clear
that ρX = πρG is G-invariant. For any G-invariant measure ν on X, the unique K-right
invariant measure µ on G with πµ = ν, given by (1.15), is clearly left invariant, and so
µ = cρG for some constant c ≥ 0. This implies ν = cρX and proves (b). 2
In the literature (for example, in [29]), the convolution of measures in X = G/K has
been defined by identifying the measures on X with the K-right invariant measures in G
and then use the convolution on G. By Proposition 1.9, this definition is consistent with our
definition given in (1.13).
Convolution of functions on X = G/K have appeared in literature under various contexts,
but in the present setting, they all take the following form (see for example [88]): For
f1 , f2 ∈ B+ (X),
∫
f1 ∗ f2 (gK) =
f1 (hK)f2 (h−1 gK)ρ(dh), g ∈ G,
(1.17)
G
where ρ denotes a left Haar measure on G. This definition is compatible with our definition
of convolution of measures on X by the following proposition.
Proposition 1.11 If µ1 and µ2 are measures on X = G/K with densities f1 and f2 with
respect to πρ, then µ1 ∗ µ2 has density f1 ∗ f2 .
Proof: By Proposition 1.10 (a), ρ is K-right invarint. For f ∈ B+ (X), writing dg and dh
for ρ(dg) and ρ(dh) respectively,
µ1 ∗ µ2 (f ) =
∫
∫
f (S(x)ky)dkµ(dx)ν(dy) =
∫
=
∫
=
f (hgK)f1 (hK)f2 (gK)dhdg =
∫
∫
f (S(hK)kgK)dkf1 (hK)f2 (gK)dhdg
f (gK)[
G
G
f1 (hK)f2 (h−1 gK)dh]dg
f (gK)(f1 ∗ f2 )(gK)dg. 2
A probability measure µ on G satisfying µ ∗ µ = µ is called an idempotent. By [35,
Theorem 1.2.10], if µ is an idempotent, then µ = ρH for some compact subgroup H of G.
For a convolution semigroup µt on G, µ0 ∗ µ0 = µ0 , so µ0 = ρH for some compact subgroup
H of G. Then µt is H-bi-invariant because µt = µ0 ∗ µt = µt ∗ µ0 .
16
Proposition 1.12 (a) If a convolution semigroup µt on G is K-right invariant, that is, if
each µt is K-right invariant, then it is K-bi-invariant.
(b) If νt is a convolution semigroup on X = G/K, then each νt is K-invariant, and ν0 = πρH
for some compact subgroup H of G containing K.
(c) The map
µt 7→ νt = πµt
is a bijection from the set of K-bi-invariant convolution semigroups µt on G onto the set of
convolution semigroups νt on X = G/K. Moreover, µt is continuous if and only if so is νt .
Proof: By the preceding discussion, µ0 = ρH and µt is H-bi-invariant for some compact
subgroup H of G. The K-right invariance of µ0 implies K ⊂ H. This proves (a). Let νt
be a convolution semigroup on X, and let µt be the unique K-right invariant probability
measure on G with πµt = νt . By Proposition 1.9 and (a), µt is a K-bi-invariant convolution
semigroup on G, and hence νt = πµt is K-invariant. Moreover, µ0 = ρH for some compact
subgroup H of G. Because µ0 is K-bi-invariant, K ⊂ H. This proves (b). Now (c) follows
from (b) and Proposition 1.9. Note that to derive the continuity of µt from that of νt , (1.15)
∫
is used, where K f (S(x)k)ρK (dk) is continuous in x if f is bounded continuous on G. This
is because the integral does not depend on S, and S may be chosen to be continuous near
any x (that is, in a neighborhood of x) 2
Because a left invariant rcll Markov process in G with an infinite life time is a Lévy process
in G, a G-invariant rcll Markov process xt in X = G/K with an infinite life time will be
called a Lévy process in X. Let Pt be its transition semigroup. By the G-invariance of Pt , it
is clear that µt = Pt (o, ·), t ∈ R+ , are K-invariant measures. We now show that they form
a continuous convolution semigroup on G with µ0 = δo , which will be called the convolution
semigroup associated to the Lévy process xt . For f ∈ Bb (X) and s, t ∈ R+ ,
∫
µs+t (f ) = Ps+t f (o) = Ps Pt f (o) =
∫
=
∫X
∫
X
µs (dx)Pt f (x) =
X
µs (dx)Pt f (S(x)o)
µs (dx)Pt (f ◦ S(x))(o) (by the G-invariance of Pt )
=
X×X
µs (dx)µt (dy)f (S(x)y) = µs ∗ µt (f ).
This shows that µt = Pt (o, ·), t ∈ R+ , form a convolution semigroup on X, which is continuous by the right continuity of xt .
On the other hand, given a continuous convolution semigroup µt on X with µ0 = δo , let
∫
Pt f (x) =
f (S(x)y)µt (dy)
(1.18)
for t ∈ R+ , x ∈ X and f ∈ B+ (X). Because µt is K-invariant, this expression does not
depend on the choice for the section map S, and defines a conservative G-invariant Feller
17
transition semigroup Pt on X (recall that S may be chosen to be continuous near any point
in X). The associated Feller process xt is a G-invariant rcll Markov process with an infinite
life time, and hence is a Lévy process in X with µt as the associated convolution semigroup.
For a G-invariant rcll Markov process xt in X with a possibly finite life time, it can be
shown as on G that its transition semigroup Pt satisfies Pt 1(x) = e−λt for some λ ≥ 0, and
hence xt is equal in distribution to a Lévy process killed at an independent exponential time
of rate λ. To summarize, we have the following result.
Theorem 1.13 Let xt be a Lévy process in X = G/K, that is, a G-invariant rcll Markov
process with an infinite life time, and let Pt be its transition semigroup. Then µt = Pt (o, ·)
is a continuous convolution semigroup on X with µ0 = δo . conversely, if µt is such a
convolution semigroup on X, then there is a Lévy process xt in X with x0 = o such that µt
is the distribution of xt .
In general, a G-invariant rcll Markov process xt in X with a possibly finite life time is
identical in distribution to a Lévy process x̂t in X killed at an independent exponential time
of rate λ ≥ 0. The transition semigroup Pt of xt and P̂t of x̂t are related as Pt = e−λt P̂t .
Although there is no natural product structure on the homogeneous space X = G/K,
the integrals like
∫
∫
f (xy)µ(dy) =
∫
f (S(x)y)µ(dy),
∫
f (xy, xyz)µ(dy)ν(dz) =
∫
=
f (S(x)y, S ′ (S(x)y)z)µ(dy)ν(dz)
(1.19)
f (S(x)y, S(x)S ′ (y)z)µ(dx)ν(dy),
are well defined for K-invariant measures µ and ν on X, that is, they do not depend on the
choice for section maps S and S ′ , noting that S ′ (S(x)y) = S(x)S ′ (y)kx,y for some kx,y ∈ K.
In this notation, the formula (1.12) for the finite dimensional distributions of a Lévy process
in G holds also for a Lévy process xt in X = G/K. This follows from the Markov property
of xt and (1.18).
The following result says that a Lévy process in X may also be characterized by independent and stationary increments, just like a Lévy process in G.
Theorem 1.14 Let xt be a rcll process in X = G/K with an infinite life time and let
{Ftx } be its natural filtration. If xt is a Lévy process with associated convolution semigroup
−1
x
µt , then for any section map S and s < t, x−1
s xt = S(xs ) xt is independent of Fs and
has distribution µt−s . Consequently, this distribution is K-invariant, and depends only on
t − s and not on the choice of S. Conversely, if for some section map S and any s < t,
−1
x
x−1
s xt = S(xs ) xt is independent of Fs , and its distribution is K-invariant and depends
only on t − s, then xt is a Lévy process in X.
18
Proof Let xt be a Lévy process in X with associated convolution semigroup µt . Then from
−1
its finite dimensional distributions given in (1.12), it can be shown that x−1
s xt = S(xs ) xt
is independent of Fsx and its distribution is µt−s . For simplicity, we will prove that for
r < s < t, x−1
s xt is independent of xr and has distribution µt−s . Fix an arbitrary section
map S, we will write x1 x2 · · · xk−1 xk for S(x1 )S(x2 ) · · · S(xk−1 )xk , and x−1 for S(x)−1 . By
(1.12), for f, g ∈ B+ (X),
E[f (xr )g(x−1
s xt )]
∫
=
∫
=
µ0 (dx0 )µr (dx1 )µs−r (dx2 )µt−s (dx3 )f (x0 x1 )g((x0 x1 x2 )−1 x0 x1 x2 x3 ))
µ0 (dx0 )µr (dx1 )µs−r (dx2 )µt−s (dx3 )f (x0 x1 )g(x3 )
= E[f (xr )]E[g(xt−s )].
This shows that S(xs )−1 xt is independent of xr and has the distribution µt−s . There is no
difficulty in this proof to replace xr by (xr1 , xr2 , . . . , xrk ) for r1 < r2 < · · · < rk ≤ s, except
the expressions will be much longer. Conversely, assume for some section map S and any
−1
x
s < t, x−1
s xt = S(xs ) xt is independent of Fs , and its distribution, denoted as µs,t , is
K-invariant and depends only on t − s. Then for f ∈ B+ (X),
E[f (xt ) | Fsx ] = E[f (S(xs )S(xs )−1 xt ) | Fsx ] =
∫
=
X
∫
X
f (S(xs )y)µs,t (dy)
f (S(xs )y)µ0,t−s (dy).
Because µs,t is K-invariant, by (1.14), µr,s ∗ µs,t = µr,t for r < s < t. Because µs,t depends
only on t − s, it then is easy to show that µt = µ0,t is a continuous convolution semigroup
on G and xt is a G-invariant Markov process in X with associated convolution semigroup
µt , and hence xt is a Lévy process in X. 2
We note that by Theorem 1.14, for a Lévy process xt in X and a section map S on X,
xot = S(x0 )−1 xt
(1.20)
is a Lévy process in X with xo0 = o and is independent of x0 , and its distribution does not
depend on the choice for S. Moreover, the process xt is equal in distribution to S(x0 )xot .
Remark 1.15 Let xt be a G-invariant Markov process in X = G/K with transition semi∫
group Pt and let µt = Pt (o, ·). Then Pt f (x) = Pt (f ◦ S(x))(e) = G f (S(x)y)µt (dy). If
µt → δo weakly as t → 0, then Pt is Feller and hence xt has a rcll version. In this case, if
xt has an infinite life time, then this version of xt is a Lévy process, and µt is a continuous
convolution semigroup on X.
The following three results deal with some simple relations between processes in G and
in X = G/K.
19
Proposition 1.16 Let gt be a Markov process in G. Assume its transition semigroup Pt
is K-right invariant, that is, (Pt f ) ◦ rk = Pt (f ◦ rk ) for f ∈ Bb (G) and k ∈ K. Then
xt = π(gt ) = gt o is a Markov process in X = G/K with transition semigroup
Qt f (x) = Pt (f ◦ π)(S(x)),
x ∈ X and f ∈ Bb (X),
(1.21)
which does not depend on the choice for the section map S. Moreover, if gt is a Feller process
in G, then so is xt in X.
Proof The K-right invariance of Pt shows that Qt defined by (1.21) does not depend on
the choice for S. Note that (1.21) may be written as (Qt f ) ◦ π = Pt (f ◦ π). Then
Qs+t f (x) = Ps+t (f ◦ π)(S(x)) = Ps [Pt (f ◦ π)](S(x)) = Ps [(Qt f ) ◦ π](S(x))
= (Qs Qt f ) ◦ π(S(x)) = Qs Qt f (x).
This shows that Qt is a transition semigroup. Since S may be chosen to be continuous near
any fixed point x in X, it is easy to see that if Pt is a Feller transition semigroup on G, then
so is Qt on X. The Markov property of xt follows from E[f (xs+t ) | Fsg ] = E[f ◦ π(gs+t ) |
Fsg ] = Pt (f ◦ π)(gs ) = (Qt f ) ◦ π(gs ) = Qt f (xs ). 2
Proposition 1.17 Let gt be a right Lévy process in G as defined in Remark 1.8, that is, a
rcll Markov process with a right invariant transition semigroup Pt and an infinite life time.
Let µt = Pt (e, ·) be the associated convolution semigroup. Then for any z ∈ X = G/K,
xt = gt z is a Feller process in X with transition semigroup Qt given by (1.21). Moreover,
∫
Qt f (x) =
G
x ∈ X and f ∈ Bb (X).
f (gx)µt (dg),
(1.22)
Proof Because gt = gte g0 and gt z = gte g0 z, replacing z by g0 z, we may assume g0 = e. If
the reference point o in X is replaced by z, then the natural projection π: G → X given
by g 7→ go should be replaced by πz : G → X given by g 7→ gz. If S is a section map
on X with respect to o, then Sz (·) = S(·)S(z)−1 is a section with respect to z because
Sz (x)z = S(x)S(z)−1 z = S(x)o = x for any x ∈ X. Because Pt is right invariant, by
Proposition 1.16, xt = gt z is a Feller process in X with transition semgroup
Qt f (x) = Pt (f ◦ πz )(S(x)S(z)−1 ) = Pt (f ◦ πz ◦ rS(x)S(z)−1 )(e)
∫
=
G
Pt (e, dg)f (gS(x)S(z)−1 z) =
∫
∫
G
∫
µt (dg)f (gS(x)o) =
This proves (1.22). From above, Qt f (x) = G µt (dg)f (gS(x)o) =
∫
G Pt (S(x), dg)f (go) = Pt (f ◦ π)(S(x)). This is (1.21). 2
∫
G
G
µt (dg)f (gx).
Pt (e, dg)f (gS(x)o) =
Recall that for g ∈ G, cg : G → G is the conjugation map x 7→ gxg −1 . A Lévy process
gt in G is called K-conjugate invariant if its transition semigroup is K-conjugate invariant,
20
that is, if Pt (f ◦ ck ) = (Pt f ) ◦ ck for f ∈ Bb (G) and k ∈ k. By the left invariance of Pt , this
is equivalent to the K-right invariance of Pt . In terms of the process, this is the same as
saying that for any t ∈ R+ and k ∈ K, kgte k −1 = gte in distribution.
Theorem 1.18 Let gt be an K-conjugate invariant Lévy process in G. Then xt = gt o is a
Lévy process in X = G/K with transition semigroup Qt given by (1.21).
Note: Later we will see (Theorem 3.10) that any Lévy process in X may be obtained from
an K-conjugate invariant Lévy process in G as in Theorem 1.18 when G is a Lie group.
Proof of Theorem 1.18: By Proposition 1.16, xt = gt o is a rcll Markov process in X with
transition semigroup Qt and infinite life time. It remains to show that Qt is G-invariant.
This follows from the left invariance and K-right invariance of Pt , because for f ∈ Bb (X)
and g ∈ G, Qt (f ◦ g)(x) = Pt (f ◦ g ◦ π)(S(x)) = Pt (f ◦ π ◦ lg )(S(x)) = Pt (f ◦ π)(gS(x)) =
Pt (f ◦ π)(S(gx)k) (for some k ∈ K) = Pt (f ◦ π ◦ rk )(S(gx)) = Pt (f ◦ π)(S(gx)) = Qt f (gx).
2
1.4
Inhomogeneous Lévy processes
As before, let G be a topological group equipped with a lcscH topology. An inhomogeneous
Markov process xt in G with transition semigroup Ps,t is called left invariant if it is invariant
under the action of left translations, that is, if Ps,t (f ◦ lg ) = (Ps,t f ) ◦ lg for g ∈ G and
f ∈ Bb (G).
Assume xt has an infinite life time. By the simple Markov property and the left invariance,
for s < t and f ∈ Bb (G),
x
E[f (x−1
)(xs ) = Ps,t f (e).
s xt ) | Fs ] = Ps,t (f ◦ lx−1
s
x
This shows that x−1
s xt is independent of Fs and has the distribution Ps,t (e, ·). Thus, like a left
invariant homogeneous Markov process considered in §1.2, xt has independent increments.
However, these increments are in general not stationary.
A rcll process xt in G, with an infinite life time and independent increments, will be
called an inhomogeneous Lévy process. In contrast, the Lévy processes defined in §1.2 may
be called homogeneous. Note that the class of inhomogeneous Lévy processes includes homogeneous ones as special cases. As for homogeneous Lévy processes, it is easy to show that
an inhomogeneous Lévy process xt is a left invariant inhomogeneous Markov process. To
summarize, we have the following result.
Theorem 1.19 The class of left invariant inhomogeneous Markov processes in G, that have
rcll paths and infinite life times, coincides with the class of inhomogeneous Lévy processes in
21
G. For such a process xt with transition semigroup Ps,t , Ps,t (e, ·) is the distribution of the
increment x−1
s xt , 0 ≤ s ≤ t.
A family of probability measures µs,t on G, 0 ≤ s ≤ t, is called a two-parameter convolution semigroup on G if µr,t = µr,s ∗ µs,t for r ≤ s ≤ t. It is called right continuous
if µs,t → µu,v weakly as s ↓ u and t ↓ v. It is called continuous if µs,t is continuous in
(s, t) under the weak convergence. Note that if µt is a continuous convolution semigroup,
then µs,t = µt−s is a continuous two-parameter convolution semigroup. For simplicity, a
two-parameter convolution semigroup may also be called a convolution semigroup.
It is easy to see that if xt is an inhomogeneous Lévy process in G, then by the rcll paths,
the distributions µs,t of its increments x−1
s xt , s ≤ t, form a right continuous convolution
semigroup with µt,t = δe for all t ≥ 0, which is called the convolution semigroup associated
to process xt . Moreover, µs,t is continuous if and only if xt is stochastically continuous.
Now let µs,t be a continuous (two-parameter) convolution semigroup on G with µt,t = δe .
For u, t ∈ R+ , x ∈ G and f ∈ C0 (R+ × G), let
∫
Rt f (u, x) =
G
f (u + t, xy)µu,u+t (dy).
(1.23)
It is easy to show that this defines a conservative Feller transition semigroup Rt on the
product space R+ × G. Let zt be the associated Feller process in R+ × G. Then zt has
an infinite life time, and given z0 = (u, x), zt = (u + t, yt ) is such that for f ∈ Cc (G),
∫
E(u,x) [f (yt )] = G f (xy)µu,u+t (dy). Let xt = yt−u for t ≥ u. Then xt is a left invariant
inhomogeneous Markov process with an infinite life time, and its transition semigroup Ps,t
is given by Ps,t (e, ·) = µs,t , and hence xt is an inhomogeneous Lévy process associated to
the convolution semigroup µs,t . As a component of a Feller process, xt is stochastically
continuous.
To summarize, we have established the following result.
Theorem 1.20 Let µs,t , 0 ≤ s ≤ t, be a two-parameter convolution semigroup on G with
µt,t = δe . Then µs,t is associated to a stochastically continuous inhomogeneous Lévy process
in G if and only if µs,t is continuous.
By the Markov property, it can be shown that the finite dimensional distributions of an
inhomogeneous Lévy process xt in G with initial distribution µ0 and associated convolution
semigroup µs,t are given by
∫
E[f (xt1 , xt2 , . . . , xtn )] =
f (x0 x1 , x0 x1 x2 , . . . , x0 x1 x2 · · · xn )
µ0 (dx0 )µ0,t1 (dx1 )µt1 ,t2 (dx2 ) · · · µtn−1 ,tn (dxn )
for 0 ≤ t1 < t2 < · · · < tn and f ∈ Bb (Gn ).
22
(1.24)
Remark 1.21 Let xt be a process in G with an infinite life time. By the proof of Theorem 1.19, xt is a left invariant inhomogeneous Markov process if and only if it has independent
increments. In this case, if for any s ≥ 0, x−1
s xt → e in distribution as t → s, then the dis−1
tributions µs,t of xs xt , s ≤ t, form a continuous convolution semigroup with µt,t = δe for all
t ≥ 0. It follows that the transition semigroup Rt of the process zt = (u + t, xu+t ), given in
(1.23), is Feller, and hence xt has a rcll version that is an inhomogeneous Lévy process in G.
More generally, let xt be a left invariant inhomogeneous Markov process in G without
assuming an infinite life time, and let Ps,t be its transition semigroup. Then Ps,t f (x) =
∫
Ps,t (f ◦ lx )(e) = G f (xy)µs,t (dy), where µs,t = Ps,t (e, ·). Assume µs,t is continuous in (s, t)
under the weak convergence. Although µs,t is a sub-probability, Rt defined by (1.23) is still
a Feller transition semigroup on R+ × G, and hence xt has a rcll version.
Recall that a Radon measure has a finite charge on any compact set. A measure is called
diffuse if it does not charge points.
Theorem 1.22 Let xt be a left invariant inhomogeneous Markov process in G with a possibly
finite life time and transition semigroup Ps,t . Assume xt has rcll paths and is stochastically
continuous. Then there are a diffuse Radon measure λ on R+ and a stochastically continuous
inhomogeneous Lévy process x̂t in G with transition semigroup P̂s,t such that
Ps,t = e−λ((s, t]) P̂s,t .
(1.25)
This means that process xt is equal in distribution to x̂t killed at the random time point of an
independent Poisson random measure ξ on R+ with intensity measure λ (see Appendix A.6).
More precisely, for any starting time s, let τ be the first point of ξ on (s, ∞), and let x′t = x̂t
for s ≤ t < τ and x′t = ∆ for t ≥ τ , then x′t is an inhomogeneous Markov process with
transition semigroup Ps,t .
Proof To prove (1.25), note that by the left invariance and the semigroup property of
∫
transition semigroup, Ps,t 1 = Ps,t 1(e) and for r < s < t, Pr,t 1(e) = Pr,s (e, dx)Ps,t 1(x) =
Pr,s 1(e)Ps,t 1(e). By the right continuity of Ps,t 1(e) in t, it follows that there is a Radon
measure λ on R+ such that Ps,t 1(e) = e−λ((s, t]) . Let P̂s,t = eλ((s, t]) Ps,t . Then Ps,t is a
conservative two-parameter transition semigroup and µ̂s,t = P̂s,t (e, ·) is a two-parameter
convolution semigroup of probability measures on G. By the stochastic continuity of xt , λ
is a diffuse measure and hence µ̂s,t is continuous. By Theorem 1.20, there is a stochastically
continuous inhomogeneous Lévy process x̂t in G associated to µ̂s,t . This proves (1.25). 2
As for a homogeneous Lévy process, we define an inhomogeneous Lévy process xt to be
associated to a filtration {Ft } if xt is adapted to {Ft } and for s < t, x−1
s xt is independent of Fs . This is equivalent to saying that xt is associated to the filtration {Ft } as an
inhomogeneous Markov process.
23
Let xt be an inhomogeneous Lévy process xt in G with associated convolution semigroup
µs,t . If it is associated to a filtration {Ft }, then for fixed r ∈ R+ , the process x′t = x−1
r xr+t ,
t ∈ R+ , is an inhomogeneous Lévy process with associated convolution semigroup νs,t =
µr+s,r+t , and is independent of Fr . In a certain sense, this holds when r is replaced by a
stopping time. The following result is an inhomogeneous analog of Theorem 1.6.
Theorem 1.23 Let xt be an inhomogeneous Lévy process in G with associated convolution
semigroup µs,t . Assume it is associated to a filtration {Ft }. If τ is an {Ft }-stopping time,
then for t1 < t2 < · · · < tn and f ∈ Bb (Gn ),
−1
−1
E[f (x−1
τ xτ +t1 , xτ xτ +t2 , . . . , xτ xτ +tn )1[τ <∞] | Fτ ]
∫
=
f (x1 , x1 x2 , . . . , x1 x2 · · · xn )µτ,τ +t1 (dx1 )µτ +t1 ,τ +t2 (dx2 ) · · · µτ +tn−1 ,τ +tn (dxn )1[τ <∞] .
This implies that under the conditional distribution given τ , x′t = x−1
τ xτ +t is an inhomogeneous Lévy process in G with associated convolution semigroup νs,t = µτ +s,τ +t , and is
independent of Fτ . More precisely, this means that for any bounded Fτ -measurable H,
−1
−1
E[Hf (x−1
τ xτ +t1 , xτ xτ +t2 , . . . , xτ xτ +tn )1[τ <∞] | σ(τ )]
= E[H | σ(τ )]
∫
f (x1 , . . . , x1 x2 · · · xn )µτ,τ +t1 (dx1 ) · · · µτ +tn−1 ,τ +tn (dxn )1[τ <∞] .
Proof We may assume f ∈ Cb (Gn ). As in the proof of Theorem 1.6, first assume τ has
discrete values. Then
−1
−1
E[f (x−1
τ xτ +t1 , xτ xτ +t2 , . . . , xτ xτ +tn )1[τ <∞] | Fτ ]
=
∑
−1
−1
E[f (x−1
t xt+t1 , xt xt+t2 , . . . , xt xt+tn ) | Ft ]1[τ =t]
t<∞
=
∑∫
f (x1 , x1 x2 , . . . , x1 x2 · · · xn )
t<∞
∫
=
µt,t+t1 (dx1 )µt+t1 ,t+t2 (dx2 ) · · · µt+tn−1 ,t+tn (dxn )1[τ =t]
f (x1 , x1 x2 , . . . , x1 x2 · · · xn )µτ,τ +t1 (dx1 )µτ +t1 ,τ +t2 (dx2 ) · · · µτ +tn−1 ,τ +tn (dxn )1[τ <∞] .
For a general stopping time τ , choose discrete stopping times τm ↓ τ . The result follows
after taking the limit of the above expression with τ = τm as m → ∞ and using the right
continuity of µs,t . 2
The basic theory of inhomogeneous Lévy processes in a homogeneous space X = G/K
may be developed parallel to the case of Lévy processes in G/K discussed in §1.3 with suitable
modifications, assuming K is compact and G/K has a continuous local section. Thus, a Ginvariant inhomogeneous Markov process xt in X, that has rcll paths and an infinite life time,
will be called an inhomogeneous Lévy process in X. Let Ps,t be its transition semigroup and
24
let µs,t = Ps,t (o, ·). Then µs,t are K-invariant, and by the Markov property and the Ginvariance of Ps,t , it can be shown that the finite dimensional distributions of xt are given in
∫
(1.24) if the integral like f (xy, xyz)µ(dy)ν(dz) is understood in the sense of (1.19) with a
choice of a section map S (but not dependent on the choice).
By (1.24), it can be shown that an inhomogeneous Lévy process in X can be characterized
by independent increments just like an inhomogeneous Lévy process in G. This is similar to
the homogeneous case, see the proof of Theorem 1.14. The result is summarized below.
Theorem 1.24 Let xt be a rcll process in X = G/K with an infinite life time and let {Ftx }
be its natural filtration. If xt is an inhomogeneous Lévy process with associated transition
−1
semigroup Ps,t , then for any section map S and s < t, x−1
s xt = S(xs ) xt is independent
of Fsx , and its distribution is µs,t = Ps,t (o, ·), so is K-invariant and does not depend on the
−1
choice for S. Conversely, if for some section map S and any s < t, x−1
s xt = S(xs ) xt is independent of Fsx and its distribution is K-invariant, then xt is an inhomogeneous Lévy process
in X.
A (two-parameter) convolution semigroup µs,t on X = G/K, 0 ≤ s ≤ t, and its continuity
and right continuity, are defined just as on G. For a convolution semigroup µs,t on G, because
for any t, µt,t ∗ µt,t = µt,t , it follows that µt,t = ρH for some compact subgroup H of G (which
may depend on t). Because µs,t = µs,s ∗ µs,t = µs,t ∗ µt,t , it follows that if each µs,t is K-right
invariant, then each µs,t is K-bi-invariant. The following simple result may be derived from
Proposition 1.9 in the same way as Proposition 1.12.
Proposition 1.25 (a) If µs,t is a convolution semigroup on G such that each measure µs,t
is K-right invariant, then each µs,t is K-bi-invariant.
(b) If νs,t is a convolution semigroup on X = G/K, then each νs,t is K-invariant.
(c) The map
µs,t 7→ νs,t = πµs,t
is a bijection from the set of K-bi-invariant convolution semigroups µs,t on G onto the set
of convolution semigroups νs,t on X = G/K. Moreover, µs,t is continuous (resp. right
continuous) if and only if so is νs,t .
For an inhomogeneous Lévy process xt in X, the distribution µs,t of its increments x−1
s xt =
−1
S(xs ) xt , s ≤ t, form a right continuous convolution semigroup on X with µt,t = δo for all
t ≥ 0 (not dependent on the choice for the section map S), which is called the convolution
semigroup associated to process xt . The right continuity requires a short proof given below.
Because a lcscH space is paracompact, there is a partition of unity {ψj } on X. This
∑
is a collection of functions ψj ∈ Cc (X) with 0 ≤ ψj ≤ 1 such that j ψj = 1 is a locally
25
finite sum, that is, any point of X has a neighborhood on which only finitely many terms
ψj are nonzero. We may also assume that for each j, there is a section map Sj on X that is
continuous on the support of ψj . Then by (1.24), for s < t, f ∈ Cb (X) and a section map S
on X,
µs,t (f ) = E[f (S(xs )−1 xt )] =
=
∑
∑
E[ψj (xs )f (S(xs )−1 xt )]
j
E[ψj (xs )f (Sj (xs )−1 xt )] = E[
j
∑
ψj (xs )f (Sj (xs )−1 xt )]
(1.26)
j
This shows that µs,t (f ) is right continuous in s and t.
The following two results may be proved as for Theorems 1.20 and 1.22, respectively.
Theorem 1.26 Let µs,t be a convolution semigroup on X = G/K with µt,t = δo . Then µs,t
is associated to a stochastically continuous inhomogeneous Lévy process in X if and only if
µs,t is continuous.
Theorem 1.27 A stochastically continuous G-invariant inhomogeneous Markov process xt
in X = G/K, with rcll paths and a possibly finite life time, is identical in distribution to
a stochastically continuous inhomogeneous Lévy process x̂t killed at the random time point
of an independent Poisson random measure on R+ with a Radon intensity measure λ. The
transition semigroups Ps,t of xt and P̂s,t of x̂t are related by (1.25).
Remark 1.28 Let xt be a G-invariant inhomogeneous Markov process in X = G/K. It
can be shown as in Remark 1.21 that if µs,t = Ps,t (o, ·) is continuous in (s, t) under the weak
convergence, then xt has a rcll version. If xt also has an infinite life time, then this verion is
an inhomogeneous Lévy process in X.
As for a Lévy process, an inhomogeneous Lévy process gt in G is called K-conjugate
invariant if its transition semigroup Ps,t is K-conjugate invariant, that is, if Ps,t (f ◦ ck ) =
(Ps,t f ) ◦ ck for f ∈ Bb (G) and k ∈ K. This is equivalent to saying that for any t > s ≥ 0
d
and k ∈ K, k(gs−1 gt )k −1 = gs−1 gt (equal in distribution). For g0 = e, this is also equivalent
to saying that for any k ∈ K, the two processes gt and kgt k −1 are equal in distribution. The
following result can be easily proved as for Theorem 1.18, and its converse will be obtained
in Chapter 5 on a homogeneous space of a Lie group.
Theorem 1.29 Let gt be a K-conjugate invariant inhomogeneous Lévy process in G. Then
xt = gt o is an inhomogeneous Lévy process in X = G/K. Moreover, the transition semigroups Ps,t of gt and Qs,t of xt are related by (Qs,t f ) ◦ π = Ps,t (f ◦ π) for f ∈ Bb (X).
26
1.5
Markov processes under a non-transitive action
Let X be a topological space under the continuous action of a topological group G, both are
assumed to be lcscH, and let xt be a G-invariant rcll Markov process in X with transition
semigroup Pt . Suppose the G-action on X is non-transitive. Then X is a collection of
disjoint G-orbits. In this case, under some suitable regularity condition, we may obtain
a decomposition of the process into two components, one transversal to the G-orbits and
the other along an orbit, with the former preserving the Markov property and the latter
preserving the G-invariance.
We first derive a simple result under a general setting. Let X/G be the space of G-orbits
and let J: X → (X/G) be the projection map: x 7→ Gx. Equip X/G with the quotient
topology induced by J so that J is continuous and open. Note that if G is compact, then
X/G is lcscH.
Theorem 1.30 J(xt ) is a rcll Markov process in X/G with transition semigroup Qt given
by
Qt f (y) = Pt (f ◦ J)(x),
y = J(x) ∈ X/G and f ∈ Bb (X/G)
(1.27)
(Qt does not depend on the choice of x in Gy due to the G-invariance of Pt ). Moreover,
when G is compact, if xt is a Feller process, then so is J(xt ).
Proof For f ∈ Bb (X/G) and y ∈ (X/G),
E[f ◦ J(xt+s ) | Ftx ] = Ps (f ◦ J)(xt ) = Qs f (J(xt )).
This proves that J(xt ) is a Markov process in X/G with transition semigroup Qt . When G
is compact, if f ∈ C0 (X/G), then f ◦ J ∈ C0 (X) and (Qt f ) ◦ J = Pt (f ◦ J) ∈ C0 (X), which
implies Qt f ∈ C0 (X/G). The Feller property of J(xt ) follows from that of xt . 2
Let Y be a topological subspace of X that is transversal to the action of G in the sense
that it intersects each G-orbit at exactly one point, that is,
∀y ∈ Y,
(Gy) ∩ Y = {y}
and
X = ∪y∈Y Gy.
(1.28)
Then Y will be called a transversal subspace of X (under the G-action). Let J1 : X → Y be
the projection map J1 (x) = y for x ∈ Gy. Note that Y is naturally identified with the orbit
space X/G via the map y 7→ Gy and J1 identified with J, but the subspace topology on Y
may not be the same as the quotient topology on X/G induced by J. However, when G is
compact and Y is closed in X, then the two topologies on Y agree and J1 is continuous.
In the rest of this section, we will assume the two topologies agree even when G is
not compact. Then J1 is continuous. The following result is an immediate consequence of
Theorem 1.30.
27
Theorem 1.31 yt = J1 (xt ) is a rcll Markov process in the transversal subspace Y with
transition semigroup Qt given by
Qt f (y) = Pt (f ◦ J1 )(y),
y ∈ Y and f ∈ Bb (Y ).
(1.29)
Moreover, if G is compact and if xt is a Feller process in X, then yt is a Feller process in Y .
Now assume the isotropy subgroup of G at every point y ∈ Y is the same compact
subgroup K of G. This assumption is often satisfied if the transversal subspace Y is properly
chosen. For example, consider Rd without the origin under the action of the group O(d) of
orthogonal transformations on Rd . The O(d)-orbits are spheres centered at the origin, and
any curve from the origin to infinity is transversal if the curve intersects each of these spheres
only once. When the curve is straight, that is, if it is a half line, all its points will have the
same isotropy subgroup of O(d).
Let Z = G/K. For y ∈ Y and z ∈ Z, the product zy = gy, where z = gK, is well
defined. For y ∈ Y , zy traces out the G-orbit through y as z varies over Z, and hence Z may
be regarded as the standard G-orbit. Note that map F : (Y × Z) → X given by (y, z) 7→ zy
is bijective. Because the restricted action map G × Y → X, (g, y) 7→ gy, is continuous
and the projection map G × Y → Y × Z, (g, y) 7→ (y, gK), is open, it follows that F is
continuous. If G is compact, then it is easy to show that F has a continuous inverse and
hence F : Y × Z → X is a homeomorphism.
We will now take this to be part of our assumption. By slightly changing the notation,
our general setup may be stated as follows. Let X = Y × Z be a topological product with
Z = G/K, where G is a topological group and K is a compact subgroup. Assume both Y
and G are lcscH. Let G act on X by its natural action on G/K, that is, g(y, z) = (y, gz) for
g ∈ G, y ∈ Y and z ∈ Z. Then Y × {o} is transversal to the G-action and K is the common
isotropy subgroup of G at every point of Y × {o}. Let
J1 : X → Y,
and J2 : X → Z
(1.30)
be the natural projections (y, z) 7→ y and (y, z) 7→ z, respectively.
The processes yt = J1 (xt ) and zt = J2 (xt ) are called respectively the radial and angular
parts of process xt . By Theorem 1.31, the radial part yt is a Markov process in Y . As the
angular process zt lives in the standard G-orbit Z which is invariant under the G-action,
it is natural to expect that it should inherit the G-invariance of xt in some sense. Before
discussing the properties of the angular process zt , we present some examples of spaces under
group actions for which the topological assumptions made here are satisfied.
Example 1.32 X = Rm+n = Y × Z with Y = Rm and Z = Rn . In this case, G = Rn
(additive group), and K consists of only the origin in Rn .
28
Example 1.33 Consider the action of the orthogonal group G = O(d) on X = Rd (d ≥ 2).
Any half line Y from the origin is a transversal subspace. Except the origin, all the points in
Y have the same isotropy subgroup K of O(d), which may be identified with the orthogonal
group O(d − 1) on a (d − 1)-dimensional linear subspace of Rd . The O(d)-orbits are spheres
in Rd centered at origin.
Let X ′ be the Rd without the origin. Then X ′ = Y ′ × Z, where Y ′ = (0, ∞), and Z is
the unit sphere S d−1 = O(d)/O(d − 1), via the map (r, z) 7→ rz from Y ′ × Z to X ′ .
Example 1.34 Consider the space X of n × n real symmetric matrices (n ≥ 2) under the
action of G = O(n) by conjugation: (g, x) 7→ gxg −1 for g ∈ O(n) and x ∈ X. Because
symmetric matrices in X with the same set of eigenvalues are O(n)-conjugate to each other,
the set Y of all n×n diagonal matrices with non-ascending diagonal elements is a transversal
subspace.
Let Y ′ be the subset of Y consisting of diagonal matrices with strictly descending diagonals. Then all y ∈ Y ′ have the same isotropy subgroup K of G = O(n), which is the finite
subgroup of O(n) consisting of diagonal matrices with ±1 along diagonal. Note that an
element of Y ′ not in Y has a larger isotropy subgroup. Let X ′ be the subset of X consisting
of symmetric matrices with distinct eigenvalues. Then X ′ may be identified with Y ′ ×(G/K)
via the map (y, gK) 7→ gyg −1 from Y ′ × (G/K) to X ′ .
Example 1.35 For readers who know symmetric spaces, we mention one more example.
Let X = G/K be a symmetric space of noncompact type, with a fixed Weyl chamber a+ and
the centralizer M of a+ in K, to be discussed in §5.6. Let A+ ⊂ G be the image of a+ under
the Lie group exponential map from the Lie algebra of G to G. The set X ′ of regular points
in X, as defined before (5.47), is diffeomorphic to A+ × (K/M ) under the map: A+ × (K/M )
given by (a, kM ) 7→ kaK, so X ′ under the K-action satisfies our assumption with Y = A+
and Z = K/M . A special case is the space X of n × n real positive definite symmetric
matrices of unit determinant (see Example 5.15), and X ′ is the space of those matrices with
distinct eigenvalues. This is a subset of the space in Example 1.34.
It may occur as in Examples 1.33, 1.34 and 1.35 that not all the points of the transversal
subspace Y share the same isotropy subgroup of G, but by restricting to a slightly smaller
open subset Y ′ of Y and the G-invariant open subset X ′ of X that is the union of G-orbits
through Y ′ , our assumptions are satisfied. In this case, if xt is a G-invariant rcll Markov
process in X, then its restriction to X ′ , defined below, is a G-invariant rcll Markov process
in X ′ .
The restriction of the process xt to X ′ is the process x̂t in X ′ defined by
x̂t = xt for t < ζ
and
29
x̂t = ∆ for t ≥ ζ,
(1.31)
where ζ = inf{t ≥ 0; xt ̸∈ X ′ or xt− ̸∈ X ′ } (inf ∅ = ∞ by convention) and ∆ is the point
at infinity (see Appendix A.3). Then x̂t is a rcll process in X ′ . Moreover, ζ is a stopping
time under the natural filtration {Ftx } of process xt because [ζ = 0] = [x0 ̸∈ X ′ ] and for any
t > 0,
[t < ζ] = ∪n>1 {[xt ∈ Xn′ ] ∩ [xr ∈ Xn′ for any r ∈ Q with 0 < r < t]} ∈ Ftx ,
where Xn′ are open subsets of X with closures X̄n′ contained in X ′ and Xn′ ↑ X ′ as n ↑ ∞,
and Q is the set of rational numbers. Here and in the sequel, for a sequence of sets An ,
An ↑ A means An ⊂ An+1 and ∪n An = A, unless when explicitly stated otherwise.
Proposition 1.36 If xt is a G-invariant rcll Markov process in X, then x̂t is a G-invariant
rcll Markov process in X ′ with transition semigroup P̂t given by
P̂t f (x) = Ex [f (xt ); t < ζ]
for f ∈ B+ (X ′ ) and x ∈ X ′ .
(1.32)
Proof Let θt be the time shift (see Appendix A.3). For s < t, 1[t<ζ] = 1[s<ζ] (1[t−s<ζ] ◦ θs )
and
Ex [f (x̂t ) | Fsx ] = Ex [f (xt )1[t<ζ] | Fsx ] = Ex {[f (xt−s )1[t−s<ζ] ] ◦ θs | Fsx ]1[s<ζ]
= Exs [f (xt−s )1[t−s<ζ] ]1[s<ζ] = P̂t−s f (x̂s )1[s<ζ] = P̂t−s f (x̂s ),
under the convention that any function on X or X ′ vanishes at ∆. This shows that x̂t is a
Markov process in X ′ with transition semigroup P̂t . By Proposition 1.1, the G-invariance of
process xt implies the G-invariance of P̂t . 2
Remark 1.37 We may also define the restriction of process xt in X ′ by stopping xt at the
first time τ when it exits X ′ , where τ = inf{t > 0; xt ̸∈ X ′ }. Thus, let x′t be defined by
(1.31) with x̂t and ζ replaced by x′t and τ . However, τ in general is not a stopping time
under Ftx and x′t may not be rcll in X ′ . By the standard stochastic analysis, τ is a stopping
x
}, and if the original process
times under the filtration {Ft } that is the completion of {Ft+
xt is quasi-left-continuous, that is, if xσ = limn xσn almost surely on [σ < ∞] for any {Ft }stopping times σn ↑ σ, then the two processes x′t and x̂t are identical almost surely. To show
this, let Xn′ be the open subsets of X ′ with Xn′ ↑ X ′ as defined earlier, and let τn be the first
times when xt exits Xn′ . It is easy to see that τn ↑ ζ, and so by the quasi-left-continuity,
xζ = limn xτn almost surely. Because Xn′ is open, xτn ̸∈ Xn′ , and hence xζ ̸∈ X ′ almost surely.
This shows ζ = τ almost surely. It is well known that a Feller process is quasi-left-continuous
(see [46, Proposition 25.20]).
30
1.6
Angular part
As discussed in the previous section §1.5, when discussing an invariant Markov process under
a non-transitive action, by suitably restricting the process, it may be possible to work under
the following setup, which will be assumed throughout this section. Let X = Y × Z be a
topological product with Z = G/K, where G is a topological group with identity element
e and K is a compact subgroup. Assume both Y and G are lcscH. Let G act on X by its
natural action on G/K. Then with o = eK, Y × {o} is transversal to the G-action and K is
the common isotropy subgroup of G at every point of Y × {o}. We will also assume G/K has
a continuous local section (as defined in §1.3). As in §1.5, let J1 : X → Y and J2 : X → Z
be the natural projections.
We mention an important case when this setup holds. When X is a manifold under the
(smooth) action of a Lie group G, and Y is a submanifold of X transversal to the G-action,
if the tangent space of X at any y ∈ Y is the direct sum of the tangent space of Y at y and
that of the orbit through y, that is, if
∀y ∈ Y,
Ty X = Ty (Gy) ⊕ Ty Y
(a direct sum).
(1.33)
then by [34, II.Lemma 3.3], for any y ∈ Y , there is a neighborhood U of y in Y , and a
neighborhood V of o = eK in Z = G/K, where e is the identity element of G as before,
such that the map (u, v) 7→ vu is a diffeomorphism from U × V onto a neighborhood of y in
X. From this, it is easy to show that F : Y × Z → X is a diffeomorphism. The existence of
a continuous local section holds automatically in the Lie group case.
Let xt be a G-invariant Markov process in X. By Theorem 1.31, the radial process
yt = J1 (xt ) is a Markov process in Y . The purpose of this section is to study the angular
process zt = J2 (xt ) under the conditional distribution given the radial process yt . We now
begin with some preparation.
Let (S, ρ) be a complete and separable metric space, and let D(S) be the space of rcll
paths in S, that is, the space of rcll maps: R+ → S. Under the Skorohod metric, D(S)
is a complete and separable metric space (see [22, chapter 3]). By Proposition 5.3(c) in
[22, chapter 3], the convergence xn → x in D(S) means that up to a time change that is
asymptotically an identity, xn (t) converge to x(t) uniformly for bounded t. Precisely, this
means that for any T > 0, there are bijections λn : R+ → R+ (necessarily continuous with
λn (0) = 0) such that |λn (t) − t| → 0 and ρ(xn (t), x(λn (t))) → 0 as n → ∞ uniformly for
t ∈ [0, T ]. By Proposition 7.1 in [22, chapter 3], the Borel σ-algebra on D(S) is generated by
the coordinate maps: D(S) → S given by x 7→ x(t) for t ∈ R+ . It is easy to show that the
topology on D(S) induced by the Skorohod metric is determined completely by the topology
of (S, ρ), and does not depend on the choice of ρ.
31
Let X∆ = X ∪ {∆} be the one-point compactification of X. Because X is a lcscH space,
X∆ is metrizable (see Theorem 12.12 in [10, chapter I]), and as being compact, X∆ may
be equipped with a complete and separable metric. Let D′ (X) be the space of rcll paths
in X with possibly finite life times. These are rcll paths in X∆ such that each path x(·) is
associated to a life time ζ ∈ [0, ∞] with x(t) ∈ X for t < ζ and x(t) = ∆ for t ≥ ζ. By
Proposition 1.38 below, D′ (X) is a Borel subset of D(X∆ ), and hence D′ (X) is a Borel space
(see Appendix A.7 for the definition of Borel spaces).
Proposition 1.38 D′ (X) is a Borel subset of D(X∆ ), and hence D′ (X) is a Borel space.
Proof Let Xn be open subsets of X such that X̄n ⊂ Xn+1 and Xn ↑ X. Then D′ (X) =
∩m>0 A(m), where A(m) = ∪∞
n=1 ∪r∈Q,r≥0 A(m, n, r) and A(m, n, r) = {x ∈ D(X∆ ); x(s) = ∆
for s ∈ Q ∩ [r, ∞) and x(s) ∈ Xn for s ∈ Q ∩ [0, r − 1/m]}. 2
For b > 0, let Db′ (X) be the subset of D′ (X) consisting of all paths with a common
constant life time b. Then Db′ (X) is a closed in D′ (X) under the Skorohod metric, and
′
D∞
(X) = D(X).
Recall J1 : X → Y and J2 : X → Z are the natural projections Y × Z → Y and
Y × Z → Z. We will also use J1 and J2 to denote the maps
J1 : D′ (X) → D′ (Y ),
x(·) 7→ y(·),
and J2 : D′ (X) → D′ (Z),
x(·) 7→ z(·),
respectively, given by the decomposition x(·) = (y(·).z(·)).
Recall xt is a G-invariant rcll Markov process in X. It may be regarded as the coordinate
process on the canonical path space D′ (X), that is, xt (ω) = ω(t) for ω ∈ Ω. The radial
part yt = J1 (xt ) and the angular part zt = J2 (xt ) are thus the coordinate processes on
Y
D′ (Y ) and D′ (Z), respectively. For t > s ≥ 0, let Fs,t
= σ{yu ; u ∈ [s, t]}, the σ-algebra on
Y
Y
D′ (Y ) generated by the radial process on the time interval [s, t], and set FtY = F0,t
and F∞
=
∞
σ{∪t>0 Ft }. The distribution Px of the process xt with x0 = x is a probability kernel from X
to Ω. Let Px′ (x(·), ·) be its regular conditional distribution given the radial process yt = J1 xt ,
Y
that is, given J1−1 (F∞
), see Theorem A.5 in Appendix A.7. This is a probability kernel from
−1
Y
Y
(X ×Ω, B(X)×J1 (F∞ )) to (Ω, F). Because it is B(X)×J1−1 (F∞
)-measurable, for any y(·) ∈
−1
Y
′
′
D (Y ), Px (x(·), ·) is constant on the set J1 ({y(·)}). Let Px (y(·), ·) = Px′ ((y(·), o(·)), ·),
where o(·) ∈ D′ (Z) is the constant path o(t) = o for all t ≥ 0.
For x ∈ X with decomposition x = (y, z), let
Qy = J1 Px = Px (J1−1 (·)).
(1.34)
This is the distribution of the radial process yt = J1 xt with y0 = y, and it does not depend
on x in the orbit Gy. For z ∈ Z, y(·) ∈ D′ (Y ) and x = (y(0), z), let
Rzy(·) (·) = PxY (y(·), J2−1 (·)).
32
(1.35)
This is a probability kernel from Z × D′ (Y ) to D′ (Z) such that for any y ∈ Y , z ∈ Z and
measurable F ⊂ D′ (Z), with x = (y, z),
Y
Rzy(·) (F ) = Px [J2−1 (F ) | J1−1 (F∞
)]
for Qy -almost all y(·).
(1.36)
Thus, Rzy(·) is the conditional distribution of the angular process zt = J2 (xt ) given the radial
path y(·) under Px with x = (y(0), z).
Y
The life time ζ of process xt is equal to that of the radial process yt , and hence is J1−1 (F∞
)−1
Y
measurable. Thus, given J1 (F∞
) or under Rzy(·) , ζ is a constant. For z ∈ Z and Qy -almost
all y(·) ∈ D′ (Y ), the measure Rzy(·) is supported Dζ′ (X).
For t ∈ R+ , the time shift θtY on D′ (Y ) is defined as usual by θtY y(·) = y(· + t). Then
ys ◦ θtY = ys+t for the coordinate process ys on D′ (Y ).
Recall Lévy processes in Lie groups and homogeneous spaces, including inhomogeneous
ones, are defined to have infinite life times, but these definitions may be easily modified to
include processes defined on a finite time interval [0, T ] or [0, T ) for some constant T > 0.
The following result says that almost surely, given the radial process yt , the conditioned
angular process zt is an inhomogeneous Lévy process in Z = G/K for 0 ≤ t < ζ.
Theorem 1.39 For y ∈ Y and z ∈ Z, and Qy -almost all y(·) in D′ (Y ), the coordinate process zt on Dζ′ (Z) is an inhomogeneous Lévy process under Rzy(·) . The associated convolution
Y
semigroup µs,t , setting µs,t = 0 for t ≥ ζ, is Fs,t
-measurable for any s ≤ t (that is, µs,t (B) is
Y
Fs,t
-measurable for B ∈ B(Z)) and has the time shift property
µs,t = µ0,t−s ◦ θsY .
(1.37)
Moreover, the transition semigroup Pt of the Markov process xt is given by
∀f ∈ B+ (X) and x = (y, z) ∈ X,
∫
Pt f (x) = Qy [
µ0,t (dz1 )f (yt , zz1 )1[ζ>t] ].
(1.38)
Proof Recall Qt is the transition semigroup of the radial process yt . By the existence of a
regular conditional distribution, there is a probability kernel Rt (y, y1 , ·) from Y 2 = Y × Y
to Z such that for y ∈ Y , Pt ((y, o), dy1 × dz1 ) = Qt (y, dy1 )Rt (y, y1 , dz1 ). This is proved
by applying Theorem A.5 in Appendix A.7 to Pt ((y, o), ·), regarded as a kernel from Y
to X, to obtain its regular conditional distribution Rt′ (y, x, ·) given J1−1 (B(Y )). Then let
Rt (y, y1 , ·) = J2 Rt′ (y, (y1 , o), ·), noting Rt′ (y, (y1 , z), ·) is constant in z ∈ Z and supported by
J1−1 ({y1 }).
The G-invariance of Pt implies that Pt ((y, o), ·) is K-invariant, and hence the measure
Rt (y, y1 , ·) is K-invariant for Qt (y, ·)-almost all y1 . Modifying Rt on the exceptional set
{(y, y1 ); Rt (y, y1 , ·) is not K-invariant}, its y-section has zero Qt (y, ·)-measure for all y ∈
Y , we may assume Rt (y, y1 , ·) is K-invariant for all y, y1 ∈ Y . Therefore, for z ∈ Z, it
33
∫
is meaningful to write Rt (y, y1 , z −1 dz1 ) = Rt (y, y1 , S(z)−1 dz1 ) and f (zz1 )Rt (y, y1 , dz1 ) =
∫
f (S(z)z1 )Rt (y, y1 , dz1 ) because they are independent of choice of section map S. We then
have, for x = (y, z) ∈ X,
∀y ∈ Y and z ∈ Z,
Pt (x, dz1 × dy1 ) = Qt (y, dy1 )Rt (y, y1 , z −1 dz1 ).
(1.39)
For 0 < s1 < s2 < · · · < sk < ∞, y ∈ Y , z ∈ Z, h ∈ Cb (Y k ) and f ∈ Cb (Z k ),
Ex [h(ys1 , . . . , ysk )f (zs1 , . . . , zsk )1[ζ>sk ] ]
∫ ∫
=
∫
=
∫
Ps1 (x, dz1 × dy1 )Ps2 −s1 ((y1 , z1 ), dz2 × dy2 ) · · · Psk −sk−1 ((yk−1 , zk−1 ), dzk × dyk )
h(y1 , y2 , . . . , yk )f (z1 , z2 , . . . , zk )
Qs1 (y, dy1 )Qs2 −s1 (y1 , dy2 ) · · · Qsk −sk−1 (yk−1 , dyk )h(y1 , y2 , . . . , yk )
Rs1 (y, y1 , dz1 )Rs2 −s1 (y1 , y2 , dz2 ) · · · Rsk −sk−1 (yk−1 , yk , dzk )f (zz1 , zz1 z2 , . . . , zz1 · · · zk )
∫
= Ey [h(ys1 , ys2 , . . . , ysk )
Rs1 (y, ys1 , dz1 )Rs2 −s1 (ys1 , ys2 , dz2 ) · · ·
Rsk −sk−1 (ysk−1 , ysk , dzk )f (zz1 , zz1 z2 , . . . , zz1 · · · zk )1[ζ>sk ] ].
This implies that Px -almost surely on [ζ > sk ],
Ex [f (zs1 , . . . , zsk ) | ys1 , . . . , ysk ] =
∫
Rs1 (y, ys1 , dz1 )Rs2 −s1 (ys1 , ys2 , dz2 ) · · ·
Rsk −sk−1 (ysk−1 , ysk , dzk )f (zz1 , zz1 z2 , . . . , zz1 · · · zk )].
(1.40)
For an integer m ≥ 1, let Γm be the set of dyadic numbers i/2m for i = 0, 1, 2, . . . and let
Γ = ∪∞
m=1 Γm . For s < t ≤ T in Γm , let 0 = s0 < s1 < s2 < · · · < sk = T be a partition of
[0, T ] spaced by 1/2m with s = si and t = sj , and let
µm
s,t = Rsi+1 −si (ysi , ysi+1 , ·) ∗ Rsi+2 −si+1 (ysi+1 , ysi+2 , ·) ∗ · · · ∗ Rsj −sj−1 (ysj−1 , ysj , ·).
(1.41)
By (1.40), K-invariance of Pt ((y, o), y1 , ·) and the measurability of µm
s,t in ysi , . . . , ysj ,
−1
−1
µm
s,t (f ) = Ex [f (zs zt ) | ys1 , . . . , ysk ] = Ex [f (zs zt ) | ysi , . . . , ysj ]
(1.42)
Px -almost surely on [ζ > T ] for f ∈ Cb (Z), which is independent of the choice for the section
map S to represent zs−1 zt = S(zs )−1 zt .
Y
,
By the right continuity of yt , as m → ∞, σ{ys1 , . . . , ysk } ↑ FTY and σ{ysi , . . . , ysj } ↑ Fs,t
Y
−1
Y
−1
m
it follows that as m → ∞, µs,t (f ) → Ex [f (zs zt ) | FT ] = Ex [f (zs zt ) | Fs,t ] Px -almost
surely on [ζ > T ]. The exceptional set may be chosen simultaneously for countably many
f ∈ Cb (Z) and hence for all f ∈ Cb (Z). It follows that Px -almost surely on [ζ > T ], there is
an K-invariant probability measure µs,t such that µm
s,t → µs,t weakly and for f ∈ Cb (Z),
Y
] Px -almost surely on [ζ > T ].
µs,t (f ) = Ex [f (zs−1 zt ) | FTY ] = Ex [f (zs−1 zt ) | Fs,t
34
(1.43)
Note that µm
s,t is independent of choice T ∈ Γm with T ≥ t, and hence µs,t is defined for any
Y
s, t ∈ Γ with s < t < ζ. Set µt,t = δo and µs,t = 0 for t ≥ ζ. By (1.41), µs,t is an Fs,t
measurable random measure independent of starting point x and has the time shift property
(1.37) for s, t ∈ Γ. Moreover, µr,s ∗ µs,t = µr,t for r ≤ s ≤ t in Γ. Because Γ is countable,
the exceptional set of zero Px -measure for the weak convergence µm
s,t → µs,t may be chosen
simultaneously for all s ≤ t and T in Γ.
For 0 ≤ t1 < t2 < · · · < tn ≤ T in Γ, it can be shown from (1.40) and by choosing a
partition s1 < s2 < · · · < sk of [0, T ] from Γ containing all ti , spaced by 1/2m , that almost
surely on [ζ > T ], for f ∈ Cb (Z n ),
Ex [f (zt1 , . . . , ztn ) | FTY ] = lim Ex [f (zt1 , . . . , ztn ) | ys1 , . . . , ysk ]
∫
=
lim
m→∞
m→∞
m
m
f (zz1 , zz1 z2 , . . . , zz1 · · · zn )µm
0,t1 (dz1 )µt1 ,t2 (dz2 ) · · · µtn−1 ,tn (dzn ).
This implies that for 0 ≤ t1 < · · · < tn ≤ T in Γ, y ∈ Y and z ∈ Z, and Qy -almost all
y(·) ∈ [ζ > T ],
Rzy(·) [f (zt1 , . . . , ztn )] = Ex [f (zt1 , . . . , ztn ) | FTY ]
∫
=
f (zz1 , zz1 z2 , . . . , zz1 · · · zn )µ0,t1 (dz1 )µt2 ,t1 (dz2 ) · · · µtn−1 ,tn (dzn ).
(1.44)
Then the finite dimensional distribution of the conditioned angular process zt under Rzy(·) ,
when restricted to time points in Γ ∩ [0, T ], has the form consistent with an inhomogeneous Lévy process in Z. To prove that the conditioned process zt is an inhomogeneous
Lévy processes in Z, restricted to time interval [0, T ], it remains to extend µs,t in (1.43) to
all real s ≤ t in [0, T ] and prove (1.44) for real times t1 < t2 < · · · < tn in [0, T ].
By a computation similar to the one leading to (1.26), it can be show that for s < t ≤ T
in Γ and f ∈ Cb (Z),
∑
µs,t (f ) = Ex [
ψj (zs )f (Sj (zs )−1 zt ) | FTY ]
Px -almost surely on [ζ > T ],
j
where {ψj } is a partition of unity on Z and for each j, Sj is a section map on Z = G/K that
is continuous on the support of ψj . The above expression for µs,t (f ) extends to real times
s < t, and it is clearly right continuous in s and t. Moreover, considering Px (· | FTY ) as a
regular conditional distribution of Px given FTY , no additional exceptional set is produced.
Because for real s < t, µs,t is the weak limit of µp,q for p, q ∈ Γ as p ↓ s and q ↓ t, taking limit
Y
-measurable
in (1.44) shows that it holds on real time points. It is also clear that µs,t is Fs,t
and has the time shift property (1.37).
It remains to prove (1.38) which follows from
Pt f (x) = Ex [f ((yt , zt ))1[ζ>t] ] = Ex {Rzy(·) [f ((y ′ , zt ))]y′ =yt 1[ζ>t] }
∫
= Qy [ µ0,t (dz1 )f ((yt , zz1 ))1[ζ>t] ],
35
where the last equality is due to (1.44). 2
The following result provides a converse to Theorem 1.39. Although its assumption may
not be easy to verify, a more useful condition will be provided in Theorem 9.1 in the case of
Lie group actions.
Theorem 1.40 Let yt be a rcll Markov process in Y with life time ζ and for any y ∈ Y , let
Qy be its distribution on D′ (Y ) with Qy (y0 = y) = 1. Assume for any y ∈ Y , z ∈ Z = G/K
and Qy -almost all y(·), there is a probability measure Rzy(·) on D′ (Z) such that under Rzy(·) ,
the coordinate process zt on D′ (Z) is an inhomogeneous Lévy process zt in Z for t < ζ, with
z0 = z, and the associated convolution semigroup µs,t (setting µs,t = 0 for t ≥ ζ) is FtY measurable and has the time shift property (1.37) as in Theorem 1.39. Then xt = (yt , zt ) is
a G-invariant rcll Markov process in X with transition semigroup Pt given by (1.38).
Proof For z ∈ Z and y(·) ∈ D′ (Y ), let Rzy(·) be the distribution on D′ (Z) of the inhomogeneous Lévy process zt in Z for t < ζ with z0 = z, associated to convolution semigroup µs,t .
Then for 0 ≤ t1 < t2 < · · · < tn < ζ and f ∈ Cb (Z n ),
∫
Rzy(·) [f (zt1 , zt2 , . . . , ztn )] =
f (zz1 , zz1 z2 , . . . , zz1 z2 · · · zn )µ0,t1 (dz1 )µt1 ,t2 (dz2 ) · · · µtn−1 ,tn (dzn ).
Because µs,t is FtY measurable, Rzy(·) [f (zt1 , zt2 , . . . , ztn )] is measurable on Z × D′ (Y ). Then
a simple monotone class argument shows that Rzy(·) is a kernel from Z × D′ (Y ) to D′ (Z).
For x ∈ X with x = (y, z), let Px be the measure on D′ (Y ) × D′ (Z) defined by
∀F ∈ Bb (D′ (Y ) × D′ (Z)),
Px (F ) = Qy {Rzy(·) [F (y(·), ·)]}.
Then Px is a probability measure supported by the closed subset H of D′ (Y )×D′ (Z) consisting of (y(·), z(·)) with the same life time for y(·) and z(·). The map: x(·) 7→ (J1 x(·), J2 x(·))
is continuous and bijective from D′ (X) onto H, and its inverse is also continuous under the
Skorohod metric. We will identify D′ (X) with H via this map, and then Px may be regarded
as a probability measure on D′ (X).
For 0 ≤ s1 < s2 < · · · < sn = s < t, h ∈ Cb (X n ) and f ∈ Cb (X),
Ex [h(xs1 , xs2 , . . . , xsn )f (xt )]
= Qy {Rzy(·) [h(zs1 y1 , zs2 y2 , . . . , zsn yn )f (zt y ′ )]y1 =y(s1 ),y2 =y(s2 ),...,yn =y(sn ),y′ =y(t) }
= Qy {
= Qy {
∫
∫
h(zz1 ys1 , zz1 z2 ys2 , . . . , zz1 z2 · · · zn ysn )f (zz1 z2 · · · zn z ′ yt )
µ0,s1 (dz1 )µs1 ,s2 (dz2 ) · · · µsn−1 ,sn (dzn )µs,t (dz ′ )}
h(zz1 ys1 , zz1 z2 ys2 , . . . , zz1 z2 · · · zn ysn )µ0,s1 (dz1 )µs1 ,s2 (dz2 ) · · · µsn−1 ,sn (dzn )
∫
Qy [(
µ0,t−s ◦ (dz ′ )f (zz1 z2 · · · zn z ′ yt−s )) ◦ θsY | FsY ]}
36
= Qy {
∫
h(zz1 ys1 , zz1 z2 ys2 , . . . , zz1 z2 · · · zn ysn )µ0,s1 (dz1 )µs1 .s2 (dz2 ) · · · µsn−1 ,sn (dzn )
∫
Qys [
=
µ0,t−s (dz ′ )f (zz1 z2 · · · zn z ′ yt−s )]}
∫
Qy {Rzy(·) {h(zs1 y1 , zs2 y2 , . . . , zsn yn )Qyn [
µ0,t−s (dz ′ )f (z̃z ′ yt−s )]z̃=zs }yi =y(si ),1≤i≤n }
= Px {h(xs1 , xs2 , . . . , xsn )Pt−s f (xs )}.
This shows that under Px , xt is a Markov process in X with transition semigroup Pt given
by (1.38). It follows directly from (1.38) that Pt is G-invariant. 2
37
38
Chapter 2
Lévy processes in Lie groups
It is well known that the distribution of a classical Lévy process in a Euclidean space Rd
is determined by a triple of a drift vector, a covariance matrix and a Lévy measure, which
are called the characteristics of the Lévy process. The triple appears in the Lévy -Khinchin
formula, which is the Fourier transform of the distribution, or the pathwise Lévy -Itô representation. In the latter representation, the three elements of the triple correspond respectively to a non-random drift, a diffusion part, and a pure jump part of the process.
A Lévy process in a Lie group G cannot be decomposed into three parts as in Rd , due to
the non-commutative nature of G, but the triple representation holds in an infinitesimal
sense, in the form of Hunt’s generator formula, to be discussed in §2.1. A functional form
of Lévy -Itô representation on G, in the form of a stochastic integral equation obtained by
Applebaum-Kunita [1], will be discussed in §2.3. For a Lévy process in a matrix group, this
equation takes a more direct and simpler form in §2.4.
We note that Heyer [35] generalized Hunt’s generator formula on Lie groups to locally
compact groups in the form of a sum of three maps, corresponding to a drift, a diffusion part
and a jump part. Using a projective basis of Lie algebras, Born [9] rewrote Heyer’s formula
in a form more closely resembling Hunt’s formula.
2.1
Generators of Lévy processes in Lie groups
From now on, we will assume that G is a Lie group of dimension d with Lie algebra g
unless when explicitly stated otherwise. Let e be the identity element of G. The reader
is referred to Appendices A.1 and A.2 for a summary of basic facts about Lie groups and
related homogeneous spaces.
For any integer k ≥ 0, let C k (G) be the space of the real or complex valued functions
on G that have continuous derivatives up to order k with C(G) = C 0 (G) being the space of
∩
continuous functions on G and C ∞ (G) = k>0 C k (G). Let Cck (G) denote the space of the
39
functions in C k (G) with compact supports. As before, Cb (G) and C0 (G) are respectively
the space of bounded continuous functions on G and the space of continuous functions on G
convergent to 0 at ∞ (in the one-point compactification of G).
Let gt be a Lévy process in G. Recall that this is a rcll process with independent and
stationary increments gs−1 gt for s ≤ t. Equivalently, a Lévy process gt in G may also be
defined as a left invariant rcll Markov process in G with an infinite life time, which is
equivalent to saying that its transition semigroup Pt is a left invariant and conservative
Feller transition semigroup. Hunt [41] found a complete characterization of such transition
semigroups on G by their generators. Some preparation is required to state this result.
Let {ξ1 , ξ2 , . . . , ξd } be a basis of g. There are functions ϕ1 , ϕ2 , . . . , ϕd ∈ Bb (G) such that
ϕi vanish outside a compact subset of G and are smooth near e, ϕi (e) = 0 and ξj ϕk = δjk ,
where δjk is the Kronecker delta and ξi is regarded as a tangent vector at e. These functions
may be used as local coordinates in a neighborhood of e with ξi = (∂/∂ϕi ) at e, and, hence,
will be called the coordinate functions associated to the basis {ξ1 , . . . , ξd }.
In a relatively compact neighborhood U of e which is the diffeomorphic image of some
open subset of g under the Lie group exponential map eξ = exp(ξ), ξ ∈ g, the coordinate
functions ϕi may be chosen to satisfy
g = exp[
∑
ϕi (g)ξi ]
(2.1)
i
for g ∈ U , then they will be called exponential coordinate functions associated to the basis
{ξi } of g.
It is clear that the coordinate functions are not uniquely determined by the basis, but if
ϕ′1 , . . . , ϕ′d are another set of coordinate functions associated to the same basis, then
ϕ′i = ϕi + O(∥ϕ∥2 )
(2.2)
∑
on some neighborhood of e, where ∥ϕ∥2 = ni=1 ϕ2i , and O(t) as usual denotes a function of
t > 0 such that O(t)/t is bounded as t → 0. Note that if ϕi and ϕ′i are two sets of exponential
coordinate functions associated to the same basis, then ϕi = ϕ′i near e.
For G = Rd (additive group) with standard √coordinates x1 , . . . , xd , we may choose
ξi = ∂/∂xi and ϕi (x) = xi 1[∥x∥≤1] , where ∥x∥ = x21 + · · · + x2d . Note that ϕi are exponential coordinate functions because with the identification of Rd with its Lie algebra, the
exponential map on Rd is the identity map.
Any ξ ∈ g induces a left invariant vector field ξ l on G, defined by ξ l (g) = Dlg (ξ), where
Dlg is the differential map of the left translation lg . It also induces a right invariant vector
field ξ r on G defined by ξ r (g) = Drg (ξ). For simplicity, ξ l will be written as ξ. Thus, for
f ∈ C 1 (G) and g ∈ G,
ξf (g) = ξ l f (g) =
d
f (getξ ) |t=0
dt
and
40
ξ r f (g) =
d
f (etξ g) |t=0 .
dt
It is easy to see that ξζ r f (g) = ζ r ξf (g) for ξ, ζ ∈ g.
A d × d nonnegative definite symmetric matrix ajk will be called a covariance matrix as
it is the covariance matrix of d real valued random variables.
Theorem 2.1 Let L be the generator of a left invariant and conservative Feller transition
semigroup Pt on G. Then its domain D(L) contains Cc∞ (G), and ∀f ∈ Cc∞ (G) and g ∈ G,
d
1 ∑
Lf (g) =
ajk ξj ξk f (g) + ξ0 f (g)
2 j,k=1
∫
+
G
[f (gh) − f (g) −
d
∑
ϕi (h)ξi f (g)]η(dh),
(2.3)
i=1
where ajk is a covariance matrix, ξ0 ∈ g and η is a measure on G satisfying
d
∑
η({e}) = 0, η(
ϕ2i ) < ∞ and η(U c ) < ∞
(2.4)
i=1
for any neighborhood U of G with U c being the complement of U in G. Moreover, the triple
(ξ0 , aij , η) is completely determined by L.
Conversely, given such a triple (ξ0 , aij , η), there exists a unique left invariant conservative
Feller transition semigroup Pt on G whose generator L restricted to Cc∞ (G) is given by (2.3).
Note that the condition (2.4) on η is independent of the choice of the basis {ξ1 , . . . , ξd } of
g and the associated coordinate functions ϕ1 , . . . , ϕd ∈ Cc∞ (G). A measure η on G satisfying
this condition will be called a Lévy measure on G. Note that any finite measure on G that
does not charge e is a Lévy measure.
For a Lévy process gt in G with generator L Pt , let aij , ξ0 and η be associated to L as in
Theorem 2.1. Then they will be called respectively the covariance matrix, the drift vector
and the Lévy measure of gt . Together, ξ0 , aij and η determine, and are also determined by,
the distribution of the Lévy process gt if g0 is given, are called the characteristics of gt .
The reader is referred to Theorem 5.1 in [41] for Hunt’s original proof. Some obscure
points in Hunt’s paper was clarified by Ramaswami [77]. A complete proof of Hunt’s result
can also be found in [35] and in [55]. A clearly written partial proof may be found in [5]. In
this work, Theorem 2.1 will be derived from a more general result in Chapter 6. Later it will
be shown that the domain of L contains a larger function space than Cc∞ (G). The following
lemma deals with some natural questions about the integral in (2.3).
Lemma 2.2 For a bounded f ∈ C 2 (G), the η-integral in (2.3) exists and is continuous in
g. Moreover, if f ∈ Cc2 (G), then it belongs to C0 (G), and if f ∈ Cc∞ (G), then it is smooth
in g.
41
Proof Let U be a neighborhood of e that is a diffeomorphic image of an open neighborhood
of 0 in g under the exponential map. First assume the exponential coordinates ϕi are
∑
chosen so that g = exp[ di=1 ϕi (g)ξi ] for g ∈ U . Applying Taylor’s expansion to ϕ(t) =
∑
f (g exp(t di=1 ϕi ξi )), we obtain
f (gh) − f (g) −
d
∑
ϕi (h)ξi f (g) =
i=1
d
1 ∑
ϕj (h)ϕk (h)ξj ξk f (gh′ )
2 j,k=1
(2.5)
∑
for g ∈ G and h ∈ U , where h′ = exp[t di=1 ϕi (h)ξi ] for some t ∈ [0, 1]. In particular,
for fixed g ∈ G, the left hand side of (2.5) is O(∥ϕ∥2 ). Write the η-integral in (2.3) as
∫
∫
∫
c
G = U + U c , as the sum of the integrals taken over U and U respectively. By (2.5) and
∫
the condition (2.4) imposed on η, U is finite and converges to 0 as U ↓ {e} uniformly for g
in a compact subset of G. Because f is bounded, η(U c ) < ∞ and ϕi have compact support,
∫
∫
it follows that U c is continuous in g, and hence so is G . If f has a compact support, then by
∫
∫
(2.5), U → 0 as g → ∆ (the point at infinity), and because η(U c ) < ∞, U c → 0 as g → ∆.
∫
It follows that G ∈ C0 (G).
∫
Now assume f ∈ Cc∞ (G). To prove the smoothness of G in g, let F (g, h) denote its
integrand. For ζ ∈ g, by Taylor’s expansion of ψ(t) = F (etζ g, h) around t = 0,
F (etζ g, h) = F (g, h) + tζ r F (esζ g, h)
for some s = s(g, h, t) ∈ (0, t). Then, because ζ r ξi = ξi ζ r ,
∫
∫
1∫
[ F (etζ g, h)η(dh) − F (g, h)η(dy)] =
ζ r F (esζ g, h)η(dy)
t G
G
G
∫
=
U
∫
=
U
[ζ r f (esζ gh) − ζ r f (esζ g) −
1∑
2
d
∑
∫
ϕi (y)ξi ζ r f (esζ g)]η(dy) +
i=1
r
sζ
∫
′
ϕj (y)ϕk (y)ξj ξk ζ f (e gh )η(dy) +
j,k
Uc
Uc
ζ r F (esζ g, h)η(dy)
ζ r F (esζ g, h)η(dy)
(by (2.5) with f (g) replaced by ζ r f (esζ g))
∫
∫
The second term above, U c ζ r F (esζ g, h)η(dy), will converge to U c ζ r F (g, h)η(dy) as t → 0,
∫
whereas the first term U (. . .) will converge to 0 as U ↓ {e} uniformly in t. It follows that
∫
∫
ζ r G F (g, h)η(dh) exists and is equal to G ζ r F (g, h)η(dh), which is just the integral in (2.3)
∫
when f is replaced by ζ r f . This proves that G F (g, h)η(dh) belongs to C 1 (G). Repeating
∫
the same argument with f replaced by ζ1r f for ζ1 ∈ g shows that G F (g, h)η(dh) ∈ C 2 (G),
∫
and continuing in this way shows that G F (g, h)η(dh) is smooth in g.
Now assume general coordinate functions ϕ′i are chosen in (2.3). We will let F (g, h)
denote the integrand of the η-integral with exponential coordinate functions ϕi . Then by
∑
(2.2), the integrand of the η-integral with ϕ′i can be written as F (g, h)− i ri (h)ξi f (g), where
42
|ri (h)| ≤ c∥ϕ∥3 for some constant c > 0. From this, it is easy to see that Lemma 2.2 holds
for general coordinate functions. 2
Let A be a closed linear operator on a Banach space B with domain D(A) dense in B.
A linear subspace D of D(A) is called a core of A if the closure of A restricted to D is equal
to A, that is, if for any f ∈ D(A), there are fn ∈ D such that fn → f and Afn → Af in B
as n → ∞. It is clear that A is completely determined by its restriction on a core. By [46,
Lemma 19.8], for the generator A of a Feller transition semigroup on a lcscH space X, with
B = C0 (X) equipped with the supremum norm, D is a core if and only if (λ − A)D is dense
in C0 (X) for some λ > 0.
Proposition 2.3 Cc∞ (G) is a core of the generator L in Theorem 2.1.
Proof: This proof is taken from [5] with some minor modification. It suffices to show
(1 − L)Cc∞ (G) is dense in C0 (G). Suppose it is not. Then there is a nonzero continuous
linear functional ρ on C0 (G) such that ρ((1 − L)f ) = 0 for any f ∈ Cc∞ (G). By the Riesz
representation, ρ may be identified with a finite signed measure on G. Let f ∈ Cc∞ (G). For
∫
x ∈ G, let ρ ∗ f (x) = G ρ(dy)f (yx) and rx f = f ◦ rx . Because ρ ∗ f ∈ C0 (G), there is z ∈ G
such that ρ ∗ f (z) = maxG (ρ ∗ f ). Because rz f ∈ Cc∞ (G), ρ((1 − L)rz f ) = 0, and hence
1
1
ρ(rz f ) = ρ(L(rz f )) = ρ{lim [Pt (rz f ) − (rz f )]} = lim [ρ(Pt (rz f )) − ρ(rz f )]
t→0 t
t→0 t
1
= lim [Pt (ρ ∗ (rz f ))(e) − ρ ∗ (rz f )(e)] (by the left invariance of Pt )
t→0 t
≤ 0
because ρ ∗ (rz f )(x) = ρ ∗ f (xz) is maximized at x = e. This shows ρ(rz f ) ≤ 0. Because
ρ(rz f ) = ρ ∗ f (z) = maxG (ρ ∗ f ), ρ(f ) ≤ 0 for any f ∈ Cc∞ (G). Replacing f by −f shows
ρ(f ) = 0, and hence ρ = 0, but ρ is assumed to be nonzero. 2
We note that the covariance matrix {aij } depends on the choice of a basis {ξ1 , . . . , ξd }
but is independent of the associated coordinate functions ϕi , however, the drift vector ξ0
depends on both. The operator D on C 2 (G) defined by
D=
d
1 ∑
ajk ξj ξk ,
2 j,k=1
(2.6)
will be called the diffusion part of the generator L,.
Proposition 2.4 The operator D and the Lévy measure η are completely determined by the
generator L restricted to Cc∞ (G), and are not dependent on the choice of the basis {ξ1 , . . . , ξd }
of g and the associated coordinate functions ϕi .
43
Proof By (2.3), for any f ∈ Cc∞ (G) that vanishes in a neighborhood of e, Lf (e) = η(f ).
Since η({e}) = 0, this proves that η is determined by L.
Since ϕi (e) = 0 and ξj ϕk (e) = δjk , L(ϕj ϕk )(e) = ajk + η(ϕj ϕk ). Let ϕ′1 , . . . , ϕ′d be
another set of coordinate functions associated to the same basis {ξ1 , . . . , ξd } of g and let
∑
D′ = (1/2) dj,k=1 a′jk ξj ξk . Then
ajk + η(ϕ′j ϕ′k ) = L(ϕ′j ϕ′k )(e) = a′jk + η(ϕ′j ϕ′k ).
This implies ajk = a′jk , hence, D is not dependent on the choice of the coordinate functions
if the basis of g is fixed.
∑
Now let {ξ1′ , . . . , ξd′ } be another basis of g with ξj′ = dp=1 bjp ξp . Let {cjk } be the inverse
∑
matrix of {bjk }. Then ϕ′k = dq=1 cqk ϕq for 1 ≤ k ≤ d can be used as coordinate functions
∑
associated to the new basis {ξ1′ , . . . , ξd′ }. Let D′ = (1/2) dj,k=1 a′jk ξj′ ξk′ . Then
a′jk = L(ϕ′j ϕ′k )(e) − η(ϕ′j ϕ′k ) =
d
∑
cpj cqk [L(ϕp ϕq )(e) − η(ϕp ϕq )] =
p,q=1
and
d
∑
apq cpj cqk
p,q=1
d
d
d
∑
1 ∑
1
1 ∑
′
′ ′
D =
a ξξ =
apq cpj cqk bju bkv ξu ξv =
apq ξp ξq = D.
2 j,k=1 jk j k 2 j,k,p,q,u,v=1
2 p,q=1
′
This proves that D is not dependent on the basis of g. 2
If the Lévy measure η satisfies the following finite first moment condition:
∫ ∑
d
|ϕi (g)| η(dg) < ∞,
(2.7)
i=1
then the integral
∫
G [f (gh)
− f (g)] η(dh) exists and the formula (2.3) simplifies to:
∫
d
1 ∑
′
Lf (g) =
ajk ξj ξk f (g) + ξ0 f (g) + [f (gh) − f (g)]η(dh)
2 j,k=1
G
(2.8)
∑ ∫
for f ∈ C02,l (G), where ξ0′ = ξ0 − i [ G ϕi (h)η(dh)]ξi . In this case, the coordinate functions
ϕi do not appear in the expression of L. Note that the condition (2.7) is not dependent on
the choice of the basis and the associated coordinate functions, and is satisfied if η is finite.
We note that in the special case of G = Rd (additive group), the generator formula (2.3)
may take the following form. For f ∈ Cc∞ (Rd ) and x ∈ Rd ,
Lf (x) =
d
d
∑
1 ∑
ajk ∂i ∂j f (x) +
bi ∂i f (x)
2 j,k=1
i=1
∫
+
Rd
[f (x + y) − f (x) −
d
∑
i=1
44
xi 1[∥x∥≤1] ∂i f (x)]η(dy),
(2.9)
where ∂i = (∂/∂xi ). The condition (2.4) on the Lévy measure η may be written as, with o
denoting the origin of Rd ,
d
∑
η({o}) = 0, η(
x2i 1[∥x∥≤1] ) < ∞ and η([∥x∥ > 1]) < ∞.
(2.10)
i=1
Remark 2.5 Let H be a closed subgroup of G with Lie algebra h and let the basis {ξ1 , . . . , ξd }
of g be chosen so that {ξ1 , . . . , ξm } is a basis of h for some m ≤ d. Assume the associated
coordinate functions ϕi on G are chosen so that ϕi = 0 on H for i > m, such as exponential
coordinate functions on G suitably defined outside a neighborhood of e. If (ξ0 , aij , η) are the
characteristics of a Lévy process gt in G such that ξ0 ∈ h, aij = 0 for i > m or j > m, and
supp(η) ⊂ H, where supp(η) is the support of measure η defined as the smallest closed set F
such that η(F c ) = 0, then (ξ0 , aij , η) may be naturally identified with a set of characteristics
on H, under the basis {ξ1 , . . . , ξm } and associated coordinate functions ϕ1 , . . . , ϕm . Because
ϕi = 0 on H for i > m, the generator L of gt may also be naturally regarded as an operator
on H. In this case, the Lévy process gt in G, with g0 ∈ H, is also a Lévy process in H with
the same characteristics.
2.2
Lévy measure
Let gt be a Lévy process in G with Lévy measure η. Its jump counting measure N is the
random measure on R+ × G defined by
−1
−1
N ([0, t] × B) = #{s ∈ (0, t]; gs−
gs ̸= e and gs−
gs ∈ B}
(2.11)
for t ∈ R+ and B ∈ B(G). So N ([0, t]×B) is the number of jumps of the process gt contained
in B during the time interval [0, t]. Because gt is rcll, for any closed B not containing e,
N ([0, t] × B) is finite almost surely.
The reader is refered to Appendix A.6 for the definition of Poisson random measures and
other related definitions in the following result. Recall that a Lévy process xt is said to be
associated to a filtration {Ft }, or an {Ft }-Lévy process, if it is adapted to Ft , and for s < t,
x−1
s xt is independent of Fs . A Lévy process is clearly associated to its natural filtration.
Theorem 2.6 The jump counting measure N of an {Ft }-Lévy process, defined by (2.11), is
a homogeneous Poisson random measure on R+ × G associated to {Ft } and its characteristic
measure is the Lévy measure η of gt .
Proof It is easy to see that for each t ∈ R+ , N ({t}×G) = 0 almost surely, and almost surely,
N ({t}×G) ≤ 1 for all t ∈ R+ . Moreover, by the independent increments of the Lévy process,
the measure-valued process Nt = N ([0, t] × ·) has independent increments. If we can show
45
that E[N ([0, t] × ·)] is a σ-finite measure on G for any t ∈ R+ , then by the discussion in
Appendix A.6, N is a Poisson random measure on R+ × G. It is also homogeneous because
Nt has stationary increments. It is also clear that Nt is Ft -measurable and for s < t, Nt − Ns
is independent of Fs . This shows that N is associated to the filtration {Ft }.
It remains to prove E[N ([0, 1] × B)] = η(B) for B ∈ B(G). For any ϕ ∈ Cc∞ (G) with
0 ≤ ϕ ≤ 1 and vanishing in a neighborhood of e, let
∫
Fϕ =
1
∫
ϕ(g)N (dt dg) =
0
G
∑
−1
gu ).
ϕ(gu−
u≤1
It suffices to show E(Fϕ ) = η(ϕ). Let U be a neighborhood of e such that the interior of
U c contains supp(ϕ) (the support of the continuous function ϕ). Then the set Λ of u ≤ 1
−1
such that gu−
gu ∈ U c is finite almost surely. Let 0 = t0 < t1 < t2 < · · · < tn = 1 with
ti+1 − ti = δ = 1/n for all 0 ≤ i ≤ n − 1, and let
fδ =
n−1
∑
ϕ(gt−1
gti+1 ).
i
i=0
−1
Fix a sample path of gt . If (ti , ti+1 ] contains some u ∈ Λ, then gt−1
gti+1 → gu−
gu as δ → 0,
i
−1
and if not, then gti gti+1 ̸∈ supp(ϕ) when δ is sufficiently small. It follows that fδ → Fϕ
almost surely as δ → 0. However,
E(fδ2 ) =
n−1
∑ n−1
∑
E[ϕ(gt−1
gti+1 )ϕ(gt−1
gtj+1 )]
i
j
i=0 j=0
=
∑
E[ϕ(gδe )] E[ϕ(gδe )] +
n−1
∑
E[ϕ(gδe )2 ] =
i=0
i̸=j
→ [Lϕ(e)] + L(ϕ )(e)
2
2
(as δ → 0)
1
1−δ
2
[P
ϕ(e)]
+
Pδ (ϕ2 )(e)
δ
δ2
δ
= η(ϕ)2 + η(ϕ2 ) < ∞,
where the first half of Theorem 2.1 is used to justify the convergence and the equality. This
proves that the family {fδ ; δ > 0} is L2 (P )-bounded, hence, it is uniformly integrable. It
follows that
1
E(Fϕ ) = lim E(fδ ) = lim Pδ ϕ(e) = Lϕ(e) = η(ϕ). 2
δ→0
δ→0 δ
By Theorem 2.6, a Lévy process gt is continuous if and only if its Lévy measure η vanishes.
∑
In this case, its generator L becomes a differential operator T = (1/2) aij ξi ξj + ξ0 , and gt
becomes a diffusion process in G with generator T as defined in Appendix A.5. If {aij } also
vanishes, then the process is just a non-random drift gt = g0 etξ0 .
We now consider adding or removing jumps from a given Lévy process gt in G. For a rcll
path xt in a Lie group G, a possible jump at time u > 0 may be removed to obtain a new
path x′t defined by
x′t = xt
for t < u and x′t = xu− x−1
u xt
46
for t ≥ u.
(2.12)
Similarly, a jump σ ∈ G may be added to the path xt at time u > 0, replacing a possible
existing jump at u, to obtain a new path x′t defined by
x′t = xt
for t < u and x′t = xu− σx−1
u xt
for t ≥ u.
(2.13)
Several jumps at different times may be removed from or added to the path xt to obtain a
new rcll path, which does not depend on the order in which the jumps are removed or added.
Let η ′ be a finite Lévy measure on G, and let N ′ be a homogeneous Poisson random
measure on R+ × G with characteristic measure η ′ , and assume N ′ is independent of the
jump counting measure N of the Lévy process gt . Because η ′ is finite, by the discussion in
Appendix A.6, there are an iid sequence of exponential random variables τi of rate λ = η ′ (G)
and an independent sequence of G-valued random variables σi of distribution η ′ (·)λ such
∑
that with Tn = ni=1 τi and T0 = 0, (Tn , σn ), n ≥ 1, exhaust all the points of N ′ . We may
successively add the jumps σn to the process gt at times Tn to obtained a new process gt′ ,
called the process obtained from gt after adding jumps from N ′ , or from η ′ .
Proposition 2.7 Let gt be a Lévy process in G with generator L, let η ′ be a finite Lévy measure
on G, and let gt′ be the process obtained from gt after adding jumps from η ′ . Then gt′ is a
Lévy process in G and its generator L′ , restricted on Cc∞ (G), is given by
L′ f (g) = Lf (g) +
∫
G
[f (gx) − f (g)]η ′ (dx),
g ∈ G.
Consequently, if (ξ0 , aij , η) are the characteristics of gt , then (ξ0′ , aij , η + η ′ ) are the charac∑
teristics of gt′ , where ξ0′ = ξ0 + i η ′ (ϕi )ξi .
Proof: For s < t, gs′−1 gt′ can be constructed from gs−1 gu , u ∈ (s, t], and the points of N ′
′
in (s, t] × G, in the same way as g0′−1 gt−s
from g0−1 gu , u ∈ (0, t − s], and the points of N ′
in (0, t − s] × G, it follows that gt′ has independent and stationary increments, and so is a
Lévy process.
Let Pt and Pt′ be respectively the transition semigroups of gt and gt′ . Assume g0 = e.
Then in the the notation introduced before Proposition 2.7, for f ∈ Cc∞ (G) and g ∈ G,
Pt′ f (g) = E[f (ggt′ )] = E[f (ggt ); t < T1 ] + E[f (ggT1 − σ1 gT′−1
gt′ ; T1 ≤ t]
1
= Pt f (g)e−λt +
∫
t
0
E[f (ggs σ1 gs′−1 gt′ )]λe−λs ds
= [Pt f (g) − f (g)]e−λt +
∫
t
0
E[f (ggs σ1 gs′−1 gt′ ) − f (g)]λe−λs ds + f (g),
1
L′ f (g) = lim [Pt′ f (g) − f (g)] = Lf (g) + E[f (gσ1 ) − f (g)]λ
t→0 t
∫
= Lf (g) +
G
[f (gx) − f (g)]η ′ (dx). 2
47
and hence
Let gt be a Lévy process in G with Lévy measure η and jump counting measure N . Let
A ∈ B(G) with η(A) < ∞. Then for any t ∈ R+ , N has finitely many points (u, σ) in
−1
[0, t] × A almost surely. Each (u, σ) corresponds to a jump gu−
gu = σ of the process gt . We
may successively remove these jumps from gt to obtain a new process gt′ , called the process
obtained from gt after jumps in A are removed.
For any measure µ on a measurable space (S, S), its restriction to A ∈ S is the measure
µ|A defined by µ|A (B) = µ(A ∩ B), B ∈ S.
Proposition 2.8 Let gt be a Lévy process in G with characteristics (ξ0 , aij , η), let A ⊂ B(G)
with η(A) < ∞, and let gt′ be the process obtained from gt after jumps in A are removed. Then
∑
gt′ is a Lévy process in G with characteristics (ξ0′ , aij , η − η|A ), where ξ0′ = ξ0 − i η(ϕi 1A )ξi .
Proof: Let yt be a Lévy process in G with characteristics (ξ0′ , aij , η − η|A ) and y0 = g0 . By
Proposition 2.7, the process yt′ obtained from yt after adding jumps from η|A is a Lévy process
in G with characteristics (ξ0 , aij , η). Because yt can be clearly obtained from yt′ after jumps
in A are removed, by the uniqueness of distribution in Theorem 2.1, one sees that the pair
of the processes (yt , yt′ ) have the same joint distribution as (gt′ , gt ). 2
2.3
Stochastic integral equations
Let {ξ1 , . . . , ξd } be a basis of g and let ϕ1 , . . . , ϕd be the associated coordinate functions
introduced earlier. The reader is referred to Appendices A.4 and A.6 for the definitions
of a Brownian motion Bt with covariance matrix {aij }, a homogeneous Poisson random
measure N on R+ × G with characteristic measure η, its compensated form Ñ (dt, dg) =
N (dt, dg) − η(dg)dt, and the stochastic integrals with respect to Bt and Ñ . Recall that a
Brownian motion Bt is said to be associated to a filtration Ft if Bt is adapted to Ft and
for any s < t, Bt − Bs is independent of Fs , and a Poisson random measure N is said to
be associated to Ft if the measure-valued process Nt = N ([0, t] × ·) is so. If Bt and N are
both associated to the same filtration Ft , then they are independent under Ft in the sense
that for any s < t, Bt − Bs , Nt − Ns and Fs are independent. Moreover, if {Bt } and N are
independent, then they are automatically independent under the filtration FtB,N generated
by Bt and N (see Appendix A.6), and so are associated to FtB,N .
The following result, due to Applebaum and Kunita [1], characterizes a Lévy process
in G by a stochastic integral equation driven by a Brownian motion and a homogeneous
Poisson random measure. For simplicity, we will assume gt starts at e. The result for a
general Lévy process can be easily obtained from this as it is just an independent product
g0 gte . Recall that a Lévy process gt in G is said to be associated to a filtration Ft , or called
an {Ft }-Lévy process, if it is adapted to Ft and for s < t, gs−1 gt is independent of Fs . Note
48
that if gt , Bt or N is associated to a filtration {Ft }, then it is also associated to {F̄t }, the
completion of {Ft }.
Theorem 2.9 Let gt be a Lévy process in G with g0 = e, associated to a filtration {Ft }, and
has characteristics (ξ0 , aij , η) as defined after Theorem 2.1. Then there exist a d-dimensional
Brownian motion Bt = (Bt1 , . . . , Btd ) with covariance matrix aij and a homogeneous Poisson
random measure N on R+ × G with characteristic measure η, such that both {Bt } and N
are associated to {F̄t }, and ∀f ∈ Cc∞ (G),
f (gt ) = f (e) +
∫
t
d ∫
∑
∫
i=1 0
+
0
G
t
ξi f (gs− ) ◦
∫
dBsi
+
[f (gs− h) − f (gs− ) −
∫
t
0
d
∑
ξ0 f (gs− )ds +
t
0
∫
G
[f (gs− h) − f (gs− )]Ñ (ds dh)
ϕi (h)ξi f (gs− )]ds η(dh) (almost surely).
(2.14)
i=1
Moreover, the pair of {Bt } and N are unique almost surely (see Note 1 below), and N is
almost surely the jump counting measure of gt given by (2.11).
Conversely, let Bt be a Brownian motion with covariance matrix aij and N be a homogeneous Poisson random measure on R+ × G with characteristic measure η, and let ξ0 ∈ g.
Assume both Bt and N are associated to a filtration {Ft }, and η is a Lévy measure. Then
there is a rcll process gt in G with g0 = e, adapted to {F̄t }, such that (2.14) holds for any
f ∈ Cc∞ (G). Moreover, gt is unique almost surely (see Note 1 below), and is an {F̄t }Lévy process with characteristics (ξ0 , aij , η).
∫
∫
Note 1: Here, as in the rest of this work, the convention st = (s, t] is used when integrating
on R. The almost sure uniqueness of Bt and N in the first part of Theorem 2.9 means
that if (Bt′ , N ′ ) is another pair of a Brownian motion Bt′ with covariance matrix aij and a
homogeneous Poisson random measure N ′ of characteristic measure η, both are associated to
a complete filtration {Ft′ }, such that gt is {Ft′ }-adapted and (2.14) holds for any f ∈ Cc∞ (G)
with (Bt , N ) replaced by (Bt′ , N ′ ), then almost surely, Bt = Bt′ for all t ≥ 0 and N = N ′ . The
almost sure uniqueness of gt in the second part of Theorem 2.9 means that if gt′ is another
rcll process in G with g0′ = e, such that gt′ is adapted to a complete filtration {Ft′ } to which
both Bt and N are associated, and (2.14) holds for any f ∈ Cc∞ (G) with gt replaced by gt′ ,
then almost surely, gt = gt′ for all t ≥ 0.
Note 2: Note that gs− may be replaced by gs for the two integrals taken with respect
to the Lebesgue measure ds on R+ in (2.14). See Appendix A.4 for the existence of the
∫
Stratonovich stochastic integral 0t ξi f (gs− ) ◦ dBsi . The existence of the stochastic integral
taken with respect to Ñ (ds dh) is guaranteed by the discussion in Appendix A.6 and the
condition (2.4). This condition also ensures that the last integral in (2.14) exists and is
finite. Therefore, all the integrals in (2.14) exist and are finite.
49
Theorem 2.9 will be proved at the end of this chapter.
We note that if gt is a Lévy process that satisfies (2.14) for any f ∈ Cc∞ (G), then for any
f ∈ C 1 (G),
∫ t
∫ t
d ∫ t
1∑
f (gs− ) ◦ dBsj =
f (gs− )dBsj +
ξk f (gs )ajk ds.
(2.15)
2 k=1 0
0
0
To prove this, as in Appendix A.5, we may use the Whittney embedding and the tubular
neighborhood theorems to embed G into an Euclidean space Rn , and extend any f ∈ Cc∞ (G)
and smooth vector fields ξ on G to f ∈ Cc∞ (G) and a smooth vector field ξ on Rn , respectively.
Then gt may be regarded as a process in Rn , and by (2.14), which now may be regarded as
to hold on Rn , gt is a semi-martingale. Any f ∈ C 1 (G) may be extended to be a function
f ∈ C 1 (Rn ). By (A.8) in Appendix A.4 and (A.20) in Appendix A.6, we have
∫
t
0
∫
t
=
0
f (gs− ) ◦ dBsj =
f (gs− )dBsj +
∫
t
0
d
∑
1
f (gs− )dBsj + [f (g· ), B j ]ct
2
∫ t
1
ξk f (gs ) d[B j , B k ]cs .
2 k=1 0
This proves (2.15) because [B j , B k ]t = ajk t.
Replacing f by ξj f in (2.15), we see that the stochastic integral equation (2.14) can also
be written as
∫ t
f (gt ) = f (e) + Mtf +
Lf (gs )ds
(2.16)
0
for f ∈
Cc∞ (G),
where
Mtf =
d ∫
∑
t
j=1 0
∫
ξj f (gs− )dBsj +
t
∫
0
G
[f (gs− h) − f (gs− )]Ñ (ds dh)
is an L2 -martingale, and Lf is the expression on the right hand side of (2.3) without using
the fact that f ∈ D(L).
In fact, the stochastic integral equation (2.14) holds for f contained in a larger function
space. Let f ∈ Cb (G) ∩ C 2 (G), the space of bounded functions on G possessing continuous
second order derivatives. Then the first integral in (2.14), a Stratonovich stochastic integral,
exists and is finite by the discussion in Appendix A.4. The same is true for the second
integral because it is a path-wise Lebesgue integral. For the last two integrals, let U be a
relatively compact open neighborhood of e and let
τU = inf{t ≥ 0; gt ∈ U c or gt− ∈ U c }.
(2.17)
It is proved in §1.5 that τU is a stopping time under Ft . Then U U = {gh; g, h ∈ U } is
relatively compact and τU ↑ ∞ as U ↑ G. Each of the last two integrals in (2.14), taken with
∫ ∫
∫ ∫
respect to Ñ (ds dh) and ds η(dh), can be written as a sum 0t U + 0t U c . The first term of
50
this sum is finite when t is replaced by t ∧ τU = min(t, τU ) because of the Taylor expansion
(2.5) and also because gs− h ∈ U U , and so is the second term because f is bounded and
η(U c ) < ∞. Therefore, all the integrals in (2.14) exist and are finite for f ∈ Cb (G) ∩ C 2 (G).
Theorem 2.10 The stochastic integral equation (2.14) in Theorem 2.9 holds for any f ∈
Cb (G) ∩ C 2 (G).
Proof Let Rt (f ) denote the right hand side of (2.14). Fix f ∈ Cb (G) ∩ C 2 (G). Let U
be a relatively compact open neighborhood of e such that the coordinate functions ϕi are
supported by U , and let τU be the stopping time defined by (2.17). Choose ϕ ∈ Cc∞ (G) such
that 0 ≤ ϕ ≤ 1 and ϕ = 1 on U U . Since f ϕ ∈ Cc2 (G) ⊂ C02,l (G), (2.14) holds when f is
replaced by f ϕ. If t < τU , Y ∈ g and h ∈ U , then (f ϕ)(gt ) = f (gt ), Y (f ϕ)(gt− ) = Y f (gt− )
and (f ϕ)(gt− h) = f (gt− h). It follows that for t < τU ,
f (gt ) = f (e) +
∫
t
d ∫
∑
∫
i=1 0
+
∫
0
t
∫
G
+
∫
0
t
+
0
∫
t
G
= Rt (f ) +
t
+
0
ξ0 f (gs )ds
[f (gs− h) − f (gs− )]Ñ (dsdh) +
[f (gs h) − f (gs ) −
d
∑
∫
t
0
∫
Uc
[(f ϕ)(gs− h) − f (gs− h)]Ñ (dsdh)
ϕi (h)ξi f (gs )]dsη(dh)
i=1
Uc
= Rt (f ) +
ξi f (gs− ) ◦
∫
dBsi
[(f ϕ)(gs h) − f (gs h)]dsη(dh)
∫
t
∫
0
∑
Tn ≤t
Uc
[(f ϕ)(gs− h) − f (gs− h)]N (dsdh)
[(f ϕ)(gTn − σn ) − f (gTn − σn )],
where (Tn , σn ) are the points of the homogeneous Poisson random measure N on R+ × U c
defined by N ′ ([0, t] × ·) = N ([0, t] × (· ∩ U c )) as described in Appendix A.6. Since τU ↑ ∞
and T1 ↑ ∞ as U ↑ G, f (gt ) = Rt (f ) almost surely for any t ∈ R+ . 2
Let Cb2 (G) be the space of functions f ∈ Cb (G) ∩ C 2 (G) such that ξf and ξξ ′ f belong to
Cb (G) for ξ, ξ ′ ∈ g. We note that for f ∈ Cb2 (G), the expression Lf given by (2.3) makes
sense and is bounded, but f ∈ Cb2 (G) may not belong to the domain of the generator L.
Theorem 2.11 For f ∈ Cb2 (G) and g ∈ G,
d
Pt f (g) |t=0 = Lf (g),
dt
(2.18)
where Lf denote the the right hand side of (2.3). Moreover, the domain D(L) of the generator
L of a Lévy process in G contains Cb2 (G) ∩ C0 (G), and for f ∈ Cb2 (G) ∩ C0 (G), (2.3) holds.
51
Proof For f ∈ Cb2 (G), Mtf in (2.16) is an L2 -martingale. Taking the expectation on (2.16)
∫
yields E[f (gt )] = E[f (g0 )] + E[ 0t Lf (gs )ds]. Taking the derivative of Pt f (e) = E[f (gt )]
at t = 0, we obtain (2.18). Because Lf is bounded, it follows that (1/t)[Pt f − f ] =
∫
E[(1/t) 0t Lf (gs )ds] converges to Lf uniformly on G. It follows that D(L) contains Cb2 (G) ∩
C0 (G) and for f ∈ Cb2 (G) ∩ C0 (G), Lf is given by (2.3). 2
In Theorem 2.9, if the Lévy measure η satisfies the finite first moment condition (2.7),
then the integral
∫
t
∫
0
G
[f (gs− h) − f (gs− )]N (ds dh)
exists and the stochastic integral equation (2.14) can be simplified as follows.
f (gt ) = f (g0 ) +
∫
t
∫
d ∫
∑
i=1 0
+
0
t
G
ξi f (gs− ) ◦
∫
dBsi
t
+
0
ξ0′ f (gs )ds
[f (gs− h) − f (gs− )]N (ds dh)
∑ ∫
for f ∈ Cb (G) ∩ C 2 (G), where ξ0′ = ξ0 − i [
to Cb2 (G) ∩ C0 (G), is now given by (2.8).
G
(2.19)
ϕi (h)η(dh)]ξi . The generator L of gt , restricted
If we assume the Lévy measure η is finite, then N is determined by a sequence of random
times Tn ↑ ∞, and an independent sequence of iid (independent and identically distributed)
random variables σn in G as described in Appendix 5, where Tn − Tn−1 are iid of the exponential distribution of rate η(G), and σn has distribution η/η(G). In this case, the last term
∑
in (2.19) may be written as Tn ≤t [f (gTn − σn ) − f (gTn − )]. Therefore, the stochastic integral
equation (2.19) is equivalent to the following stochastic differential equation
dgt =
d
∑
ξi (gt ) ◦ dBti + ξ0′ (gt )dt
(2.20)
i=1
on G together with the jump conditions gt = gt− σn at t = Tn for n = 1, 2, . . .. More precisely,
the following result (whose simple proof is omitted) holds.
Theorem 2.12 Let gt be a Lévy process in G with g0 = e and a finite Lévy measure η, and
let N be the counting measure of its jumps given by (2.11). Let random times Tn ↑ ∞ and
G-valued random variables σn be determined by the homogeneous Poisson measure N on
R+ × G as in Appendix A.6. Then gt may be obtained by successively solving the stochastic
differential equation (2.20) on the random time intervals [Tn , Tn+1 ) with initial conditions
g0 = e and g(Tn ) = g(Tn −)σn for n ≥ 1. In particular, a continuous Lévy process gt in G
with g0 = e is the unique solution of stochastic differential equation (2.14) starting at e.
52
By this theorem, a Lévy process gt with a finite Lévy measure may be regarded as a
continuous Lévy process, determined by (2.20), interlaced with iid random jumps at exponentially distributed random time intervals. It can be shown (see Applebaum [2]) that a
general Lévy process can be obtained as a limit of such processes.
If aij = 0 and ξ0 = 0, and η is finite, then the Lévy process gt is a discrete process
consisting of a sequence of iid jumps at exponentially spaced random time intervals. More
precisely, gt = g0 for 0 ≤ t < T1 , and
gt = g0 σ1 σ2 · · · σi for Ti ≤ t < Ti+1 and i ≥ 1,
(2.21)
where (Tn , σn ) are the points of the Poisson random measure N , the jump counting measure
of gt as defined by (2.11). Therefore, the random walks in G, which are defined as the
products of iid G-valued random variables, can be regarded as discrete time Lévy processes.
Sometimes it is convenient to work with stochastic integral equations driven by a standard Brownian motion. The following result allows the conversion of equation (2.14) to an
equation driven by a standard Brownian motion.
Theorem 2.13 In Theorem 2.9, let m be the rank of the covariance matrix a = {aij }. Then
there are an m-dim standard Brownian motion Wt = (Wt1 , . . . , Wtm ) and an m×d real matrix
σ such that a = σ ′ σ and Bt = Wt σ, where σ ′ is the transpose of σ, and the natural filtrations
∑
of Bt and Wt are the same. Let ζi = dj=1 σij ξj for 1 ≤ i ≤ m. Then the stochastic integral
equation (2.14) for gt may be written as
f (gt ) = f (e) +
∫
t
m ∫
∑
j=1 0
∫
+
∫
0
t
∫
G
+
0
G
t
ζj f (gs− ) ◦ dWsj +
∫
0
t
ξ0 f (gs )ds
[f (gs− h) − f (gs− )]Ñ (ds dh)
[f (gs− h) − f (gs− ) −
d
∑
ϕi (h)ξi f (gs− )]ds η(dh).
(2.22)
i=1
Moreover, the generator L of gt given by (2.3) may be written as
Lf (g) =
∫
m
d
∑
1∑
ζj ζj f (g) + ξ0 f (g) + [f (gh) − f (g) −
ϕi (h)ξi f (g)]η(dh).
2 j=1
G
i=1
(2.23)
Proof If m = d, then a is invertible. Let σ be the square root of a and Wt = Bt σ −1 . Then
the conclusions follow from a simple computation. Suppose m < d. Let ai· = (ai1 , . . . , aid )
be the ith row vector of matrix a. Then a has m linearly independent row vectors and the
rest are their linear combinations. Say the first m row vectors of a are linearly independent.
∑m
∑
j
i
′
For j > m, aj· = m
i=1 λi Bt .
i=1 λi ai· for some constants λ1 , . . . , λm , and let Bt = Bt −
Because aij t = E(Bti Btj ), the above implies E(Bt′ Btk ) = 0 for 1 ≤ k ≤ d. Hence Bt′ has
53
zero covariance and so must be 0. This proves that for j > m, Btj is a linear combination
of Bt1 , . . . , Btm . The submatrix ã of a formed by the first m rows and first m columns is
invertible, otherwise, by the preceding argument, some Bti with i ≤ m would be a linear
combination of the rest of Bt1 , . . . , Btm , which would imply rank(a) < m. Let σ̃ be the square
root of the positive definite symmetric matrix ã. Then Wt = (Bt1 , . . . , Btm )σ̃ −1 is a standard
Brownian motion such each Bti is a linear combination of Wt1 , . . . , Wtm for 1 ≤ i ≤ d. The
matrix σ is obtained from the coefficients of these linear combinations. 2
Note that by Theorems 2.10 and 2.11, (2.22) and (2.23) in Theorem 2.13 hold for f ∈
Cb (G) ∩ C 2 (G) and f ∈ Cb2 (G) ∩ C0 (G) respectively. If η satisfies the finite first moment
condition (2.7), then they simplify as follows:
f (gt ) = f (e) +
∫
t
m ∫
∑
j=1 0
∫
+
0
G
t
ζj f (gs− ) ◦
∫
dWsj
t
+
0
ξ0′ f (gs )ds
[f (gs− h) − f (gs− )]N (ds dh)
(2.24)
for f ∈ Cb (G) ∩ C 2 (G), and
m
1∑
Lf (g) =
ζj ζj f (g) + ξ0′ f (g) +
2 j=1
∫
G
[f (gh) − f (g)]η(dh)
(2.25)
∑ ∫
for f ∈ Cb2 (G) ∩ C0 (G), where ξ0′ = ξ0 − i [ G ϕi (h)η(dh)]ξi .
By Theorem 2.13, a continuous Lévy process gt in G, for which η = 0, is the solution of
the following sde (stochastic differential equation),
dgt =
m
∑
ζj (gt ) ◦ dWtj + ξ0 (gt )dt.
(2.26)
j=1
In fact, for any ξ0 , ζ1 , . . . , ζm ∈ g, the solution of sde (2.26) is a diffusion process in G with
generator
L = (1/2)
m
∑
ζj ζj + ξ0 ,
(2.27)
j=1
as defined in Appendix A.5. Moreover, because ξ0 and ζj are left invariant vector fields on G,
gt is a Feller process with an infinite life time (see Appendix A.5), and so is a Lévy process
in G.
2.4
Lévy processes in a matrix group
In this section, let G be the general linear group GL(d, R), the group of the d × d real
invertible matrices. This is a d2 -dimensional Lie group with Lie algebra g being the space
54
gl(d, R) of all the d × d real matrices and the Lie bracket given by [X, Y ] = XY − Y X. See
Appendix A.1 for more details.
2
We may identify g = gl(d, R) with the Euclidean space Rd and G = GL(d, R) with a
2
2
dense open subset of Rd . For any X = {Xij } ∈ Rd , its Euclidean norm
∑
∥X∥ = (
Xij2 )1/2 = [Tr(XX ′ )]1/2
i,j
satisfies ∥XY ∥ ≤ ∥X∥ ∥Y ∥ for any X, Y ∈ Rd , where Tr(X) is the trace of a matrix X and
XY is the matrix product.
Let Eij be the matrix that has 1 at place (i, j) and 0 elsewhere. Then the family {Eij ;
i, j = 1, 2, . . . , d} is a basis of g. Let {ϕij ; i, j = 1, 2, . . . , d} be a set of associated coordinate
functions and let ϕ = {ϕij }. One may take ϕ(g) = g − Id for g contained in a neighborhood
of e = Id (the d × d identity matrix). In general, by (2.2), any set of coordinate functions
satisfy ϕ(g) = g − Id + O(∥g − Id ∥2 ) for g contained in a neighborhood of Id .
2
For g ∈ G = GL(d, R), the tangent space Tg G of G at g may be identified with Rd ,
therefore, any X of Tg G may be represented by a d × d real matrix {Xij } in the sense that
2
∀f ∈ C(G),
Xf =
d
∑
Xij
i,j=1
∂
f (g),
∂gij
where gij , for i, j = 1, 2, . . . , d, are the standard coordinates on Rd . It can be shown that for
g, h ∈ GL(d, R) and X ∈ Te G = gl(d, R), Dlg ◦ Drh (X) is represented by the matrix product
gXh, where X is identified with its matrix representation {Xij }. Therefore, we may write
gXh for Dlg ◦ Drh (X). Thus, X(g) = gX and X r (g) = Xg. This short-hand notation may
even be used on a general Lie group G.
Under a suitable condition, the stochastic integral equation (2.22) for a Lévy process gt
in G = GL(d, R) has a more explicit expression. Let f be the matrix-valued function on G
defined by f (g) = gij for g ∈ G. Although f is not contained in Cb (G) ∩ C 2 (G), at least
informally, (2.22), with g0 = e replaced by a general g0 , leads to the following stochastic
integral equation in matrix form:
2
gt = g0 +
∫
t
m ∫
∑
i=1 0
∫
+
0
G
t
gs− Yi ◦ dWsi +
∫
∫
t
0
gs Zds +
0
t
∫
G
gs− (h − Id )Ñ (ds dh)
gs [h − Id − ϕ(h)]ds η(dh) (almost surely)
(2.28)
for some Y1 , . . . , Ym , Z ∈ gl(d, R).
For any process yt taking values in a Euclidean space, let
yt∗ = sup ∥ys ∥.
0≤s≤t
55
(2.29)
Theorem 2.14 Let gt be a Lévy process in G = GL(d, R) with g0 = Id and let N be its
jump counting measure. Assume gt is associated to a filtration {Ft }, and the characteristic
measure η of N satisfies
∫
∥h − Id ∥2 η(dh) < ∞.
(2.30)
G
Then for any t > 0,
E[(gt∗ )2 ] < ∞.
Moreover, there are Y1 , . . . , Ym , Z ∈ g = gl(d, R), and an m-dimensional standard Brownian
motion Wt = (Wt1 , . . . , Wtm ) associated to {F̄t }, such that (2.28) holds.
Conversely, let Wt be an m-dimensional standard Brownian motion and let N be a homogeneous Poisson random measure on R+ × G with characteristic measure η. Assume
both Wt and N are associated to a filtration {Ft }, and η is a Lévy measure and satisfying
(2.30). Let Y1 , . . . , Ym , Z ∈ g. Then there is a rcll process gt in G with g0 = Id , adapted to
{F̄t }, such that (2.28) hold. Moreover, such a process gt is unique almost surely and is an
{F̄t }-Lévy process in G with N being its jump counting measure almost surely. Furthermore,
E[(gt∗ )2 ] < ∞ for any t > 0.
Note: The existence of the Ñ -integral in (2.28) is guaranteed by (2.30). Because [h − Id −
ϕ(h)] = O(∥h − Id ∥2 ) for h near Id , the existence of the last integral in (2.28) is implied by
(2.30). Hence all the integrals in (2.28) exist and are finite.
Proof Let gt be an {Ft }-Lévy process in G with g0 = Id . Fix two positive constants α < β.
Let f be a G-valued function with components contained in Cc2 (G) such that f (g) = g for
any g ∈ G with ∥g∥ ≤ β, and ∥f (g) − f (h)∥ ≤ c∥g − h∥ for any g, h ∈ G and some c > 0, let
τ = inf{t ≥ 0; ∥gt ∥ ≥ α or ∥gt− ∥ ≥ α} (an {Ft }-stopping time as shown in §1.5), and let
U = {h ∈ G; ∥gh∥ ≤ β for any g ∈ G with ∥g∥ ≤ α}.
We may assume that the ϕij are supported by U . By (2.22), for some Y1 , Y2 , . . . , Ym , Z ∈ g,
f (gt∧τ ) = Id +
∫
t∧τ
∫
+
∫
0
U
t∧τ
∫
+
0
U
m ∫
∑
t∧τ
i=1 0
∫
gs− Yi ◦ dWsi +
gs− (h − Id )Ñ (ds dh) +
∫
gs [h − Id − ϕ(h)]dsη(dh) +
gs Zds
0
t∧τ
0
t∧τ
∫
∫
[f (gs− h) − f (gs− )]Ñ (ds dh)
Uc
t∧τ
∫
Uc
0
[f (gs h) − f (gs )]dsη(dh).
(2.31)
Since ∥f (gh) − f (g)∥ ≤ c∥gh − g∥ ≤ c∥g∥ ∥h − Id ∥,
∫
t∧τ
E{∥
0
≤ α ct
∫
2 2
Uc
∫
Uc
[f (gs− h) − f (gs− )]Ñ (ds dh)∥2 } = E
∥h − Id ∥2 η(dh) → 0
56
∫
t∧τ
0
∫
Uc
∥f (gs− h) − f (gs− )∥2 dsη(dh)
as β ↑ ∞ because U ↑ G. Similarly,
∫
t∧τ
∫
E{∥
Uc
0
[f (gs h) − f (gs )]dsη(dh)∥} ≤ αct
∫
Uc
∥h − Id ∥η(dh) → 0
as β ↑ ∞. Since f (g) → g as β ↑ ∞ for any g ∈G, we have, for any t ∈ R+ ,
gt∧τ = Id + Jt + Kt + Lt ,
where
Jt =
∫
Kt =
and
∫
Lt =
t∧τ
0
m ∫
∑
t∧τ
i=1 0
t∧τ
∫
0
G
gs− Yi dWsi ,
gs− (h − Id )Ñ (ds dh)
∫
m
1∑
{
gs Yi Yi + gs Z + gs [h − Id − ϕ(h)]η(dh)}ds.
2 i=1
G
2
Applying Doob’s norm inequality (A.4) to the Rd -valued martingale Jt and using the basic
property of stochastic integrals, we obtain
E[(Jt∗ )2 ]
≤ 4E[∥Jt ∥ ] ≤ c1 E[
2
m ∫
∑
∫
t∧τ
∥gs Yi ∥ ds] ≤ c2 E[
i=1 0
for positive constants c1 and c2 . Similarly, because
E[(Kt∗ )2 ]
∫
≤ 4E[
t∧τ
0
∫
t∧τ
2
∫
G
0
∥h − Id ∥2 η(dh) is finite,
∫
∥gs (h − Id )∥ η(dh)ds] ≤ c3 E[
2
G
(gs∗ )2 ds]
t∧τ
0
(gs∗ )2 ds]
for some constant c3 > 0. Using the Schwartz inequality and the fact that
∫
G
∥h − Id − ϕ(h)∥η(dh) < ∞,
we obtain
∫
E[(L∗t )2 ] ≤ c4 tE[
t∧τ
0
(gs∗ )2 ds]
for some constant c4 > 0. It follows that
∗
)2 ]
E[(gt∧τ
∫
≤ b + ct E[
t∧τ
0
(gs∗ )2 ds] ≤ b + ct tα2
∗
)2 ] is finite.
for some positive constants b and ct with the latter depending on t, hence, E[(gt∧τ
Moreover,
∫
∗
)2 ] ≤ b + ct
E[(gt∧τ
t
0
∗
)2 ]ds.
E[(gs∧τ
The Gronwall inequality (see for example [46, Lemma 21.4], or simply apply the inequality
∗
)2 ] ≤ bect t . Since τ ↑ ∞ as α ↑ ∞, this
displayed here repeatedly to itself) yields E[(gt∧τ
proves that E[(gt∗ )2 ] < ∞ and (2.28) holds for any t ∈ R+ .
57
2
If we repeat the preceding argument with g0 = 0 (the origin in Rd ), then b = 0. This
proves the uniqueness of the process gt satisfying (2.28). To prove the existence of such
a process, by Theorem 2.13, there is a Lévy process gt in G satisfying (2.31). Then the
computation after (2.31) shows that gt is the unique rcll process satisfying (2.28). It also
shows E[(gt∗ )2 ] < ∞. 2
Note that if the Lévy measure η in Theorem 2.14 also satisfies
then the stochastic integral equation (2.28) becomes
m ∫
∑
gt = g0 +
where Y0 = Z −
2.5
t
i=1 0
∫
G
gs− Yi ◦ dWsi +
∫
0
∫
t
gs Y0 ds +
t
∫
0
G
∫
G
∥h − Id ∥η(dh) < ∞,
gs− (h − Id )N (ds dh),
(2.32)
ϕ(h)η(dh).
Proof of Theorem 2.9, part 1
We will use “sie” as the abbreviation for “stochastic integral equation”. Recall that in the
sie (2.14), Bt is a Brownian motion with covariance matrix aij , N is a homogeneous Poisson
random measure on R+ × G with characteristic measure η being a Lévy measure, and ξ0 ∈ g.
Let {Ft } be a filtration and let {F̄t } be its completion. A rcll process gt in G, with g0 = e,
will be called a solution of the sie (2.14) under {Ft } if Bt and N are associated to {Ft }, gt is
adapted to {F̄t } and satisfies (2.14) for any f ∈ Cc∞ (G). The sie (2.14) will be said to have
a unique solution under a filtration {Ft } if it has a solution gt under {Ft }, and if any other
solution, possibly under a different filtration, is equal to gt almost surely.
Theorem 2.15 Let {Ft } be a filtration to which both Bt and N are associated. Then the sie
(2.14) has a unique solution gt under {Ft }. Moreover, gt is a Lévy process in G associated
to {F̄t }, and N is its jump counting measure almost surely.
By (2.16), for gt in Theorem 2.15 and f ∈ Cc∞ (G),
∫
f (gt ) = f (e) +
∫
d
t∑
0 i=1
∫
ξi f (gs− )dBsi +
t
0
∫
G
[f (gs− h) − f (gs− )]Ñ (ds dh)
t
+
0
Lf (gs )ds,
(2.33)
where Lf (g) is the expression on the right hand side of (2.3). Here we are not using the fact
that L is the generator of a Lévy process. Of course, because
1 ∫t
1
{E[f (gt )] − f (e)} = E[ Lf (gs )ds] → Lf (e)
t
t
0
58
as t → 0, by the left invariance, L on Cc∞ (G) is in fact the generator of gt , restricted on
Cc∞ (G), as given in Theorem 2.1. This shows that Theorem 2.15 implies the second half of
Theorem 2.9. In the rest of this section, Theorem 2.15 will be proved by a series of lemmas,
and this proof will not depend on Theorem 2.1.
Lemma 2.16 If gt is a solution of the sie (2.14) under some filtration Ft , then N is its
jump counting measure almost surely.
Proof On the right hand side of (2.14), the Ñ -integral is the only term that is discontinuous
in t. For any neighborhood U of e, this integral may be written as
∫
∫
t
∫
Uc
0
t
∫
+
0
U
[f (gs− h) − f (gs− )]N (ds dh) −
∫
0
t
∫
Uc
[f (gs− h) − f (gs− )]dsη(dh)
[f (gs− h) − f (gs− )]Ñ (ds dh),
where the first term is a pathwise integral, the second term is continuous in t, and the last
∫ ∫
term 0t U [· · ·] converges to 0 in L2 as U ↓ {e}. By Doob’s norm inequality (A.4), almost
surely, the last term converges to 0 uniformly pathwise over any finite time interval. It
follows that for any t > 0, if (t, h) is a point of the Poisson random measure N for some
h ̸= e, then f (gt ) − f (gt− ) = f (gt− h) − f (gt− ), and if otherwise, f (gt ) − f (gt− ) = 0. This
being true for countably many f ∈ Cc∞ (G) that separate points on G shows that gt = gt− h,
and hence N is the counting measure of the jumps of gt almost surely. 2
Let Ft f (g; B, N, ξ0 ) be the sum of integrals on the right hand side of (2.14). Then the
sie (2.14) may be written as
f (gt ) = f (e) + Ft f (g; B, N, ξ0 ).
(2.34)
For s < t, let Fs,t f (g; B, N, ξ0 ) = Ft f (g; B, N, ξ0 ) − Fs f (g; B, N, ξ0 ). Then
f (gt ) = f (gs ) + Fs,t f (g; B, N, ξ0 ).
(2.35)
Note that Ft f (g; B, N, ξ0 ) is left invariant in the sense that
Ft (f ◦ lx )(g; B, N, ξ0 ) = Ft f (xg; B, N, ξ0 )
for x ∈ G. Similarly, Fs,t f (B, N, ξ0 ) is left invariant. It follows that if gt is a solution of the
sie (2.34), then for any fixed r > 0, gtr = gr−1 gr+t is a solution of the r-time shifted sie
f (gtr ) = f (e) + Ft f (g r ; B r , N r , ξ0 ),
where Btr = Br+t − Bs and N r ([0, t] × ·) = N ([r, r + t] × ·).
59
(2.36)
Let τ be a stopping time under a filtration Ft to which both B and N are associated.
We may consider a τ -time shifted sie driven by Btτ and N τ . Then under the conditional
probability P (· | τ < ∞), Btτ is a Brownian motion with the same covariance matrix as Bt ,
and N τ is a homogeneous Poisson random measure with the same characteristic measure as
N . Moreover, B τ , N τ and Fτ are independent. To show this, let A1 , A2 , . . . , Ak ∈ B(G) and
apply Theorem 1.6 to the Lévy process (Bt , N ([0, t] × A1 ), . . . , N ([0, t] × Ak )) in Rd+k .
For Λ ⊂ G, let NΛ = N |R+ ×Λ be the restriction of N to R+ ×Λ. Let U be a neighborhood
of e. Because η(U c ) < ∞, as mentioned in Appendix 5, there are stopping times τ1 ≤ τ2 ≤
τ3 < . . . of NU c , with iid exponential differences (τj − τj−1 ) of rate η(U c ), and an independent
sequence of iid G-valued random variables σ1 , σ2 , σ3 , . . . of distribution η(· ∩ U c )/η(U c ), such
that (τn , σn ) exhaust all the points of the Poisson random measure NU c . Then
∫
Ft f (g; B, NU , ξ0U )
Ft f (g; B, N, ξ0 ) =
∫
Uc
η(dx)
∑
i
∫
+
= Ft f (g; B, NU , ξ0U ) +
where ξ0U = ξ0 −
t
0
∑
τn ≤t
Uc
[f (gs− x) − f (gs− )]N (ds dx)
[f (gτn − σn ) − f (gτn − )],
(2.37)
ϕi (x)ξi .
Lemma 2.17 Assume that the sie
f (gtU ) = f (g0U ) + Ft f (g U ; B, NU , ξ0U ),
f ∈ Cc∞ (G),
(2.38)
has a unique solution gtU under FtB,NU . Then sie (2.34) has a unique solution gt under
FtB,N . Moreover, gt is obtained from gtU , that has only jumps in U , interlaced with jumps in
U c given by NU c , that is,
gt = gtU for t < τ1 and recursively gt = (gτn − )σn (gτUn )−1 gtU for τn ≤ t < τn+1 .
(2.39)
Proof Because gtU is adapted to F̄tB,NU , it is adapted to F̄tB,N . Let gt1 = gtU , and recursively
for n ≥ 1, let gtn+1 = gtn for t < τn and gtn+1 = (gτnn − )σn (gτUn )−1 gtU for τn ≤ t. Then all gtn are
adapted to F̄tB,N . Let gt be defined by (2.39). Then gt = gtn for t < τn . This shows that gt
is adapted to F̄tB,N .
Because gtU is independent of NU c , τi and σi may be regarded as non-random for gtU . The
process gt clearly satisfies (2.34) for t < τ1 . For τ1 ≤ t < τ2 , with τ = τ1 and σ = σ1 ,
f (gt ) = f (gτ − ) + [f (gτ ) − f (gτ − ) + [f (gt ) − f (gτ )]
= f (gτU ) + [f (gτU σ) − f (gτU )] + [f (gτU σgτU −1 gtU ) − f (gτU σ)]
= f (e) + Fτ f (g U ; B, NU , ξ0U ) + [f (gτU σ) − f (gτU )]
+Fτ,t f (xg U ; B, NU , ξ0U ) (where x = gτU σgτU −1 )
= f (e) + Ft f (g; B, N, ξ0 ) (by (2.37)).
60
This proves that gt satisfies (2.34) for t < τ2 . In the same way, we prove recursively that for
any n > 1, gt satisfies (2.34) for t < τn , and hence is a solution of the sie (2.34) under FtB,N .
Conversely, if gt is a solution of the sie (2.34) under some Ft , then by Lemma 2.16, N is
its jump counting measure, so gt has jumps σ1 , σ2 , . . . at times τ1 , τ2 , . . .. Let yt1 = gt , and
recursively for n ≥ 1, let ytn+1 = ytn for t < τn and ytn+1 = yτnn − gτ−1
gt for τn ≤ t < τn+1 . Then
n
ytn is adapted to F̄t . Let
gtU = gt for t < τ1 , and recursively gtU = gτUn − gτ−1
gt for τn ≤ t < τn+1 .
n
Then gtU = ytn for t < τn , and hence gtU is adapted to F̄t . Moreover, gt may be obtained
from gtU through (2.39). By (2.37), gtU satisfies (2.38) for t < τ1 . Let τ = τ1 . For τ ≤ t < τ2 ,
gtU = gτ − gτ−1 gt , and then with y = gτ − gτ−1 ,
f (gtU ) = (f ◦ ly )(gt ) = (f ◦ ly )(gτ ) + Fτ,t (f ◦ ly )(g; B, N, ξ0 )
= f (gτU ) + Fτ,t f (g U ; B, NU , ξ0U ).
This shows that gtU satisfies (2.38) for τ ≤ t < τ2 . In the same way, it can be shown
recursively that gtU satisfies (2.38) for τn ≤ t < τn+1 , and hence is a solution of sie (2.38)
under Ft . The uniqueness of gtU as a solution to sie (2.38) implies the uniqueness of gt as a
solution to sie (2.34). 2
By Lemma 2.17, to prove Theorem 2.15, we may assume η is supported by any neighborhood U of e. We now establish a lemma about filtrations before continuing the proof of
Theorem 2.15.
Lemma 2.18 For any right continuous filtration Ft , let τ be a {Ft }-stopping time and let
σ be a stopping time under the filtration Gt = Fτ +t . Then τ + σ is a {Ft }-stopping time and
Gσ = Fτ +σ .
Proof We first note that the filtration Gt = Fτ +t is also right continuous because for any
stopping times σn ↓ σ, Fσ = ∩n Fσn , in particular, Fτ +t = ∩n>1 Fτ +t+1/n . For any t > 0,
[τ + σ < t] = ∪r≤t [σ < r] ∩ [τ < t − r], where r ranges over rationals < t. Because
[σ < r] ∈ Gr = Fτ +r and Fτ +r ∩ [τ + r < t] ⊂ Ft , [τ + σ < t] ∈ Ft . This shows that τ + σ is
a {Ft }-stopping time. For A ∈ Gσ , A ∩ [σ < t] ∈ Gt = Fτ +t . Then
A ∩ [τ + σ < t] =
∪
A ∩ [σ < r] ∩ [τ + r < t] ∈
r<t
∪
Fτ +r ∩ [τ + r < t] ⊂ Ft .
r<t
This shows A ∈ Fτ +σ . Next, let A ∈ Fτ +σ . Then
A ∩ [σ < t] =
∪
A ∩ [τ + σ < t + r] ∩ [r < τ ] ∈
r<t
∪
r<t
61
Ft+r ∩ [t + r < τ + t] ⊂ Fτ +t .
This shows A ∈ Gσ . 2
A rcll process gt in G with g0 = e will be called a local solution of sie (2.34) in an open
neighborhood U of e under a filtration {Ft } if both B and N are associated to {Ft }, and if
gt is adapted to F̄t and satisfies (2.34) for t ≤ τU (g· ), where
τU (g· ) = inf{t ≥ 0; gt ∈ U c or gt− ∈ U c }
(2.40)
is a stopping time under the natural filtration of gt (shown in §1.5). The local solution is
called unique if it agrees with any other local solution, possibly under a different filtration,
for t ≤ τU .
Lemma 2.19 Let U be an open neighborhood of e. If the sie (2.34) has a unique local
solution in U under FtB,N , then it has a unique solution under FtB,N for all time t > 0.
Proof Let zt1 be a local solution of sie (2.34) in U under FtB,N and let τ = τU (z·1 ). Define
gt = zt1 . Then gt is adapted to F̄tB,N and satisfies (2.34) for t ≤ τ . Let τ1 = τ . We will
successively modify the process gt for t ≥ τn , where τn are a sequence of stopping times ↑ ∞,
so that gt becomes a solution of the sie (2.34) under FtB,N .
As before, let Btτ = Bτ +t −Bτ and N τ ([0, t]×·) = N ([0, τ +t]×·)−N ([0, τ ]×·), and let zt2
τ
τ
be a local solution of the τ -time shifted sie (2.34) under FtB ,N , driven by (B τ , N τ ) instead
of by (B, N ). Let {Ft } = {F̄tB,N }, the completion of {FtB,N }, which is right continuous (see
τ
τ
Appenix A.3). Because FtB ,N ⊂ Fτ +t , zt2 is adapted to Gt = Fτ +t . Fix s ≥ 0. Because
for any t ≥ 0, [(s − τ ) ∨ 0 ≤ t] = [s ≤ τ + t] ∈ Gt , where a ∨ b = max(a, b), (s − τ ) ∨ 0
is a {Gt }-stopping time. By Lemma 2.18, τ ∨ s = τ + (s − τ ) ∨ 0 is a {Ft }-stopping time
2
and G(s−τ )∨0 = Fτ ∨s . It follows that z̃t = z(t−τ
)∨0 is Fτ ∨t -measurable. Because z̃t = e for
t < τ , and Fτ ∨t ∩ [τ ≤ t] ⊂ Ft , it follows that z̃t is in fact {Ft }-adapted. Let σ = τU (z·2 ).
Because for f ∈ Cc∞ (G), f (zt2 ) = f (e) + Ft f (z 2 ; B τ , N τ , ξ0 ) for t ≤ σ, after shifting time in
the stochastic integrals contained in Ft f (z 2 ; B τ , B τ , ξ0 ), we obtain, for τ ≤ t ≤ τ + σ,
f (z̃t ) = f (e) + Fτ,t f (z̃; B, N, ξ0 ).
This is equivalent to f (z̃t ) = f (e) + Fτ ∧t,t f (z̃; B, N, ξ0 ) for t ≤ τ + σ.
Let τ2 = τ + σ = τ1 + σ. Modify gt for t ≥ τ1 by setting gt = gτ ∧t z̃t . For τ ≤ t ≤ τ2 ,
f (gt ) = f (gτ ) + [f (gτ z̃t ) − f (gτ )] = f (e) + Fτ f (g; B, N, ξ0 ) + Fτ,t f (gτ z̃; B, N, ξ0 )
= f (e) + Fτ f (g; B, N, ξ0 ) + Fτ,t f (g; B, N, ξ0 ) = f (e) + Ft f (g; B, N, ξ0 ).
This shows that gt satisfies (2.34) for τ1 ≤ t ≤ τ2 . Because it satisfies (2.34) for t ≤ τ1 ,
we have obtained an {Ft }-adapted rcll process gt that satisfies the sie (2.34) for t ≤ τ2 .
62
Now let zt3 be a local solution of the sie (2.34), τ2 time shifted, and repeating the above
argument, we may modify gt for t ≥ τ2 so that it is {Ft }-adapted and satisfies the sie (2.34)
for t ≤ τ3 = τ2 + τU (z·3 ). In this way, gt may be successfully modified for t ≥ τn , where
τn+1 − τn are iid so τn ↑ ∞ almost surely, to obtain a solution of the sie (2.34) under Ft .
By the uniqueness of local solution, the solution gt of the sie (2.34) is unique up to time
τ = τ1 . For t > 0 and f ∈ Cc∞ (G),
f (gτ +t ) = f (gτ ) + Fτ,τ +t f (g; B, N, ξ0 ) = f (gτ ) + Ft (f ◦ lx )(gτ−1 gτ +· ; B τ , N τ , ξ0 ),
where x = gτ . Replacing f by f ◦ lx , noting that B τ and N τ are independent of F̄τB,N ,
yields f (gτ−1 gτ +t ) = f (e) + Ft f (gτ−1 gτ +· ; B τ , N τ , ξ0 ). By the uniqueness of the local solution,
gτ−1 gτ +t is unique up to time τU (gτ−1 gτ +· ). This shows that gt is unique up to time τ2 . In the
same way, the uniqueness of gt can be successively extended to any τn ↑ ∞. 2
Fix a relatively compact and open neighborhood V of e on∑which the exponential coordinate functions ϕi form a set of local coordinates, and x = e i ϕ(x)ξi for x ∈ V . Let U be
an open neighborhood of e such that U U = {uv; u, v ∈ U } is contained in V . The map
ϕ = (ϕ1 , . . . , ϕd ): G → Rd is diffeomorphic on V . Let W = ϕ(U ), an open neighborhood
of 0 = ϕ(e) in Rd . For i = 0, 1, . . . , d, let ξi′ (x) = Dϕ(ξi (ϕ−1 (x)), x ∈ Rd , be the vector
fields on W obtained by transferring the vector fields ξi on U , and let z: W × W → ϕ(V ),
z(x, y) = ϕ(ϕ−1 (x)ϕ−1 (y)), be the map that transfers the product structure on U to W .
Let yt = ϕ(gt ), N ′ = (id[0, ∞) × ϕ)N and η ′ = ϕη (a Lévy measure on Rd ). Assume η is
supported by U . Then the sie (2.34) or (2.14) for t ≤ τU (g· ) becomes, for f ∈ Cc∞ (Rd ),
∫
f (yt ) = f (0) +
∫
t
∫
t
∫
+
0
∫
W
+
0
W
d
t∑
0 i=1
ξi′ f (ys− ) ◦ dBsi +
∫
t
0
ξ0′ f (ys )ds
[f (z(ys− , x)) − f (ys− )]Ñ ′ (ds dx)
[f (z(ys , x)) − f (ys ) −
d
∑
xi ξi′ f (ys )]η ′ (dx)ds.
(2.41)
i=1
The vector fields ξi′ may be extended from W to Rd , and the map z from W × W to
Rd × Rd , so that they are still smooth and have compact supports. More precisely, we should
have introduced a slightly larger open set U ′ containing the closure of U with U ′ U ′ ⊂ V ,
and defined ξi on W ′ = ϕ(U ′ ) and z on W ′ × W ′ . Then the extensions can be made so that
ξi and z are unchanged on W and W × W respectively. It is clear that, assuming η(U c ) = 0,
if the sie (2.41) has a unique local solution yt in W under FtB,N , then the sie (2.34) has a
unique local solution in U under FtB,N .
∑
σ (x)∂j ,
There are σij ∈ Cc∞ (Rd ), for 0 ≤ i ≤ d and 1 ≤ j ≤ d, such that ξi′ (x) = dj=1
∑ ij
x ∈ W , where ∂j = ∂/∂ϕj . Because ϕi are exponential coordinate functions, ϕ(e i ui ξi ) = u
63
for u = (u1 , . . . , ud ) ∈ W , and for f ∈ Cc∞ (Rd ) and x ∈ W ,
∂
d
(f ◦ ϕ)(ϕ−1 (x)etξi ) |t=0 =
f (z(x, u)) |u=0
dt
∂ui
ξi′ f (x) = ξi (f ◦ ϕ)(ϕ−1 (x)) =
d
∑
=
∂j f (x)
j=1
∂
zj (x, u) |u=0 .
∂ui
It follows that for x ∈ W ,
σi· (x) =
∂
z(x, u) |u=0 .
∂ui
(2.42)
The function f in (2.41) may be vector valued. In particular, we may take f (x) = x for
x ∈ W . Then (2.41) becomes
∫
yt =
d
t∑
0 i=1
∫
t
σi· (ys− ) ◦ dBsi +
∫
+
0
W
∫
[z(ys , x) − ys −
∫
t
0
σ0· (ys )ds +
d
∑
0
t
∫
W
[z(ys− , x) − ys− ]Ñ ′ (ds dx)
xi σi· (ys )]η ′ (dx)ds,
(2.43)
i=1
where σi· = (σi1 , . . . , σid ). A rcll process yt in Rd will be called a local solution of the sie
(2.43) in W under some filtration {Ft } if both Bt and N are associated to {Ft }, and if yt is
{F̄t }-adapted and satisfies (2.43) for t ≤ τW (y· ). It is said to be a unique local solution if it
agrees with any other local solution, possibly under a different filtration, for t ≤ τW (y· ).
Lemma 2.20 Assume η(U c ) = 0. If the sie (2.43) has a unique local solution yt in W under
FtB,N , then the sie (2.34) has a unique local solution in U under FtB,N .
Proof Because local solutions of (2.34) in U correspond to local solutions of (2.41) in W , and
different local solutions of (2.41) lead to different local solutions of (2.43), hence it suffices
to show that if yt is a local solution of (2.43), then it satisfies (2.41) for any f ∈ Cc2 (Rd ) and
t ≤ τW (y· ).
Let yt be a local solution of sie (2.43) in W under FtB,N and let t ≤ τW (y· ). We will write
fi for (∂/∂ϕi )f and yi (t) for the i-th component of yt . Let h(s, x) = [zi (ys , x)) − yi (s)] and
∫ ∫
(h.Ñ ′ )t = 0t W h(s−, x)Ñ ′ (ds dx). By (A.19) and (A.20) in Appendix A.6, it can be shown
that [yi (·), (h.Ñ ′ )· ]ct = 0, and then by (A.8), [fi (y· ), (h.Ñ ′ )· ]ct = 0. It follows that
∫
0
t
fi (ys− ) ◦
d(h.Ñ ′ )s
∫
=
0
t
fi (ys− )d(h.Ñ ′ )s
∫
t
∫
=
0
W
fi (ys− )[zi (ys− , x) − yi (s−)]Ñ ′ (ds dx).
By the Itô formula (A.9), for any f ∈ Cc2 (Rd ),
f (yt ) = f (0) +
d ∫
∑
i=1 0
t
fi (ys− ) ◦ dyi (s) +
∑
[f (ys ) − f (ys− ) −
0<s≤t
64
d
∑
i=1
fi (ys− )∆s yi (·)].
(2.44)
By (2.43), N ′ is the counting measure of the jumps of yt with ∆s y = z(ys− , x) − ys− , hence
∑
∑d
0<s≤t [f (ys ) − f (ys− ) −
i=1 fi (ys− )∆s yi ] is equal to
∫
∫
t
0
W
{f (z(ys− , x)) − f (ys− ) −
d
∑
fi (ys− )[zi (ys− , x) − yi (s−)]}N ′ (ds dx).
i=1
Let Wε = {x ∈ W ; ∥x∥ > ε}. Then by (2.44) and (2.43),
f (yt ) = f (0) +
+
+
d ∫
∑
t
i=1 0
d ∫ t
∑
∫
W
∫
i=1 0
∫
W
t∫
+
0
W
d
∑
i=1
−
+
lim{
t
ξj′ f (ys− ) ◦ dBsj +
∫
ε→0
∫
ε→0
−
0
d ∫
∑
Wε
t
0
t
∫
t
Wε
j=1 0
t
∫
+
∫
0
t
∫
W
+
0
W
Wε
∫
i=1 0 Wε
d ∫ t
∑
∫
∫
0
= f (0) +
i=1 0
t
fi (ys )σ0i (ys )ds
d
∑
xj σji (ys )]dsη ′ (dx)
∫
d
∑
fi (ys− )[zi (ys− , x) − yi (s−)]}N ′ (ds dx)
i=1
t
0
ξ0′ f (ys )ds
fi (ys− )[zi (ys− , x) − yi (s−)]N ′ (ds dx)
fi (ys )[zi (ys , x) − yi (s)]dsη ′ (dx)}
Wε
lim{
+ lim{
t∫
∫
i=1 0
d
∑
i=1
∫
d ∫
∑
j=1
{f (z(ys− , x)) − f (ys− ) −
ε→0
d ∫
∑
i,j=1 0
fi (ys− )σji (ys− ) ◦ dBsj +
fi (ys )[zi (ys , x) − yi (s) −
j=1 0
+
t
fi (ys− )[zi (ys− , x) − yi (s−)]Ñ ′ (ds dx)
d ∫ t
∑
= f (0) +
d ∫
∑
fi (ys )[zi (ys , x) − yi (s)]dsη ′ (dx) −
∫
0
t
∫
Wε
fi (ys )
d
∑
xj σij (ys )dsη ′ (dx)}
j=1
[f (z(ys− , x)) − f (ys− )]N ′ (ds dx)
fi (ys− )[zi (ys− , x) − yi (s−)]N ′ (ds dx)}
ξj′ f (ys− )
◦
∫
dBsj
+
0
t
ξ0′ f (ys− )ds
[f (z(ys− , x)) − f (ys− )]Ñ ′ (ds dx)
[f (z(ys , x)) − f (ys ) −
d
∑
xj ξj′ f (ys )]dsη ′ (dx)
j=1
(after some cancellation and rearrangement). This is (2.41) and the lemma is proved. 2
Lemma 2.21 Assume η(U c ) = 0. Then the sie (2.43) has a unique local solution in W
under FtB,N .
65
Proof By (A.7) and (A.20), the sie (2.43) may be written as
∫
yt =
∫
d
t∑
σi· (ys− )dBsi
0 i=1
∫
t
σ0· (ys )ds +
+
0
t
∫
0
W
[z(ys− , x) − ys− ]Ñ ′ (ds dx)
∫ t∫
d
d
∑
1∫ t ∑
+
aij σjk (ys− )∂k σi· (ys− )ds +
[z(ys , x) − ys −
xi σi· (ys )]η ′ (dx)ds. (2.45)
2 0 i,j=1
0 W
i=1
When t < τW (y· ), yt ∈ W , then by (2.42), the integrand of the η ′ (dx)ds-integral in (2.45)
∑
is z(ys , x) − z(ys , 0) − di=1 xi (∂/∂xi )z(ys , x) |x=0 , which is the first order remainder of the
Taylor expansion of z(ys , x) at x = 0. We may modify σij outside W so that (2.42) holds
for all x ∈ Rd . Then the above statement for the Taylor expansion remainder holds for all
t > 0. Although (2.45) and (2.43) are no longer the same, but up to time τW (y· ), they are
identical, so that a solution of (2.45) is still a local solution of (2.43).
The existence of rcll process yt in Rd , adapted to FtB,N and satisfying (2.45) may be
proved by a routine argument of successive approximation, also called Picard iteration, as
in [3, §6.2]. Let yt0 = 0, and inductively after ytn has been obtained, let ytn+1 be given by the
right hand side of (2.45)with ys = ysn . Then
yin+1 (t)
−
∫
yin (t)
=
d
t∑
0 i=1
n
[σi· (ys−
)
−
−[ysn − ysn−1 ] − [
+ ··· +
n−1
σi· (ys−
)]dBsi
d
∑
xi σi· (ysn ) −
i=1
d
∑
∫
t
∫
0
W
{[z(ysn , x) − z(ysn−1 , x)]
xi σi· (ysn−1 )]}η ′ (dx)ds.
(2.46)
i=1
Note that the integrand of the last integral, the η ′ (dx)ds-integral, is the first order remainder
of the Taylor expansion of z(ysn , x) − z(ysn−1 , x), which is equal to
d
∂2
1 ∑
xi xj
[z(ysn , rx) − z(ysn−1 , rx)]
2 i,j=1
∂xi ∂xj
for some r ∈ (0, 1). Because η ′ is a Lévy measure, this may be used to estimate the integral
in terms of sup0≤s≤t ∥ysn − ysn−1 ∥, where ∥y∥ is the Euclidean norm of y ∈ Rd .
Now square both sides of (2.46). Using the inequality
(
q
∑
q
∑
ap )2 ≤ c
p=1
a2p
p=1
for any a1 , a2 , . . . , aq ≥ 0 and some c > 0, applying Doob’s norm inequality (A.4) for square
integrable martingales to the two stochastic integrals, the dBs - and Ñ (ds, dx)- integrals, and
applying Schwartz inequality to other integrals, we obtain, for some constant c > 0,
E[ sup
0≤s≤t
∥ysn+1
≤ cn (1 + t)n
∫
0
t
−
∫
0
ysn ∥2 ]
t1
···
≤ c(1 + t)
∫
tn−1
0
∫
t
0
E[ sup ∥ysn − ysn−1 ∥2 ]dt1
0≤s≤t1
E[ sup ∥ys1 − ys0 ∥2 ]dtn · · · dt1 ≤
0≤s≤tn
66
cn (1 + t)n tn
E[ sup ∥ys1 ∥2 ].
n!
0≤s≤t
Because all the integrands in (2.45) are bounded, it follows that E[sup0≤s≤t ∥ys1 ∥2 ] < ∞, and
from this it can be shown that almost surely ytn converges uniformly on finite t-intervals to
a solution yt of the sie (2.45). From its construction, yt is adapted to F̄tB,N , and hence is a
local solution of the sie (2.43) under FtB,N .
To prove the uniqueness of the local solution of the sie (2.43), let yt and yt′ be two local
solutions of (2.43) under possibly two different filtrations. Then they are also local solutions
of (2.45). Let τ = τW (y· ) ∧ τW (y·′ ). By a computation similar to the above,
′
E[ sup ∥ys∧τ − ys∧τ
∥2 ] ≤ c(1 + t)
0≤s≤t
∫
t
0
′
E[ sup ∥ys∧τ − ys∧τ
∥2 ]du.
0≤s≤u
′
Because both E[sup0≤s≤t ∥ys∧τ ∥2 ] and E[sup0≤s≤t ∥ys∧τ
∥2 ] are finite, using Gronwall’s in′
equality, it can be shown that yt∧τ = yt∧τ
. 2
Proof of Theorem 2.15 To prove that the sie (2.34) has a unique solution gt under FtB,N ,
by Lemma 2.17, we may assume η(U c ) = 0 for an arbitrarily chosen neighborhood U of e,
and then by Lemma 2.19, we just have to prove that (2.34) has a unique local solution in U
under FtB,N , which is guaranteed by Lemma 2.21. We have proved that the sie (2.34) has a
unique solution under FtB,N .
We have seen that gtr = gr−1 gr+t satisfies the r-time shifted sie (2.36), and hence gtr is
r
r
adapted to F̄tB ,N . It follows that gr−1 gr+t is independent of F̄r for any filtration {Ft } to
which both Bt and N are associated. This proves that gt is a Lévy process in G associated
to F̄t . 2
2.6
Proof of Theorem 2.9, part 2
As mentioned at the end of the proof of Theorem 2.15, the Lévy process gt in Theorem 2.15
is associated to the completion of any filtration to which both Bt and N are associated.
Because N is the jump counting measure of gt almost surely, by Lemma 2.22 below, up to
a modification on a null set, N is also associated to any filtration to which gt is associated.
By Lemma 2.23, this holds essentially for Bt too.
Lemma 2.22 Let gt be a Lévy process in G associated to a filtration {Ft }, and let N be its
jump counting measure. Then N is also associated to {Ft }.
Proof: For r < s, N ((r, s]×·) is determined by the jumps of gt in the time interval (r, s], so
it is independent of Fr . It remains to show that N ([0, t] × ·) is Ft -measurable for any t > 0.
It suffices to show that for any open B ⊂ G and integer n > 0, the event [N ([0, t] × B) ≥ n]
is Ft -measurable. Choose open Bk ↑ B with B k ⊂ B. The claim is true because the event
67
occurs if and only if there are an integer k > 0 and a rational δ > 0 such that for all rational
ε > 0, there are r1 < s1 < r2 < s2 < · · · < rn < sn in Q ∩ [0, t] satisfying
gr−1
gsi ∈ Bk and si − ri < ε for 1 ≤ i ≤ n,
i
and ri+1 − si > δ for 1 ≤ i ≤ n − 1. 2
Lemma 2.23 In Theorem 2.15, suppose gt is associated to a filtration {Ft′ }, then Bt is
associated to {F̄t′ }, the completion of {Ft′ }.
Proof: Because ξi ϕj (e) = δij , the smooth symmetric matrix function {ξi ϕj } is invertible on
a neighborhood U of e. Its inverse {bij } may be extended to be a smooth symmetric matrix
valued function on G. Let τ1 = τU (g· ) be the exit time of the process gt from U as defined
by (2.40), and recursively for n ≥ 1, let τn+1 be the exit time of gτ−1
gt , t ≥ τn , from U . It
n
is shown in §1.5 that τ1 is a stopping time under the natural filtration Ftg of gt . In fact, all
τn are stopping times under Ftg . For simplicity, we will only prove for n = 2. Let Un be a
sequence of open subsets of U such that the closure U n of Un is contained in Un+1 and Un ↑ U
as n ↑ ∞. For t > 0, [τ2 > t] = [τ1 ≥ t] ∪ [τ1 < t, τ2 > t]. Because τ1 is an {Ftg }-stopping
time, [τ1 ≥ t] ∈ Ftg . It is easy to see that the event [τ1 < t, τ2 > t] occurs if and only if
for some integer m > 0 and for any integer n ≥ 1, there is a rational r = r(n) ∈ (0, t)
such that r − 1/n < τ1 < r, gr−1 gt ∈ Um and gr−1 gs ∈ Um for all s ∈ Q ∩ [r, t]. This shows
[τ1 < t, τ2 > t] ∈ Ftg , and hence τ2 is an {Ftg }-stopping time.
Let g̃t = gt for t < τ1 , and recursively for n ≥ 1, let g̃t = g̃τ−1
gt for τn ≤ t < τn+1 . Then
n
g
∞
g̃t is {Ft }-adapted. For f ∈ Cc (G) and t ≥ 0, let
Mt f =
Let τ = τn . For τ < t ≤ τn+1 ,
∫t
τ
d ∫
∑
t
i=1 0
ξi f (g̃u− )dBui .
ξi f (g̃u− )dBui =
∫t
τ
ξi f (g̃τ−1 gu− )dBui , and hence by (2.14),
Mt f − Mτ f = f (g̃τ−1 gt ) − f (g̃τ−1 ) −
−
−
∫
∫
t
0
t
τ
∫
∫
G
G
[f (g̃τ−1 gs− h)
[f (g̃τ−1 gs− h)
∫
τ
t
ξ0 f (g̃τ−1 gs− )ds
−
f (g̃τ−1 gs− )]Ñ (ds dh)
−
f (g̃τ−1 gs− )
−
(2.47)
d
∑
ϕi (h)ξi f (g̃τ−1 gs− )]ds η(dh).
i=1
{F̄tg },
Because N is associated to
the completion of {Ftg }, [Mt f − Mτ ]1[τ <t≤τn+1 ] is F̄tg measurable. It is also clear that for k ≤ n,
(Mτk f − Mτk−1 f )1[τn <t≤τn+1 ] = (Mτk f − Mτk−1 f )1[τk <t] 1[τn <t≤τn+1 ] ,
is F̄tg -measurable. Because for τn < t ≤ τn+1 ,
Mt f =
n
∑
(Mτk f − Mτk−1 f ) + (Mt f − Mτn f ) on [τk < t ≤ τn+1 ],
k=1
68
it follows that Mt f is {F̄tg }-adapted.
Because {bij } is the inverse matrix of {ξi ϕj } on U , by (2.47),
Bti =
d ∫
∑
i=1 0
t
bji (g̃u− )dMu ϕj .
(2.48)
This shows that Bt is {F̄tg }-adapted.
For fixed s > 0, the above proof may be applied to Bts = Bs+t − Bs , N s = N ((s, s + t] × ·)
and gts = gs−1 gs+t , the s-time shifts of Bt , N and gt . It then follows that Bts is measurable
under the completed filtration generated by gts , and hence is independent of Fs′ for any
filtration {Ft′ } to which gt is associated. This shows that Bt is associated to {Ft′ }. 2
Proof of Theorem 2.9: We will prove Theorem 2.9 based on Theorem 2.1. We have seen
that the second half of Theorem 2.9 is implied by Theorem 2.15. To prove the first half of
Theorem 2.9, assume gt is a Lévy process in G with g0 = e, with characteristics (ξ0 , aij , η)
and associated to a filtration {Ft }. Let N be its jump counting measure. By Theorem 2.6, N
is a homogeneous Poisson random measure with characteristic measure η. By Lemma 2.22,
N is associated to {Ft }. Let Bt′ be a Brownian motion with covariance matrix aij . By
Theorem 2.15, there is a Lévy process gt′ in G with g0′ = e, and the sie (2.34) holds when
gt and Bt are replaced by gt′ and Bt′ . Then (2.16) holds for gt′ , and so the generator of gt′ ,
restricted to Cc∞ (G), has the same form as the generator of gt . By Theorem 2.1, the two
processes gt′ and gt are equal in distribution. By the proof of Lemma 2.23, (2.48) holds
when Bt and gt are replaced by Bt′ and gt′ . This allows us to define Bt by (2.48) from the
Lévy process gt , and then Bt is a Brownian motion with covariance matrix aij . Moreover,
the sie (2.34) holds. Now by Lemmas 2.16 and (2.48), the pair (B, N ) are almost surely
unique, and by Lemma 2.23, Bt associated to F̄t . This proves the first half of Theorem 2.9.
2
69
70
Chapter 3
Lévy processes in homogeneous spaces
Hunt’s generator formula for Lévy processes in a Lie group G holds also for Lévy processes
in a homogeneous space G/K, this is discussed in §3.1. Using this formula, it is shown in
§3.2 that a Lévy process in G/K may be obtained as a projection of a Lévy process in G.
As an application, some results on convolution semigroups on G and on G/K are derived
in §3.3, including embedding an infinitely divisible distribution in a continuous convolution
semigroup. A special class of Lévy processes in G or G/K are Riemannian Brownian motions.
Some related stochastic differential equations, and relations between Brownian motions in
G and in G/K, are considered in §3.4.
3.1
Generators on homogeneous spaces
The basic theory of Lévy processes in a homogenous space X = G/K under the general
setting of a topological group G is developed in §1.3. We now assume that G is a Lie group
with the identity element e and K is a compact subgroup, and let X = G/K. We will need
some preparation before developing a representation theory for Lévy processes in G/K.
Let g be the Lie algebra of G as before. The differential of the conjugation map cg at e
will be denoted by Ad(g): g → g, given by ξ 7→ Ad(g)ξ = Dcg (ξ) = Dlg ◦ Drg−1 (ξ). The
map
G × g ∋ (g, ξ) 7→ Ad(g)ξ ∈ g
is a linear action of G on g, called the adjoint action of the Lie group G on g and is denoted
by Ad(G). For a subgroup G′ of G, depending on the context, Ad(G′ ) will denote either
the adjoint action of G′ on its Lie algebra or the action Ad(G) restricted to G′ given by
G′ × g ∋ (g ′ , ξ) 7→ Ad(g ′ )ξ ∈ g.
We note that from the standard Lie group theory,
∀ξ ∈ g,
Ad(eξ ) = ead(ξ) ,
71
(3.1)
where ad(ξ): g → g is the linear map given by ad(ξ)ξ ′ = [ξ, ξ ′ ] (Lie bracket) and ead(ξ) =
∑∞
n
tξ ′
′
n=0 (1/n!)ad(ξ) is the exponential of ad(ξ). In particular, (d/dt)Ad(e )ξ |t=0 = [ξ, ξ ].
Let T be an operator on G with domain D(T ) = C ∞ (G). If it is K-right invariant, that
is, if T (f ◦ rk ) = (T f ) ◦ rk for f ∈ C ∞ (G) and k ∈ K, then it induces a linear operator T π
on X = G/K with domain C ∞ (X), defined by
∀f ∈ C ∞ (X) and x ∈ X,
T π f (x) = T (f ◦ π)(g),
(3.2)
where π: G → X is the natural projection and g ∈ G is chosen to satisfy x = π(g). By
the K-right invariance of T , T π f (x) does not depend on the choice of g. We note that
T π f (x) = T (f ◦ π)(S(x)) for any section map S. Because S may be chosen to be smooth
near any x ∈ X, it follows that if T (f ◦ π) is smooth, then so is T π f .
For a left invariant operator T on G with domain C ∞ (G), its K-right invariance is
equivalent to its K-conjugate invariance. In this case, the induced operator T π on X is
G-invariant, that is, T π (f ◦ g) = (T π ) ◦ g for f ∈ C ∞ (X) and g ∈ G. This can be proved
easily from the fact that π(gh) = gπ(h) for any g, h ∈ G.
For simplicity, T π may be denoted as T , so a left invariant and K-conjugate invariant
operator T on G may be regarded as a G-invariant operator on X. The meaning of T ,
whether as an operator on G or on X, should be clear from the context.
A second order linear differential operator T on a manifold X is understood as an operator
with domain C ∞ (X) and the following form in local coordinates x1 , . . . , xn on X,
T =
n
n
∑
∂2
1 ∑
∂
aij (x)
+
bi (x)
+ c(x),
2 i,j=1
∂xi ∂xj i=1
∂xi
for some smooth coefficients aij (x), bi (x), c(x) with aij (x) = aji (x). Note that T 1 = 0 if and
only if c(x) = 0. A linear differential operator of any order is defined in a similar fashion.
Let X = G/K. Because K is compact, there is a subspace p of g that is complementary
to the Lie algebra k of K and is Ad(K)-invariant in the sense that Ad(h)p ⊂ p for h ∈ K.
Indeed, one may take p to be the orthogonal complement of k under an Ad(K)-invariant inner
product on g. The existence of an Ad(K)-invariant inner product ⟨·, ·⟩ may be obtained by
∫
taking any inner product ⟨·, ·⟩′ on g and then setting ⟨ξ, η⟩ = K ⟨Ad(k)ξ, Ad(k)η⟩′ ρK (dk) for
ξ, η ∈ g, where ρK is the normalized Haar measure on K.
Let {ξ1 , . . . , ξn } be a basis of To X, the tangent space of X at the base point o = eK.
The differential Dπ of π: G → X, restricted to p, is a bijection: p → To X, so p and To X
may be identified under this bijection, and {ξ1 , . . . , ξn } may be regarded as a basis of p. For
k ∈ K, let [Ad(k)]ij be the matrix representing Ad(k): p → p under the basis {ξ1 , . . . , ξn }
∑
of p, that is, Ad(k)ξj = ni=1 [Ad(k)]ij ξi . An n × n symmetric real matrix {aij } is called
Ad(K)-invariant if
aij =
n
∑
apq [Ad(k)]ip [Ad(k)]jq .
p,q=1
72
∑
∑
Then ni,j=1 aij [Ad(k)ξi ][Ad(k)ξj ] = ni,j=1 aij ξi ξj for any k ∈ K.
For any n × n symmetric matrix {aij } and ξ0 ∈ p, let
T = (1/2)
n
∑
aij ξi ξi + ξ0 .
(3.3)
i,j=1
Then T is clearly a left invariant operator on G. Because for any ξ ∈ g, k ∈ K and
f ∈ C ∞ (G), ξ(f ◦ ck ) = [Ad(k)ξ]f , it is easy to see that T given in (3.3) is K-conjugate
invariant if and only if both {aij } and ξ0 are Ad(K)-invariant. In this case, T can be regarded
as a G-invariant operator on X.
We note that with choice of a section map S on X = G/K, any ξ ∈ g may be regarded
as an operator on X with domain C ∞ (X) by setting
ξf (x) = ξ(f ◦ π)(S(x)) =
d
f (S(x)etξ o) |t=0 .
dt
When ξ is Ad(K)-invariant, this definition does not depend on the choice of S, and so is
a smooth differential operator on X as S can be chosen to be smooth near any point x.
Similarly, for ξ, η ∈ g, ξη may be regarded as an operator on X by setting
∂2
ξηf (x) = ξη(f ◦ π)(S(x)) =
f (S(x)esξ etη o) |s=t=0 .
∂s∂t
∑
Although this definition is dependent on the choice of S, i,j aij ξi ξj is not when aij is
Ad(K)-invariant.
We will see that any G-invariant second order linear differential operator T on X with
T 1 = 0 is given by (3.3), but we first establish a formula for T in (3.3) as an operator on G.
∑
For f ∈ C ∞ (G) and g ∈ G, with ξ0 = ni=1 bi ξi , we have
T f (g) = [
n
n
∑
∑
∑
1 ∑
∂
∂2
aij
f (ge p tp ξp ) +
bi f (ge p tp ξp )]t1 =0,...,tn =0 .
2 i,j=1 ∂ti ∂tj
i=1 ∂ti
(3.4)
It then follows that when aij and ξ0 are Ad(K)-invariant, T regarded as an operator on X
can be expressed as follows: For f ∈ C ∞ (X) and x ∈ X,
T f (x) = [
n
n
∑
∑
∑
1 ∑
∂2
∂
f (ge p tp ξp o) +
aij
bi f (ge p tp ξp o)]t1 =0,...,tn =0 ,
2 i,j=1 ∂ti ∂tj
i=1 ∂ti
where g ∈ G is chosen to satisfy x = π(g) = go. To prove (3.4), first note that
ξ0 f (g) =
Because
∑n
i,j=1
n
∑
∑
∑
d
∂
f (get p bp ξp ) |t=0 =
bi f (ge p tp ξp ) |t1 =0,...,tn =0 .
dt
i=1 ∂ti
aij [ξi , ξj ] = 0, (3.4) now follows from Lemma 3.1 below.
73
(3.5)
Lemma 3.1 For ξ, η ∈ g, f ∈ C ∞ (G) and g ∈ G,
ξηf (g) = [
∂2
1
f (getξ+sη ) |s=0,t=0 ] + [ξ, η]f (g),
∂t∂s
2
where [ξ, η] is the Lie bracket.
Proof Let ηξ be the tangent vector to the curve t 7→ ξ + tη in g at t = 0. We will write
D expξ (η) for D exp(ηξ ). By Helgason [33, II.Theorem 1.7],
D expξ (η) = Dlexp(ξ) {[
where
idg − e−ad(ξ)
]η},
ad(ξ)
∞
∞
∑
idg − e−ad(ξ) ∑
(−1)j
1
1
=
ad(ξ)j = idg − ad(ξ) +
ad(ξ)j
ad(ξ)
(j
+
1)!
2
(j
+
1)!
j=0
j=2
(3.6)
(3.7)
is a linear endomorphism on g which is invertible when ξ is close to 0 with
[
idg − e−ad(ξ) −1
1
] = idg + ad(ξ) + O(∥ad(ξ)∥2 ).
ad(ξ)
2
(3.8)
It follows that
etξ esη = etξ+sη+st[ξ,η]/2+r1 (s,t) = etξ+sη est[ξ,η]/2+r2 (s,t) = etξ+sη est[ξ,η]/2 er3 (s,t)
with ri (s, t)/(st) → 0 as (s, t) → (0, 0) for i = 1, 2, 3. We have
∂2
∂2
f (getξ esη ) |s=0,t=0 =
f (getξ+sη est[ξ,η]/2 ) |s=0,t=0
∂t∂s
∂t∂s
∂
∂ ∂
[ f (getξ+sη ) |s=0 +
f (getξ est[ξ,η]/2 ) |s=0 ] |t=0
=
∂t ∂s
∂s
t
∂ ∂
[ f (getξ+sη ) |s=0 + [ξ, η]f (getξ )] |t=0
=
∂t ∂s
2
2
∂
1
=
f (getξ+sη ) |s=0,t=0 + [ξ, η]f (g). 2
∂t∂s
2
ξηf (g) =
Proposition 3.2 Let {aij } and ξ0 ∈ p be Ad(K)-invariant. Then T given by (3.3) is a left
invariant operator on G that is K-conjugate invariant, so can be regarded as a G-invariant
operator on X = G/K. Conversely, any second order G-invariant linear differential operator
T̃ on X with T̃ 1 = 0 is T in (3.3) for a unique pair of an Ad(K)-invariant symmetric matrix
{aij } and an Ad(K)-invariant vector ξ0 ∈ p.
Proof We just have to prove the second part. Note that t = (t1 , . . . , t∑
n ) in (3.5) may be
∞
used as local coordinates on X near o, so for any f ∈ C (X) and x = e i ti ξi o near o,
T̃ f (x) =
n
n
∑
∑
∑
1 ∑
∂2
∂
f (e p tp ξp o) +
aij (t)
bi (t) f (e p tp ξp o)
2 i,j=1
∂ti ∂tj
∂ti
i=1
74
for some smooth coefficients aij (t) and bi (t) with aij (t) = aji (t). Let aij = aij (0) and
bi = bi (0). Then T̃ f (o) has the same form as T f (o) given in (3.5). By the G-invariance of
T̃ , T̃ (f ◦ k)(o) = T̃ f (o) for any k ∈ K. From this one obtains easily the Ad(K)-invariance
∑
of aij and ξ0 = ni=1 bi ξi . Now by the G-invariance of T̃ and T , one obtains T̃ f = T f on X.
The aij and bi are clearly unique as they are determined by T̃ at o. 2
There are ϕ1 , . . . , ϕn ∈ Bb (X) such that ϕi vanish outside a compact subset of X and are
smooth near o, ϕi (o) = 0 and
x = exp(
n
∑
ϕi (x)ξi )o
(3.9)
i=1
for x contained in a neighborhood of o = eK. By the compactness of K, kx is close to o
uniformly for k ∈ K when x is close to o, it follows from (3.9) that
∀k ∈ K,
n
∑
ϕi (x)Ad(k)ξi =
i=1
n
∑
ϕi (kx)ξi
(3.10)
i=1
for x in a neighborhood U of o. Replacing ϕi by ψϕi for a K-invariant ψ ∈ Cc∞ (X) supported
by U with ψ = 1 near o, we may assume (3.10) holds for all x ∈ G. The functions ϕ1 , . . . , ϕn
thus modified form a set of local coordinates on X around o, satisfying (3.9) near o and (3.10)
on X, and will be called exponential coordinate functions on X = G/K associated to the
basis {ξ1 , . . . , ξn } of p. Note that if ϕ′1 , . . . , ϕ′n are a different set of exponential coordinate
functions associated to the same basis, then ϕ′i = ϕi on a neighborhood of o.
The following theorem is an extension of Hunt’s generator formula (Theorem 2.1) for
Lévy processes from Lie groups to homogeneous spaces, obtained also by Hunt [41]. The
result presented here is a slightly different version of Hunt’s original result.
Theorem 3.3 Let L be the generator of a G-invariant conservative Feller transition semigroup on X = G/K. Then its domain D(L) contains Cc∞ (X), and for any f ∈ Cc∞ (X) and
x ∈ X,
∫
n
n
∑
1 ∑
Lf (x) =
aij ξi ξj f (x) + ξ0 f (x) + [f (xy) − f (x) −
ϕi (y)ξi f (x)]η(dy),
2 i,j=1
X
i=1
(3.11)
where {aij } is an Ad(K)-invariant covariance matrix, ξ0 ∈ p is Ad(K)-invariant, and η is
an K-invariant measure on X satisfying
η({o}) = 0,
η(∥ϕ∥2 ) < ∞
∑
and
η(U c ) < ∞
(3.12)
for any neighborhood U of o, where ∥ϕ∥2 = ni=1 ϕ2i . The integral in (3.11) is not dependent
on the choice of a section map S to represent f (xy) = f (S(x)y) and ξi f (x) = ξi (f ◦π)(S(x)).
Moreover, the triple (ξ0 , aij , η) is completely determined by L.
Conversely, given such a triple (ξ0 , aij , η), there is a unique G-invariant conservative
Feller transition semigroup on X with generator L restricted to Cc∞ (X) given by (3.11).
75
Note that the condition (3.12) is not dependent on the choice of the basis {ξ1 , . . . , ξn }
of p, the associated coordinate functions ϕi and the neighborhood U of o. Any K-invariant
measure η on X satisfying (3.12) is called a Lévy measure on X. If xt is a Lévy process in
X and its generator L is represented by a triple (ξ0 , aij , η) as in Theorem 3.3, then η will be
called the Lévy measure of xt . Together, ξ0 , aij and η determine, and are also determined
by, the distribution of the Lévy process xt if x0 is given, and are called the characteristics
of xt . We note that although the covariance matrix aij may depend on the choice of basis
∑
{ξ1 , . . . , ξn } of p, the operator ni,j=1 aij ξi ξj does not. This can be proved in the same way
as the corresponding statement on G, see Proposition 2.4.
By Theorem 3.3, the generator L of a Lévy process in X is determined by its restriction
on Cc∞ (X). It will be shown in the next section that Cc∞ (X) is in fact a core of L (see §2.1).
A proof of Theorem 3.3 is contained in [55]. In the present work, it will be derived from
a general result for inhomogeneous Lévy processes in homogeneous spaces in Chapter 8. For
the moment, we will deal with some natural questions about the integral in (3.11) in the
following Lemma, which is an extension of Lemma 2.2 from G to X = G/K.
Lemma 3.4 For a bounded f ∈ C 2 (X), the integral in (3.11) is continuous in x. If f ∈
Cc2 (X), then it belongs to C0 (X). Moreover, if f ∈ Cc∞ (X), then it is smooth in x.
Proof Let f ∈ C 2 (X) be bounded, and let U be a K-invariant compact neighborhood of o
∫
∫
∫
on which (3.9) holds. Write the integral in (3.11) as X = ∑U + U c , as the sum of∑
the integrals
t i ϕi (y)ξi
t i ϕi (y)ξi
c
over U and U . By the Taylor expansion of ψ(t) = f (xe
o) = f (S(x)e
o) at
t = 0,
n
∑
∑
1∑
(3.13)
f (xy) − f (x) −
ϕi (y)ξi f (x) =
ϕj (y)ϕk (y)ξj ξk f (xes i ϕi ξi )
2 j,k
i=1
∫
for y ∈ U and for some s ∈ [0, 1]. By (3.12), U → 0 as U ↓ {o} uniformly for x contained in
a compact set. The three terms of the integrand of this integral may be integrated separately
∫
over U c . By the K-invariance of η and (3.10), U c is independent of the choice of S, and is
∫
continuous in x. Letting U ↓ {o} shows that X is independent of S and is continuous in
∫
x. If f ∈ Cc2 (X), then by (3.13), U → 0 as x → ∆ (the point at infinity on X). Because
∫
∫
η(U c ) < ∞, U c → 0 as x → ∆. It follows that X ∈ C0 (X).
∫
Now assume f ∈ Cc∞ (X). To prove the smoothness of X in x, let F (g, y) = f (gy) −
∫
∑
f (go) − i ϕi (y)ξi f (go) for g ∈ G and y ∈ X. Then the integrand of X , which depends on
the choice of a section map S, may be written as F (S(x), y). For ζ ∈ g, let
ζ ∗ f (x) =
d
f (etζ x) |t=0 .
dt
(3.14)
Note that for g ∈ G and x ∈ X, gS(x) = S(gx)k for some k = k(g, x) ∈ K, and x 7→ S(x)k
∫
is another section map. Because the integral X F (S(x), y)η(dy) has been shown to be
76
independent of S,
∫
∫
F (S(gx), y)η(dy) =
X
Then
F (gS(x), y)η(dy).
X
1∫
[F (etζ S(x), y)η(dy) − F (S(x), y)]η(dy).
t→0 t X
X
Now using Taylor’s expansion of ψ(t) = F (etζ S(x), y) at t = 0 as in the proof of Lemma 2.2,
∫
∫
it can be shown that ζ ∗ X F (S(x), y)η(dy) = X ζ r F (g, y)η(dy) |g=S(x) , which is just the
∫
integral in (3.11) when f is replaced by ζ ∗ f . This proves X ∈ C 1 (X). Repeating the same
∫
argument with f replaced by ζ ∗ f proves X ∈ C 2 (X), and continuing in this way proves the
∫
smoothness of X . 2
ζ∗
∫
F (S(x), y)η(dy) = lim
By the theorem below, a Lévy process xt is continuous if and only if its Lévy measure η
vanishes. We note that if S is a section map, then S(x)−1 y may be used as a measurement
for the difference between x and y in X, because under a G-invariant Riemannian metric on
X (see §3.4), the distance d(x, y) between x and y is equal to d(S(x)o, y) = d(o, S(x)−1 y).
Theorem 3.5 Let xt be a Lévy process in X with Lévy measure η. Then for any f ∈ Bb (X)
vanishing on a neighborhood of o and t ∈ R+ ,
tη(f ) = E[
∑
f (S(xs− )−1 xs )],
(3.15)
0<s≤t
where S is any section map on X and the summation
many nonzero terms almost surely.
∑
0<s≤t
f (S(xs− )−1 xs ) has only finitely
Proof Because f = 0 near o and the process xt has rcll paths, it is easy to see that almost
∑
surely there are only finitely many nonzero terms in 0<s≤t f (S(xs− )−1 xs ).
Take a partition 0 = t0 < t1 < · · · < tm = t of [0, t] with ti − ti−1 = t/m, and let
∑
−1
−1
Fm = m
i=1 f (S(xi−1 ) xti ). By Theorem 1.14, f (S(xi−1 ) xti ) are iid of distribution µt/m ,
where µt is the K-invariant convolution semigroup associated to the Lévy process xt . For
simplicity, we may assume f ∈ Cc∞ (X) and f ≥ 0. This will make the convergence, and the
interchange of summation and integration in the rest of the proof easy to justify. Then
E(Fm ) =
m
∑
E[f (S(xti−1 )−1 xti )] = mµt/m (f ) → tLf (o)
(as m → ∞),
i=1
and as in the proof of Theorem 2.6,
E(Fm2 ) =
m ∑
m
∑
E[f (S(xti−1 )−1 xti )f (S(xtj−1 )−1 xtj )]
i=1 j=1
=
∑
E[f (S(xti−1 )−1 xti )]E[f (S(xtj−1 )−1 xtj )] +
m
∑
i=1
2
i̸=j
E[f (S(xti−1 )−1 xti )2 ]
= (m − m)[µt/m (f )] + mµt/m (f ) → [tLf (o)] + tL(f 2 )(o)
2
2
2
77
(as m → ∞).
This shows that Fm have a bounded L2 -norm and hence are uniformly integrable.
Let ψj , j = 1, 2, . . ., be a partition of unity on G/K, that is, 0 ≤ ψj ∈ Cc (G/K) and
∑
j ψj = 1 is a locally finite sum. We may assume that for each j, there is a section map Sj
continuous on supp(ψj ). Then
Fm =
∑∑
j
ψj (xti−1 )f (Sj (xti−1 )−1 xti ) →
i
∑ ∑
ψj (xs− )f (Sj (xs− )−1 xs )
j 0<s≤t
almost surely as m → ∞. By the uniform integrbility of Fm ,
E(Fm ) →
∑
E[
j
This proves
tη(f ) = tLf (o) =
∑
ψj (xs− )f (Sj (xs− )−1 xs )].
0<s≤t
∑
j
E[
∑
ψj (xs− )f (Sj (xs− )−1 xs )].
0<s≤t
This holds for any section maps Sj continuous on supp(ψj ). Therefore, we may replace
Sj (·) by Sj (·)hj (·) in the above expression for any continuous maps hj : X → K. A simple
monotone class argument shows that this holds for any measurable maps hj : X → K, and
hence Sj may be replaced by any section map S. 2
On a homogeneous space X = G/K, the action of K fixes the point o. This induces
a linear action of K on the tangent space To X given by (k, ξ) 7→ Dk(ξ) for k ∈ K and
ξ ∈ To X. The space X is called irreducible if the K-action on To X is irreducible, that is, if
it has no nontrivial invariant subspace of To X. Because p is mapped bijectively onto To X
by the differential Dπ of the natural projection π: G → G/K, and Ad(k) on p corresponds
to Dk under this bijection, it follows that X is irreducible if and only if the adjoint action
Ad(K) on p is irreducible.
The following lemma is Theorem 1 in [50, Appendix 5].
Lemma 3.6 Any symmetric bilinear form on Rn is a multiple of the standard inner product
on Rn if it is invariant under a subgroup of O(n) that acts irreducibly on Rn .
Suppose X = G/K is irreducible, such as the (n − 1)-dim sphere S n−1 = O(n)/O(n − 1).
By Lemma 3.6, any Ad(K)-invariant symmetric matrix aij is a constant multiple of the
identity matrix, that is, aij = aδij for some some constant a. Because Ad(K) acts irreducibly
on p, if dim(p) = dim(X) > 1, there is no Ad(K)-invariant vector in p except 0. Moreover,
by (3.10), which holds for any x ∈ X, and the K-invariance of the Lévy measure η, for
∫ ∑
any K-invariant neighborhood U of o, U c ni=1 η(dy)ϕi (y)ξi is an Ad(K)-invariant vector
in p, and hence it must be zero. It follows that the η-integral in (3.11) may be written as
∫
G [f (xy) − f (x)]η(dy), which is understood as the principal value, that is, as the limit of
78
∫
U c [f (xy) − f (x)]η(dy)
as the K-invariant neighborhood U of o shrinks to o. To summarize,
we obtain a simple form of Hunt’s generator formula on an irreducible X = G/K as follows,
which does not directly involve the coordinate functions ϕi .
Theorem 3.7 Assume X = G/K is irreducible and dim(X) > 1. Then in Theorem 3.3,
the generator formula (3.11) may be written as
Lf (x) =
∫
n
a∑
ξi ξi f (x) + [f (xy) − f (x)]η(dy)
2 i=1
X
(3.16)
for some constant a ≥ 0, where the η-integral in (3.16) is understood as the principal value
described above. Moreover, given any pair (a, η) of a constant a ≥ 0 and a Lévy measure
η on X, there is a unique G-invariant conservative Feller transition semigroup on X with
generator L restricted to Cc∞ (X) given by (3.16).
Note: By Theorem 3.7, a Lévy process xt in an irreducible X = G/K, with dim(X) > 1
∑
and x0 = o, is completely determined in distribution by the operator D = a ni=1 ξi ξi and
the Lévy measure η, which does not depend on the choice of the basis {ξ1 , . . . , ξn } (see the
discussion after Theorem 3.3).
3.2
From Lie groups to homogeneous spaces
Recall a Lévy process gt in G is called K-conjugate invariant if its transition semigroup is
d
K-conjugate invariant. This is equivalent to kgte k −1 = gte (equal in distribution) for any
k ∈ K and t > 0, where gte = g0−1 gt , and is also equivalent to saying that the two processes
kgte k −1 and gt have the same distribution for k ∈ K. By Theorem 1.18, xt = π(gt ) = gt o is a
Lévy process in X = G/K. We will establish the converse of this result in this section, that
is, any Lévy process xt in X may be obtained in this way. As an application, some results on
the convolution semigroups on G and X are obtained. We will also consider the problem of
embedding an infinite divisible probability measure on a Lie group or a homogeneous space
into a continuous convolution semigroup.
As before, let {ξ1 , . . . , ξn } be a basis of p and ϕ1 , . . . , ϕn be the associated exponential
coordinate functions on X satisfying (3.10) on X as in §3.1. Now extend {ξ1 , . . . , ξn } to
become a basis {ξ1 , . . . , ξd } of g by adding a basis {ξn+1 , . . . , ξd } of k. Let ψ1 , ∑
. . . , ψd be
d
associated coordinate functions. We will assume ψi are exponential, that is, g = e i=1 ψi (g)ξi
for g near e. Recall cg is the conjugation map x 7→ gxg −1 on G. Then
∀k ∈ K,
d
∑
d
∑
i=1
i=1
(ψi ◦ ck )ξi =
79
ψi [Ad(k)ξi ]
(3.17)
holds near e. Replacing ψi by ψψi for a K-conjugate invariant ψ ∈ Cc∞ (G) such that ψ = 1
near e and ψ = 0 outside a small neighborhood of e, we may assume (3.17) holds on G.
For k ∈ K, recall [Ad(k)] has been used in §3.1 to denote the n × n matrix representing
Ad(k); p → p under a chosen basis of p. We will also let [Ad(k)] be the d × d matrix that
∑
represents Ad(k) : g → g under the basis {ξ1 , . . . , ξd } of g, that is, Ad(k)ξj = di=1 [Ad(k)]ij ξi ,
and the meaning of [Ad(k)] should be clear from the context. A d × d matrix bij will
be called Ad(K)-invariant if for any k ∈ K, {bij } = [Ad(k)]{bij }[Ad(k)]′ , that is, bij =
∑d
p,q=1 [Ad(k)]ip bpq [Ad(k)]jq .
Let gt be a Lévy process in G with generator L. Because for f ∈ Cc∞ (G), Lf =
limh→0 [Pt f − f ]/h, by the uniqueness in Hunt’s Theorem 2.1, gt is K-conjugate invariant if
and only if for any f ∈ Cc∞ (G) and k ∈ K, L(f ◦ ck ) ◦ c−1
k = Lf . By (3.17) on G, it is easy to
−1
show that L(f ◦ck )◦ck is equal to Lf in (2.3) when ξ0 , {aij } and η are replaced, respectively,
by Ad(k)ξ0 , [Ad(k)]{aij }[Ad(k)]′ and ck η. Therefore, gt is K-conjugate invariant if and only
if for any k ∈ K,
Ad(k)ξ0 = ξ0 ,
[Ad(k)]{aij }[Ad(k)]′ = {aij },
ck η = η.
(3.18)
Proposition 3.8 Assume the bases of p and g, and the associated coordinate functions are
chosen as above. Let gt be a Lévy process in G with characteristics (ξ0 , aij , η). Then gt is
K-conjugate invariant if and only if ξ0 and aij are Ad(K)-invariant, and η is K-conjugate
invariant, as expressed in (3.18). Moreover, xt = gt o is a Lévy process in X = G/K, its
transition semigroup Qt is related to the transition semigroup Pt of gt as (Qt f )◦π = Pt (f ◦π)
for f ∈ Bb (X), and its characteristics (ξ˜0 , ãij , η̃) satisfy
ãij = aij for i, j = 1, 2, . . . , n,
and
η̃ = πη.
Proof: The first half of this proposition is already proved in the preceding discussion. By
Theorem 1.18, xt = gt o is a Lévy process in X with (Qt f ) ◦ π = Pt (f ◦ π). It remains to
prove the expressions for ãij and η̃. Let L̃ be the generator of xt . By Theorem 3.3, it suffices
to show that there is an Ad(K)-invariant ξ˜0 ∈ p such that for f ∈ Cc∞ (X) and g ∈ G, with
x = π(g),
∫
n
n
∑
1 ∑
˜
L̃f (x) =
aij ξi ξj f (x) + ξ0 f (x) + [f (π(gh)) − f (x) −
ϕi (π(h))ξi f (x)]η(dh).
2 i,j=1
G
i=1
Let
∫
d
n
n
∑
∑
1 ∑
˜
ξ0 = (ξ0 )p +
aij [ξi , ξj ] + η(dh) [ϕi (π(h)) − ψi (h)]ξi ,
2 i=n+1 j=1
G
i=1
(3.19)
where (ξ0 )p is the p-component of ξ0 under the direct sum decomposition g = p ⊕ k. Note
∑
∑
that ni=1 dj=n+1 aij [ξi , ξj ] ∈ p is Ad(K)-invariant due to the Ad(K)-invariance of {aij }.
80
Recall (3.6) provides an expression for Dlexp(ξ) , the differential of a left translation. By its
version for Drexp(ξ) , the differential of a right translation, noting ad(ξ)ζ ∈ p for ξ ∈ k and
∑
∑
∑
ζ ∈ p, we obtain exp[ di=1 ψi (g)ξi ] = exp{ ni=1 [ψi (g) + O(∥ψ∥2 )]ξi } exp[ di=n+1 ψi (g)ξi ] for g
∑
∑
near e, It follows that exp[ ni=1 ϕi (π(g))ξi ]o = π(g) = exp{ ni=1 [ψi (g) + O(∥ψ∥2 )]ξi }o, and
hence ϕi (π(g))−ψi (g) = O(∥ψ∥2 ). Therefore, the η-integral in (3.19) is absolutely integrable.
Moreover, by (3.17) on G and (3.10) on X, and K-conjugate invariance of η, this integral
is an Ad(K)-invariant element of p. This shows that ξ˜0 in (3.19) is well defined and is an
Ad(K)-invariant element of p.
Now the expression of L̃f given above follows easily from the expression for the generator
L of gt , given in (2.3), with f replaced by f ◦ π. 2
Remark 3.9 (3.19) provides an explicit expression for ξ˜0 is Proposition 3.8.
The following theorem provides a converse to the second part of Proposition 3.8.
Theorem 3.10 If xt is a Lévy process in X = G/K with x0 = o, then there is a K-conjugate
invariant Lévy process gt in G with g0 = e such that the processes gt o and xt are equal in
distribution.
Proof: Let {ξ1 , . . . , ξn } and {ξ1 , . . . , ξd } be bases of p and g with associated coordinate
functions ϕ1 , . . . , ϕn and ψ1 , . . . , ψd introduced before Proposition 3.8 so that (3.10) holds on
X and (3.17) holds on G.
∑n
Choose a section map S on X such that S(x) = e i=1 ϕi (x)ξi for x contained in a Kinvariant neighborhood U of o. By (3.10),
kS(x)k −1 = S(kx).
∀k ∈ K and x ∈ U,
Because for x near o, e
∑n
i=1
ϕi (x)ξi
∑d
= S(x) = e
ψi ◦ S = ϕi , 1 ≤ i ≤ n,
and
i=1
ψi (S(x))ξi
(3.20)
, it follows that
ψi ◦ S = 0, n + 1 ≤ i ≤ d
(3.21)
hold near o. The ψi may be modified outside a neighborhood of e so that (3.21) holds on X
and (3.17) still holds on G. This is accomplished by replacing ψi by (ϕ◦π)ψi +(1−ϕ◦π)(ϕi ◦π)
for i ≤ n and by (ϕ ◦ π)ψi for i > n, where ϕ is a K-invariant smooth function on X such
that ϕ = 1 near o and ϕ = 0 outside a neighborhood of o on which (3.21) originally holds.
The generator L of xt , restricted to Cc∞ (X), is given by (3.11). Its differential operator
∑
part is (1/2) ni,j=1 aij ξi ξj + ξ0 , which may also be regarded as a differential operator on G.
Define a measure η̄ on G by
∀f ∈ Bb (G),
∫
∫
η̄(f ) =
X
K
81
f (kS(x)k −1 )ρK (dk)η(dx),
(3.22)
where ρK is the normalized Haar measure on K. Then η̄ is K-conjugate invariant, and
because of the K-invariance of η, π η̃ = η.
Let V = π −1 (U ). Then V is a K-conjugate invariant neighborhood of e, and η̄(V c ) =
∑
∑
η(U c ) < ∞. By (3.20) and (3.21), η̄( di=1 ψi2 1V ) = η( ni=1 ϕ2i 1U ) < ∞, we see that η̄ is a
Lévy measure on G.
The Ad(K)-invariant n × n covariance matrix aij may be regarded as an Ad(K)-invariant
d × d matrix by setting aij = 0 for i > n or j > n. Let gt be a Lévy process in G with
characteristics (ξ0 , aij , η̄). By Proposition 3.8, gt is K-conjugate invariant. Its generator L̄,
restricted to Cc∞ (G), is given by
∫
n
d
∑
1 ∑
aij ξi ξj f (g) + ξ0 f (g) + [f (gh) − f (g) −
L̄f (g) =
ψi (h)ξi f (g)]η̄(dh).
2 i,j=1
G
i=1
Let x′t = gt o. Then x′t is a Lévy process in X with x′0 = o, We will show that the process
x′t has the same distribution as the process xt . The generator L′ of x′t , restricted to Cc∞ (X),
is given at o by
L′ f (o) =
=
d
d
E[f (x′t )] |t=0 = E[(f ◦ π)(gt )] |t=0 = L̄(f ◦ π)(e)
dt
dt
∫
n
d
∑
1 ∑
aij ξi ξj f (o) + ξ0 f (o) + [(f ◦ π)(h) − f (o) −
ψi (h)ξi (f ◦ π)(e))]η̄(dh).
2 i,j=1
G
i=1
By (3.22),
∫
∫
G [· · ·]η̄(dh)
∫
X
∫
−1
K
∫
=
X
∫
K
∫
=
X
∫
=
X
above is
K
[(f ◦ π)(kS(x)k ) − f (o) −
d
∑
ψi (kS(x)k −1 )ξi (f ◦ π)(e)]ρK (dk)η(dx)
i=1
{f (kx) − f (o) −
{f (kx) − f (o) −
d
∑
i=1
n
∑
ψi (S(x))[Ad(k)ξi ](f ◦ π)(e)}ρK (dk)η(dx) (by (3.17))
ϕi (x)[Ad(k)ξi ]f (o)}ρ(dk)η(dx) (by (3.21))
i=1
[f (x) − f (o) −
n
∑
ϕi (x)ξi f (o)]η(dx) (by (3.10) and K-invariance of η).
i=1
This shows that
L′ f (o) =
∫
n
n
∑
1 ∑
aij ξi ξj f (o) + ξ0 f (o) + [(f (x) − f (o) −
ϕi (x)ξi f (o)]η(dx),
2 i,j=1
G
i=1
which is the same as Lf (o), the generator L of xt at o. By the G-invariance, Lf = L′ f
on G for any f ∈ Cc∞ (X). This proves that x′t = gt o and xt have the same distribution as
processes. 2
82
Remark 3.11 We note that the process gt in Theorem 3.10 is not unique in the sense of
distribution. In fact, as we will see in Theorem 3.25, for dim(K) ≥ 1, there is a family of Kconjugate invariant Lévy processes gt in G, all starting at e but with different distributions,
such that the processes xt = π(gt ) in X have the same distribution.
Remark 3.12 Let Cb2 (X) be the space of functions f on X with f ◦ π ∈ Cb2 (G) (defined in
§2.3). Let xt be a Lévy process in X = G/K with generator L and transition semigroup Pt .
By Theorem 3.10 and (2.18) in Theorem 2.11, it is easy to show that the domain D(L) of L
contains Cb2 (X) ∩ C0 (X), and (3.11) holds for f ∈ Cb2 (X) ∩ C0 (X). Moreover, the expression
Lf given by (3.11) is valid also for f ∈ Cb2 (X) and
∀f ∈ Cb2 (X) and x ∈ X,
d
Pt f (x) |t=0 = Lf (x).
dt
(3.23)
The following result is an extension of Proposition 2.3 from G to X = G/K.
Proposition 3.13 Cc∞ (X) is a core of the generator L in Theorem 3.3.
Proof: Let xt and gt be as in Theorem 3.10 with transition semigroups Pt and P̄t , and
generators L and L̄, respectively. Then for f ∈ Bb (X), (Pt f ) ◦ π = Pt (f ◦ π). It follows
that f ∈ C0 (X) belongs to D(L) if and only if f ◦ π ∈ D(L̄), and then (Lf ) ◦ π = L̄(f ◦ π).
We have to show that for any f ∈ D(L), there are fn ∈ Cc∞ (X) such that fn → f and
Lfn → Lf uniformly on X. Because f ◦ π belongs to D(L̄), and by Proposition 2.3, Cc∞ (G)
is a core of L̄, there are f¯n ∈ Cc∞ (G) such that f¯n → f ◦ π and L̄f¯n → L̄(f ◦ π) uniformly
∫
on G. Let T : C0 (G) → C0 (G) be the map defined by T f¯(g) = K f¯(gk)dk. Then T f¯ is
K-right invariant, and T f¯ ∈ Cc∞ (G) if f¯ ∈ Cc∞ (G). Because P̄t is both left invariant and
K-conjugate invariant, so it is also K-right invariant. This implies the K-right invariance
of L̄, and hence for f¯ ∈ Cc∞ (G), T L̄f¯ = L̄T f¯. The K-right invariance of T f¯n means
that T f¯n = fn ◦ π for some fn ∈ Cc∞ (X). The uniform convergence of f¯n → (f ◦ π)
and L̄f¯n → L̄(f ◦ π) implies the uniform convergence of (fn ◦ π) = T f¯n → (f ◦ π) and
(Lfn ) ◦ π = L̄(fn ◦ π) = L̄T f¯n = T L̄f¯n → L̄(f ◦ π) = (Lf ) ◦ π. This implies that fn → f
and Lfn → Lf uniformly on X. 2
3.3
Convolution semigroups
The convolutions of measures on G and on X = G/K are defined in §1.2 and §1.3 respectively.
Recall that a convolution semigroup always refers to a family µt of probability measures with
t ∈ R+ , and a convolution semigroup µt on X is necessarily K-invariant. Moreover, µt is
called continuous if µt → µ0 weakly as t → 0, and then µt → µs weakly as t → s for any
83
s > 0. By Proposition 1.12, µt 7→ πµt is a bijection from the set of (continuous) K-biinvariant convolution semigroups on G onto the set of (continuous) convolution semigroup
on X.
Recall that the distribution of a Lévy process xt in G or X = G/K, starting at e or o =
eK, is a continuous convolution semigroup µt , and conversely, given a continuous convolution
semigroup µt on G or X, with µ0 = δe or δo , there is a Lévy process xt , unique in distribution,
with µt as distribution. Moreover, the transition semigroup Pt of xt is related to µt as
∫
Pt f (x) = f (xy)µt (dy), where the product xy on X is understood as S(x)y but the integral
does not depend on the choice of the section map S. The generator L and the characteristics
(ξ0 , aij , η) of a Lévy process will also be called the generator and the characteristics of the
associated convolution semigroup.
Therefore, the study of Lévy processes in G or X, in distribution, is equivalent to the
study of continuous convolution semigroups, and many of our results for Lévy processes, such
as Theorems 2.1, 3.3 and 3.10, may be stated in terms of continuous convolution semigroups.
For example, Theorem 3.10 may be stated as follows.
Theorem 3.14 Let µt be a continuous K-conjugate invariant convolution semigroup on G,
then νt = πµt is a continuous convolution semigroup on X = G/K. Conversely, if νt is a
continuous convolution semigroup on X, then there is a continuous K-conjugate invariant
convolution semigroup µt on G with such that πµt = νt . Moreover, if ν0 = δo , then µt may
be chosen to satisfy µ0 = δe .
Recall from §1.3 that for any convolution semigroup µt on G, µ0 = ρK for some compact
subgroup of G. Let µet be a continuous K-conjugate invariant convolution semigroup on G
with µe0 = δe . Then it is easy to show that µt = ρK ∗ µet is a continuous K-bi-invariant
convolution semigroup on G. The following result says that any continuous convolution
semigroup µt on G is obtained this way for some compact subgroup K of G.
Theorem 3.15 Let µt be a continuous convolution semigroup on G. Then for some compact
subgroup K of G, there is a K-conjugate invariant continuous convolution semigroup µet on
G with µe0 = δe such that µt = ρK ∗ µet = µet ∗ ρK . Consequently, µt is K-bi-invariant.
Proof: Because µ0 = ρK for some compact subgroup K, by the K-bi-invariance of ρK and
µt = µ0 ∗ µt = µt ∗ µ0 , µt is K-bi-invariant. By Proposition 1.12, νt = πµt is a continuous
convolution semigroup on X = G/K with ν0 = πµ0 = δo . By Theorem 3.14, there is a
continuous K-conjugate invariant convolution semigroup µet on G with µe0 = δe such that
νt = πµet . Let µ′t = ρK ∗ µet = µet ∗ ρK . Then µ′t is K-bi-invariant and π(µ′t ) = νt . By the
uniqueness in Proposition 1.9, µ′t = µt . 2
84
Remark 3.16 We note that µet in Theorem 3.15 is not unique. This is because µet given in
Theorem 3.14 is not unique, see Remark 3.11. The non-uniqueness of µet reflects the following
fact: For a continuous convolution semigroup µt on G with µ0 = ρK and dim(K) ≥ 1, there
are different Lévy processes gt in G, different in the sense that they have different transition
semigroups Pt , such that if they all start with the same initial distribution ρK , then their
1-dimensional distributions are the same, and are given by µt .
Theorem 3.15 says that any continuous convolution semigroups on G is the convolution
product of the normalized Haar measure of a compact subgroup and a continuous convolution
semigroup with initial measure δe . A similar result holds also on a homogeneous space
X = G/K as stated in the next Theorem. For two compact subgroups K ⊂ H of G, the
homogeneous space H/K is a closed topological subspace of X = G/K (see Proposition 4.4
in [33, Chapter II]), and by Proposition 1.10 (b), πρH is the unique H-invariant probability
measure on G/K supported by H/K, where π is the natural projection: G → G/K as
before.
Theorem 3.17 Let νt be a continuous convolution semigroup on X = G/K. Then ν0 = πρH
for a compact subgroup H of G containing K, and νt = ν0 ∗ νto = νto ∗ ν0 for a continuous
convolution semigroup νto on X with ν0o = δo . Consequently, νt is H-invariant.
Proof By Proposition 1.12, there is a unique K-bi-invariant continuous convolution semigroup µt on G such that νt = πµt . By Theorem 3.15, µ0 = ρH for some compact subgroup
H of G, and µt = ρH ∗ µet = µet ∗ ρH is H-bi-invariant, where µet is a continuous H-conjugate
invariant convolution semigroup on G with µe0 = δe . The K-bi-invariance of ρH = µ0 implies
K ⊂ H. Let νto = πµet . Then νto is a continuous convolution semigroup on X with ν0o = δo ,
and νt = πµt = ν0 ∗ νto = νto ∗ ν0 . 2
A probability measure is also called a distribution. A distribution µ on a Lie group
G or a homogeneous space X = G/K, where K is compact, is called infinitely divisible
if for any n ∈ N (the set of natural numbers), there is a distribution ν such that µ =
ν ∗n (n-fold convolution). If µt is a convolution semigroup on G or X, then µ = µ1 is
clearly infinitely divisible. It is an interesting and quite challenging problem to determine
whether any infinitely divisible distribution µ can be embedded in a continuous convolution
semigroup, that is, whether there is a continuous convolution semigroup µt with µ = µ1 .
The embedding problem on local compact groups has been studied extensively, see Heyer
[35, chapter III] for a summary of results up to 1976. More recently, the problem was solved
by Dani-McCrudden [15, 16] for a large class of Lie groups. They say that a Lie group G
belongs to class C if G is connected and there is a linear action of G on a finite dimensional
real vector space V with a discrete kernel, that is, the set of g ∈ G that acts as the identity
85
map on V is a discrete subset of G. Note that connected compact Lie groups are of class C,
because a compact Lie group G has a faithful representation on a finite dim complex vector
space V (see [12, III.Theorem 4.1]), and such a representation is a linear action of G on the
real structure of V with a trivial kernel. Another important example of class C Lie groups
are connected semisimple Lie groups (see the related discussion in §4.4 and §5.3).
By [16], any infinitely divisible distribution on a Lie group G of class C can be embedded
in a continuous convolution semigroup on G. The following result provides its extension to
homogeneous spaces.
Theorem 3.18 Let G be a Lie group of class C and let K be a compact subgroup. Then any
infinitely divisible distribution α on X = G/K can be embedded in a continuous convolution
semigroup αt on X, that is, α = α1 .
Proof: First let µ be an infinitely divisible distribution on G. For any n ∈ N, a distribution
ν on G is called an nth root of µ if ν ∗n = µ. A root of µ is an nth root of µ for some n ∈ N.
Now choose an nth root of µ, denoted as r(n), for each n ∈ N. Then n 7→ r(n) is a map
from N to the set of roots of µ, called a root map of µ. Let Z 0 (µ, [G, G]) be the subgroup of
G defined in [16], whose precise definition is not important here, we just need to know that
it is a set of points z ∈ G that centralize points in supp(µ), the support of µ (defined to be
the smallest closed set F with µ(F c ) = 0), that is, zxz −1 = x for any x ∈ supp(µ). Let Γ be
the closure of convolution products of the measures of the following form, under the weak
convergence topology,
zr(m)z −1 = (lz ◦ rz−1 )r(m);
z ∈ Z 0 (µ, [G, G]) and m ∈ N.
It is stated in [16, Theorem 1.3] that, for some n ∈ N, there is an nth root ν of µ such that
ν is root compact and is embeddable in a convolution semigroup νt on G with νt ∈ Γ. In
fact, the proof of this theorem shows that ν can be embedded in a convolution semigroup
parameterized by rational times, and then the reader is referred to [35] for additional details
to complete the proof.
Because this feature of the embedding in [16] is important for our purpose, for the convenience of a reader who is not an expert in the embedding problem, we will provide some
additional details, according to the directions in the last paragraph in [16], to show that there
is a continuous convolution semigroup νt with ν1 = ν such that each νt is contained in Γ̃,
the closure of the set of measures of the form xγ with x ∈ G and γ ∈ Γ. It then follows that
µ, which has ν as an nth root, can be embedded in the continuous convolution semigroup
µt = νt∗n . We will do this later, but for now, will proceed to the proof of Theorem 3.18.
Let α be an infinitely divisible distribution on X = G/K. For any n ∈ N, let α′ be an nth
root of α. Let µ and µ′ be respectively the unique K-right invariant distributions on G with
86
α = πµ and α′ = πµ′ (see Proposition 1.9), and let a root map r of µ be defined by r(n) = µ′ .
By the preceding discussion, there is a continuous convolution semigroup µt on G with µ = µ1
and µt = νt∗n . Because µ is K-right invariant, for any x ∈ supp(µ) and k ∈ K, xk ∈ supp(µ).
Because z ∈ Z 0 (µ, [G, G]) centralizes points in supp(µ), xzkz −1 = z(xk)z −1 = xk, and so
zkz −1 = k. Thus, z also centralizes k. Because r(m) is K-right invariant, it is now easy to
derive the K-right invariance of zr(m)∗j z −1 from k ∈ K being centralized by z. It follows
that any measure in Γ is K-right invariant, so is any measure in Γ̃. Because νt ∈ Γ̃, µt = νt∗n
is K-right invariant. Let αt = πµt . Because the map µ 7→ πµ preserves the convolution on G
and on X = G/K (Proposition 1.9), αt is a convolution semigroup on X, and is continuous
because µt is continuous. It is clear α = α1 , and this proves Theorem 3.18.
We now provide some details for embedding ν in a continuous convolution semigroup νt
on G with νt ∈ Γ̃ as mentioned above. A family of distributions γt on G, t ∈ Q∗+ = Q∩(0, ∞),
such that γt ∗ γs = γt+s for t, s ∈ Q∗+ , is called a convolution semigroup on G parameterized
by Q∗+ . A convolution semigroup parameterized by Q+ = Q ∩ [0, ∞) is defined similarly. In
this latter case, γ0 ∗ γ0 = γ0 , and hence, as mentioned in the proof of Theorem 3.15, γ0 = ρH
for some compact subgroup H of G. Such a convolution semigroup γt is called continuous if
it is continuous in t under the weak convergence. For γt parameterized by Q+ , the continuity
is equivalent to its continuity at t = 0 ([35, Theorem 1.5.7]). A distribution γ on G is called
Q+ - (resp. Q∗+ -) embeddable if there is a convolution semigroup γt parameterized by Q+
(resp. by Q∗+ ) with γ = γ1 . The γ is called root compact if the set of distributions ξ ∗k , where
ξ is an nth root of γ for some n ∈ N and 1 ≤ k ≤ n, is relatively compact under the weak
convergence. This definition agrees with [35], but it is called strongly root compact in [16].
It is shown at the end of the proof of [16, Theorem 1.3] that the ν above is root compact,
and there is a convolution semigroup γt on G parameterized by Q∗+ with γt ∈ Γ and γ1 = ν.
Let A be the set of weak limits of γt as t → 0, t ∈ Q∗+ . We now show
∀β ∈ A,
γt ∗ β = β ∗ γ t
for t ∈ Q∗+ .
(3.24)
Let tn ∈ Q∗+ with tn → 0 such that γtn → β weakly. Because γ1 is root compact, for any
fixed and finite T > 0, the set {γt ; t ∈ Q∗+ ∩(0, T ]} is relatively weakly compact. For t ∈ Q∗+ ,
γt = γtn ∗ γt−tn = γt−tn ∗ γtn . Along a subsequence, γt−tn converges to some distribution λ
on G, so γt = β ∗ λ = λ ∗ β. Then β ∗ γt = β ∗ β ∗ λ = β ∗ λ ∗ β = γt ∗ β. This proves (3.24).
The proof also shows that any β ∈ A is a left and right convolution factor of γt .
For any closed subgroup H of G, let N (H) = {x ∈ G; xHx−1 = H}. This is a closed
subgroup of G, called the normalizer of H, which contains H as a normal subgroup. By
[35, Theorem 3.5.4] and its proof, A is a compact and connected abelian group under the
convolution product and the weak convergence topology, with identity ρH for some compact
subgroup H of G, and A = {xρH ; x ∈ H ′ }, where H ′ is a compact subgroup with H ⊂ H ′ ⊂
87
N (H). Moreover, H ′ /H and A are isomorphic as topological groups under the map: xH 7→
xρH from H ′ /H to A, and there is a continuous convolution semigroup νt′ parameterized by
Q+ with ν0′ = ρH such that for t ∈ Q∗+ , νt′ = xt γt = δxt ∗ γt for some xt ∈ H0′ , where H0′ is
the identity component of H ′ .
Note that xρH = ρH x for x ∈ H ′ . To show this, just observe that because x normalizes
H, xρH x−1 is a probability measure supported by H and is H-left invariant, so must be
equal to ρH . For x ∈ H ′ , because xρH ∈ A, by (3.24), (xρH ) ∗ γt = γt ∗ (xρH ). As shown in
the proof of (3.24), any β ∈ A is a left and right convolution factor of γt , this implies, with
β = ρH , that γt = ρH ∗ γt = γt ∗ ρH . Then for x ∈ H ′ ,
xγt = xρH ∗ γt = γt ∗ (xρH ) = γt ∗ (ρH x) = γt x.
Because A is abelian, we obtain (xγt ) ∗ (yγs ) = xyγs+t = yxγs+t for x, y ∈ H ′ and s, t ∈ Q∗+ .
It is well know that the exponential map is surjective on a compact connected Lie group,
so there is a vector ξ in the Lie algebra of H0′ such that x1 = eξ . Let νt = e−tξ νt′ = e−tξ xt γt .
′
Then νt ∗ νs = (e−tξ xt γt ) ∗ (e−sξ νs′ ) = (e−tξ xt e−sξ γt ) ∗ νs′ = (e−tξ e−sξ νt′ ) ∗ νs′ ) = e−(t+s)ξ νt+s
=
νt+s . This shows that νt is a convolution semigroup parameterized by Q+ , which is obviously
continuous. Moreover, ν1 = e−ξ x1 γ1 = γ1 = ν. By [35, Theorem 1.5.9], νt may be extended
to become a continuous convolution semigroup defined for all t ∈ R+ . Because for t ∈ Q∗+ ,
γt ∈ Γ and νt = e−tξ xt γt ∈ Γ̃, it follows from the continuity that νt ∈ Γ̃ for any t ∈ R+ . 2
3.4
Riemannian Brownian motion
Let X be a d-dimensional Riemannian manifold equipped with the Riemannian metric {⟨·, ·⟩x ;
x ∈ X}, a smooth distribution of inner products on the tangent spaces Tx X at x ∈ X, and
let Expx : Tx X → X be the Riemannian exponential map at x, defined by Expx (v) = γ(1)
for v ∈ Tx M , where γ: [0, 1] → X is the geodesic with γ(0) = x and γ ′ (0) = v. The
Laplace-Beltrami operator ∆ on X can be defined by
∆f (x) =
d
∑
d2
i=1
dt2
f (Expx (tYi )) |t=0
(3.25)
for f ∈ Cc∞ (X) and x ∈ X, where {Y1 , . . . , Yd } is a complete set of orthonormal tangent
vectors at x. This definition is independent of the choice of Yi ’s. It is well known that in
local coordinates x1 , . . . , xd , writing ∂i for ∂/∂xi , we have
∆f (x) =
d
∑
g jk (x)∂j ∂k f (x) −
j,k=1
d
∑
i,j,k=1
88
g jk (x)Γijk (x)∂i f (x)
(3.26)
for f ∈ Cc∞ (X), where {g jk } = {gjk }−1 , gjk (x) = ⟨∂j , ∂k ⟩x , and
Γijk
= (1/2)
d
∑
g ip (∂j gpk + ∂k gpj − ∂p gjk ).
p=1
The matrix {gij } is called the Riemannian metric tensor under the local coordinates.
A diffusion process xt in X with generator (1/2)∆ (see Appendix A.5) is called a Riemannian Brownian motion. When X = Rd is equipped with the standard Euclidean metric, the
Laplace-Beltrami operator is just the usual Laplace operator and a Riemannian Brownian
motion is a standard d-dimensional Brownian motion.
From its definition, it is easy to see that the Laplace-Beltrami operator with domain
∞
Cc (X) is invariant under any isometric transformation ϕ on X. Thus, if xt is a Riemannian
Brownian motion in X, then so is ϕ(xt ) for any isometric transformation ϕ on X.
The Riemannian Brownian motion has the following scaling property. If c > 0 is a
constant, ∆ is the Laplace-Beltrami operator and xt is a Riemannian Brownian motion
under the metric {⟨·, ·⟩x ; x ∈ X}, then (1/c)∆ is the Laplace-Beltrami operator and xt/c is
a Riemannian Brownian motion under the metric {c⟨·, ·⟩x ; x ∈ X}.
There is considerable interest in studying Riemannian Brownian motions in connection
with the geometric structures of the manifolds, see for example Elworthy [20, 21], Ikeda
and Watanabe [42], Émery [23] and Hsu [40]. In this section, we consider the representation of Riemannian Brownian motions in Lie groups and homogeneous spaces by stochastic
differential equations.
Let G be a Lie group and let ⟨·, ·⟩ be an inner product on the Lie algebra g of G. The
latter induces a left invariant Riemannian metric {⟨·, ·⟩g ; g ∈ G} on G under which the left
translations lg for g ∈ G are isometric. Note that a Riemannian Brownian motion gt in G
under this metric is a continuous Lévy process in G. We will express the Laplace-Beltrami
operator ∆G on G in terms of a basis of g and from this expression we will obtain a stochastic
differential equation satisfied by gt .
Let {ξ1 , . . . , ξd } be an orthonormal basis of g with respect to ⟨·, ·⟩ and let
[ξj , ξk ] =
d
∑
cijk ξi .
(3.27)
i=1
The numbers cijk are called the structure constants of g under the basis {ξ1 , . . . , ξd }. Since
[ξj , ξk ] = −[ξk , ξj ], cijk = −cikj . Recall that exponential coordinates ϕi ∈ Cc∞ (G) associated
∑
to the basis {ξ1 , . . . , ξd } satisfy g = exp[ di=1 ϕi (g)ξi ] for g near e.
Lemma 3.19 Let ϕ1 , . . . , ϕd be exponential coordinate functions on G associated to the basis
{ξ1 , . . . , ξd } of g. Then for g in a neighborhood U of e,
ξk (g) = ∂k +
d
d
∑
1 ∑
bipq (ϕ(g))∂i ,
ϕj (g)cijk ∂i +
2 i,j=1
p,q=1
89
where ∂i = ∂/∂ϕi , ϕ = (ϕ1 , . . . , ϕd ), and bipq (ϕ) = O(∥ϕ∥2 ).
Proof We may write ξk = ∂k +
∑d
aki (ϕ)∂i with aki (0) = 0. Then
i=1
ξj ξk = ∂j ∂k +
d
∑
∂j aki (0)∂i + O(∥ϕ∥).
i=1
∑
∑d
By Lemma 3.1, at e, di=1 ∂j aki (0)∂i = (1/2)[ξj , ξk ] = (1/2)
∑
(1/2)cijk . This implies aki (ϕ) = (1/2) dj=1 ϕj cijk + O(∥ϕ∥2 ). 2
i
i=1 cjk ∂i .
Then ∂j aki (0) =
∑
From Lemma 3.19, ∂i = ξi − (1/2) p,q ϕp cqpi ξq + O(∥ϕ∥2 ). The Riemannian metric tensor
of the left invariant metric on G induced by the inner product ⟨·, ·⟩ on g can be written as
gij (x) = ⟨∂i , ∂j ⟩x = δij −
1∑
1∑
ϕp cipj −
ϕp cjpi + O(∥ϕ∥2 )
2 p
2 p
for x ∈ G, and ∂k gij (e) = −(1/2)cikj − (1/2)cjki . Since gij (e) = g ij (e) = δij ,
1
Γijk (e) = [∂j gik (e) + ∂k gij (e) − ∂i gjk (e)]
2
1
1
(−cijk − ckji − cjki − cikj + ckij + cjik ) = (ckij + cjik ).
=
4
2
Because ckii = 0,
ξi ξi = [∂i +
1∑
1∑
ϕj ckji ∂k + O(∥ϕ∥2 )] [∂i +
ϕj ckji ∂k + O(∥ϕ∥2 )] = ∂i ∂i + O(∥ϕ∥).
2 j
2 j
It follows from (3.26) and the previous computation that for f ∈ Cc∞ (G),
∆G f (g) =
d
∑
ξi ξi f (g) − ξ0 f (g)
with
ξ0 =
i=1
d
∑
cjij ξi
(3.28)
i,j=1
holds for g = e. Since both sides of (3.28) are left invariant differentiable operators on G,
therefore, it holds for all g ∈ G.
We note that ξ0 in (3.28) is independent of the choice of the orthonormal basis {ξ1 , . . . , ξd }
∑
of g because the operator di=1 ξi ξi is independent of this basis.
By Theorem 2.12, the solution gt of the stochastic differential equation
dgt =
d
∑
1
ξi (gt ) ◦ dWti − ξ0 (gt )dt,
2
i=1
(3.29)
where Wt = (Wt1 , . . . , Wtd ) is a d-dimensional standard Brownian motion, is a continuous
∑
Lévy process in G. By (A.13) in Appendix A.5, its generator is (1/2) i ξi ξi − (1/2)ξ0 =
(1/2)∆G , and hence, it is a Riemannian Brownian motion in G. To summarize, we have the
following result.
90
Theorem 3.20 The Laplace-Beltrami operator on G under the left invariant metric induced
by an inner product ⟨·, ·⟩ on g is given by (3.28). Consequently, if gt solves the stochastic
differential equation (3.29), then it is a Riemannian Brownian motion in G under this metric.
Let V be a finite dimensional vector space and let F : V → V be a linear map. The trace
Tr(F ) and the determinant det(F ) of F are just the trace and the determinant of the matrix
representing F under a basis of V and are independent of choice of the basis.
Recall that a locally compact Hausdorff group is called unimodular if the left and the right
Haar measures agree, and a compact group is always unimodular. By [34, I.Corollary 1.5],
a Lie group G is unimodular if and only if | det[Ad(g)]| = 1 for all g ∈ G. Moreover, in
this case, a Haar measure is also invariant under the inverse map. We note that besides
compact groups, another important class of unimodular groups are semisimple Lie groups,
to be discussed in §4.4 and §5.3, see [34, I.Proposition 1.6].
A Lie algebra g is called unimodular if Tr[ad(ξ)] = 0 for any ξ ∈ g. If a Lie group G is
unimodular, then so is its Lie algebra g because det[Ad(eξ )] = eTr[ad(ξ)] for ξ ∈ g. It is also
clear that if g is unimodular, then so is the identity component of G.
Proposition 3.21 ξ0 in (3.28) and (3.29) vanishes if and only if Tr[ad(ξ)] = 0 for all
ξ ∈ g. Moreover, this holds if and only if g is unimodular.
∑
Proof By (3.27), Tr[ad(ξi )] = dj=1 cjij . It follows that ξ0 in (3.28) vanishes if and only if
Tr[ad(ξi )] = 0 for all i. (This useful fact was pointed out by Longmin Wang shortly after
the publication of [55], and it also appeared in the more recent preprint of Xue-Mei Li [64,
Lemma 10.1]). 2
We now consider a homogeneous space X = G/K, where K is a compact subgroup of G
with Lie algebra k. Let π: G → X = G/K be the natural projection and let o = π(e) as in
§3.1. Let p be an Ad(K)-invariant subspace of g that is complementary to k (see §3.1). The
differential Dπ of π is a linear bijection from p onto To X. Let ⟨·, ·⟩ be an Ad(K)-invariant
inner product on p. It induces a G-invariant Riemannian metric ⟨·, ·⟩x on X such that
∀ξ, η ∈ p,
⟨Dπ(ξ), Dπ(η)⟩o = ⟨ξ, η⟩.
Let {ξ1 , . . . , ξn } be a basis of p that is orthonormal under ⟨·, ·⟩. If necessary, this basis of
p may be extended to become a basis {ξ1 , . . . , ξd } of g such that {ξn+1 , . . . , ξd } is a basis of
k. Moreover, the Ad(K)-invariant inner product ⟨·, ·⟩ on p may be extended to become an
Ad(K)-invariant inner product on g such that ξ1 , . . . , ξd is orthonormal.
Let ϕ1 , . . . , ϕn be the exponential coordinate functions on X associated to the above basis
∑
of p as defined by (3.9), that is, x = exp[ ni=1 ϕi (x)ξi ]o for x near o. Recall that ϕi satisfy
(3.10) near o. In §3.1, ϕi are chosen so that (3.10) holds on G, but this is not required here.
91
By the following result, the Laplace-Beltrami operator ∆X on X takes the same form as
∆G on G. The reader is referred to §3.1, in particular, to Proposition 3.2, for differential
operators on X = G/K expressed in terms of elements of g.
∑
Proposition 3.22 ∆X = ni=1 ξi ξi − ξ0′ , where ξ0′ =
over, ξ0′ = 0 if and only if g is unimodular.
∑n
j
i,j=1 cij ξi
is Ad(K)-invariant. More∑
Proof: By Proposition 3.2, as a G-invariant differential operator, ∆X = ni,j=1 aij ξi ξj + ξ0′
for an Ad(K)-invariant matrix {aij } and an Ad(K)-invariant ξ0′ ∈ p. Because ξi = ∂i at o,
∑
1 ≤ i ≤ n, where ∂i = (∂/∂ϕi ), and are orthonormal, by (3.26), ∆X = ni=1 ξi ξi + ξ0′ at o. By
the G-invariance, this holds on X. The computations that are used to prove Lemma 3.19 and
(3.28) may be repeated, but replacing ξ1 , . . . , ξd and ϕ1 , . . . , ϕd by ξ1 , . . . , ξn and ϕ1 , . . . , ϕn ,
∑
to show that ξ0′ = ni,j=1 cjij ξi .
∑
If g is unimodular, then dj=1 cjij = Tr[ad(ξi )] = 0 for any i. For i ≤ n, by the Ad(K)∑
∑
invariance of p, cjij = 0 for j > n. Then nj=1 cjij = dj=1 cjij = 0, which implies ξ0′ = 0.
∑
∑
Conversely, if ξ0′ = 0, then for i ≤ n, Tr[ad(ξi )] = dj=1 cjij = nj=1 cjij = 0. Because K is
compact, so unimodular, and hence for ξ ∈ k, the trace of ad(ξ), as a linear map: k → k, is
zero. By the Ad(K)-invariance of p, ad(ξ) may also be regarded as a linear map: p → p.
We will show its trace is also zero. Because the inner product ⟨·, ·⟩ on p is Ad(K)-invariant,
n
∑
n
∑
i=1
i=1
⟨Ad(etξ )ξi , Ad(etξ )ξi ⟩ =
⟨ξi , ξi ⟩ = n.
∑
Taking derivative at t = 0 shows ni=1 ⟨ad(ξ)ξi , ξi ⟩ = 0. This shows that the trace of ad(ξ):
p → p is 0. Then the trace of ad(ξ): g → g is zero for ξ ∈ k. Because Tr[ad(ξi )] = 0 for
1 ≤ i ≤ n, we have Tr[ad(ξ)] = 0 for any ξ ∈ g, and hence g is unimodular. 2
By Proposition 3.22, if g is unimodular, then ξ0′ = 0. We will mention two other conditions
on X = G/K for ξ0′ = 0, and hence also for g being unimodular, and which also have other
useful implications. First we note that if X = G/K is irreducible (defined in §3.1) with
dim(X) > 1, then p has no nonzero Ad(K)-invariant element, so ξ0′ = 0.
For the second condition that implies ξ0′ = 0, we note that by [50, X.Theorem 3.3], if
⟨ξ, [ζ, η]p ⟩ + ⟨[ζ, ξ]p , η⟩ = 0 for ξ, η, ζ ∈ p,
(3.30)
where ξp denotes the p-component of ξ ∈ g under the direct sum decomposition g = k ⊕ p,
then for any ξ ∈ p, t 7→ π(etξ ) = etξ o is a geodesic in X = G/K under the induced metric.
Letting ζ = ξi and ξ = η = ξj , 1 ≤ i, j ≤ n, in (3.30) shows that cjij = 0, and hence if (3.30)
holds, then ξ0′ = 0. These two conditions are summarized below.
Proposition 3.23 If either X = G/K is irreducible with dim(X) > 1, or (3.30) holds, then
in Proposition 3.22, ξ0′ = 0, and hence g is unimodular.
92
Let F : X → Y be a smooth map between two manifolds. A differential operator S on
X is called F -related to a differential operator T on Y if
∀f ∈ Cc∞ (Y ),
(T f ) ◦ F = S(f ◦ F ).
Now suppose S and T are diffusion generators as defined by (A.15) in Appendix 5. If xt is
an S-diffusion process in X, then yt = F (xt ) is a T -diffusion process in Y . In particular,
if X and Y are Riemannian manifolds such that their Laplace-Beltrami operators ∆X and
∆Y are F -related, and if xt is a Riemannian Brownian motion in X, then yt = F (xt ) is a
Riemannian Brownian motion in Y .
Theorem 3.24 Let G be equipped with a left invariant Riemannian metric induced by an
Ad(K)-invariant inner product ⟨·, ·⟩ on g, and let X = G/K be equipped with the G-invariant
Riemannian metric induced by the restriction of ⟨·, ·⟩ to p. Assume p is the orthogonal complement of k under ⟨·, ·⟩. Then ∆G is π-related to ∆X . Consequently, if gt is a Riemannian
Brownian motion in G, then xt = gt o is a Riemannian Brownian motion in X.
Proof Let {ξ1 , . . . , ξd } be an orthonormal basis of g such that {ξ1 , . . . , ξn } is a basis of p
and {ξn+1 , . . . , ξd } is a basis of k. For f ∈ Cc∞ (X) and g ∈ G,
∆G (f ◦ π)(g) =
d
∑
ξi ξi (f ◦ π)(g) + ξ0 (f ◦ π)(g) =
i=1
n
∑
ξi ξi (f ◦ π)(g) + ξ0′ (f ◦ π)(g)
i=1
= ∆X f (π(g)). 2
Let g be equipped with an Ad(K)-invariant inner product ⟨·, ·⟩ under which p is the
orthogonal complement of k, and let {ξ1 , . . . , ξd } be an orthonormal basis of g chosen as in
the proof of Theorem 3.24. Consider the following stochastic differential equation on G.
dgt =
n
∑
ξi (gt ) ◦ dWti + c
i=1
d
∑
ξi (gt ) ◦ dWti ,
(3.31)
i=n+1
where Wt = (Wt1 , . . . , Wtd ) is a d-dim standard Brownian motion and c is an arbitrary
constant. Its solution gt is a continuous Lévy process in G. Recall a Lévy process gt in G
with g0 = e is called K-conjugate invariant if ∀k ∈ K and t > 0, kgt k −1 is equal to gt in
distribution.
Theorem 3.25 Let gt with g0 = e solve the stochastic differential equation (3.31). Then gt
is a K-conjugate invariant Lévy process in G, and hence (by Theorem 3.10) xt = gt o is a
Lévy process in X. Moreover, if g is unimodular, then xt = gt o is a Riemannian Brownian
motion in X = G/K under the G-invariant metric on X.
93
Proof For any k ∈ K, Ad(k) maps any orthonormal basis in p (resp. in k) into another
orthonormal basis in p (resp. in k). Thus, applying the K-conjugation to the stochastic
differential equation (3.31) amounts to an orthogonal transformation of the driving Brownian
motion Wt , which is also a standard Brownian motion. This shows that gt is K-conjugate
∑
∑
invariant. The generator of the diffusion process xt is L = (1/2) ni=1 ξi ξi + (c/2) di=n+1 ξi ξi .
∑
The computation in the proof of Theorem 3.24 shows that L is π-related to (1/2) ni=1 ξi ξi
on X, which is (1/2)∆X provided that g is unimodular. 2
Note that if c = 1 in (3.31) and if g is unimodular, then by Theorem 3.20 and Proposition 3.21, the solution gt is a Riemannian Brownian motion in G. We will show that for c = 0,
(3.31) becomes the so-called canonical stochastic differential equation on the orthonormal
frame bundle O(X) on X = G/K.
On a general Riemannian manifold X, let u = (ξ1 , . . . , ξn ) be an orthonormal frame at
x ∈ X. For 1 ≤ i ≤ n, let ut be the parallel displacement of u along a curve γt in X that
is tangent to ξi at t = 0. The tangent vector Hi (u) of the curve t → ut at t = 0 is a vector
in Tu O(X) and is independent of the choice for γt . The following stochastic differential
equation on O(X),
dut =
n
∑
Hi (ut ) ◦ dWti ,
(3.32)
i=1
is called the canonical stochastic differential equation on O(X), where Wt = (Wt1 , . . . , Wtn )
is an n-dim standard Brownian motion. Let p: O(X) → X be the natural projection. It
is well known (Eells and Elworthy [19]) that if ut solves the canonical stochastic differential
equation (3.32), then xt = p(ut ) is a Riemannian Brownian motion in X.
We now return to a homogeneous space X = G/K as before. We will identify an
orthonormal basis {ξ1 , . . . , ξn } of p with an orthonormal frame u at o via Dπ, and will
write gu, g ∈ G, for the orthonormal frame (Dg(ξ1 ), . . . , Dg(ξn )) at x = go. For any two
subspaces g1 and g2 of g, let [g1 , g2 ] be the subspace spanned by [ξ, η], ξ ∈ g1 and η ∈ g2 .
By Corollary 2.5, Theorem 2.10 and Theorem 3.3 in [50, chapter X], if
[p, p] ⊂ k,
(3.33)
then for ξ ∈ p, ut = etξ u is the parallel displacement of an orthonormal frame u at o along
the curve t 7→ etξ o in X. Note that (3.33) implies (3.30). (In the language of [50, chapter X],
under (3.33), the following three connections on G/K are the same: the canonical connection, the tosion-free connection with the same geodesics, and the Riemannian connection).
Later we will see that (3.33) holds on symmetric spaces X = G/K, an important class of
homogeneous spaces, that include spheres S n = O(n + 1)/O(n).
94
Theorem 3.26 Under the assumptions of Theorem 3.24 and the condition (3.33), if gt with
g0 = e solves the following stochastic differential equation on G,
dgt =
n
∑
ξi (gt ) ◦ dWti ,
(3.34)
i=1
where {ξ1 , . . . , ξn } is an orthonormal basis of p and Wt = (Wt1 , . . . , Wtn ) is an n-dim standard
Brownian motion, then for any orthonormal frame u at o and g ∈ G, ut = ggt u solves the
canonical stochastic differential equation (3.32) with u0 = gu.
Proof Let F : G × [p−1 (o)] → O(X) be the map (g, u) → gu. Because F (gesξi , u) = gesξi u
is the parallel displacement of u along the curve s 7→ gesξi o, so Dg F (g, u)ξi (g) as the tangent
vector of the curve s 7→ F (gesξi , u) at s = 0 is equal to Hi (gu). It follows
d(ggt u) = dF (ggt , u) =
n
∑
Dg F (ggt , u)ξi (ggt ) ◦
i=1
dWti
=
n
∑
Hi (ggt u) ◦ dWti . 2
i=1
Let G be a Lie group that acts transitively on a manifold X and let ⟨·, ·⟩ be an Ad(G)invariant inner product on the Lie algebra g of G. Then ⟨·, ·⟩ induces a bi-invariant Riemannian metric on G. Fix o ∈ X and let K be the isotropy subgroup of G at o with Lie
algebra k. Then X is identified with G/K via the map G/K ∋ gK 7→ go ∈ X. Because
⟨·, ·⟩ is Ad(K)-invariant, its restriction to the orthogonal complement p of k induces a Ginvariant Riemannian metric on X = G/K as described before. Note that there may not be
an Ad(G)-invariant inner product on g, but it always exists for a compact or abelian G.
Proposition 3.27 The G-invariant metric on X induced by an Ad(G)-invariant inner product on g defined above is independent of the choice for point o ∈ X.
Proof Let x ∈ X be a point different from o and let g ∈ G with x = go. The isotropy
subgroup of G at x is K ′ = gKg −1 . Now X is also identified with G/K ′ with natural
projection π ′ : G → X = G/K ′ . Any tangent vector at x can be represented by Dg ◦ Dπ(ξ)
for some ξ ∈ g and its norm under the G-invariant metric defined with respect to point o is
⟨ξ, ξ⟩. The vector is tangent to the curve getξ o = etAd(g)ξ x at t = 0, it follows that it is also
represented by Dπ ′ (Ad(g)ξ). Its norm under the G-invariant metric defined with respect to
point x is ⟨Ad(g)ξ, Ad(g)ξ⟩. Because ⟨·, ·⟩ is Ad(G)-invariant, the norms of a tangent vector
at x under the G-invariant metrics defined with respect o and x are the same. This implies
that the two metric agrees at point x and hence agree on X by the G-invariance. 2
By Theorem 3.24, ∆G is π-related to ∆X , where π: G → X is given by g 7→ go. In view
of Proposition 3.27, o may be replaced by any point x ∈ X. To summarize, we have the
following result.
95
Theorem 3.28 Let G be a Lie group that acts transitively on a manifold X and ⟨·, ·⟩ be
an Ad(G)-invariant inner product on the Lie algebra g of G. Let G be equipped with the biinvariant Riemannian metric and let X be equipped by the G-invariant Riemannian metric
induced by ⟨·, ·⟩. Then ∆G is πx -related to ∆X for any x ∈ X, where πx is the map G → X
given by g 7→ gx. consequently, if gt is a Riemannian Brownian motion in G, then xt = gt x
is a Riemannian Brownian motion in X for any x ∈ X.
Example 3.29 The orthogonal group G = O(n) acts transitively on the unit sphere X =
S n−1 = O(n)/O(n − 1) in Rn with the isotropy subgroup K = diag{1, O(n − 1)} at point
o = (1, 0, . . . , 0). The Lie algebra g = o(n) of G = O(n) is the space of n × n skew-symmetric
matrices, and for X, Y ∈ o(n),
′
⟨X, Y ⟩ = −(1/2)Tr(XY ) = (1/2)
n
∑
Xij Yij
i,j=1
is an Ad(O(n))-invariant inner product on o(n). The orthogonal complement p of the Lie
algebra k = diag{0, o(n−1)} of K in o(n) is the subspace consisting of the following matrices


0 y 
ζy = 
−y ′ 0
for row vectors y ∈ Rn−1 . It can be shown that etζy is a rotation of S n−1 in the plane
containing o and (0, y) at a rate of ∥y∥ radians per unit time. In particular, if {e1 , . . . , en−1 }
is the standard basis of Rn−1 , then ξ1 = ζe1 , . . . , ξn−1 = ζen−1 form an orthonormal basis of
p under ⟨·, ·⟩, and by the identification of p with To X via Dπ, they may also be regarded
as an orthonormal basis of To S n−1 under the metric of Rn . From this, it is easy to see that
the O(n)-invariant metric on S n−1 induced by ⟨·, ·⟩ is just the usual Riemannian metric on
S n−1 as the unit sphere in Rn . Therefore, if gt is a Riemannian Brownian motion in O(n)
under the bi-O(n)-invariant metric induced by ⟨·, ·⟩, then for any x ∈ S n−1 , xt = gt x is a
Riemannian Brownian motion in S n−1 under the usual metric.
We note that the condition (3.30) is satisfied, so for any y ∈ Rn−1 , t 7→ etζy is a geodesic
in the sphere S n−1 .
96
Chapter 4
Lévy processes in compact Lie groups
In this chapter, we apply Fourier analysis to study the distributions of Lévy processes in
compact Lie groups. Similar study will be done for Lévy processes in symmetric spaces in
the next chapter. After a brief review of the Fourier analysis on compact Lie groups, we
discuss in §4.2 the Fourier expansion of the distribution density pt of a Lévy process gt in
terms of matrix elements of irreducible unitary representations of G. It is shown that if gt
has an L2 density pt , then the Fourier series converges absolutely and uniformly on G, and
the convergence to the uniform distribution (the normalized Haar measure) is obtained. In
Section 4.3, for Lévy processes invariant under the inverse map, the L2 density is shown to
exist under some conditions, and the exponential convergence to the uniform distribution is
obtained. The same results are proved in §4.4 for bi-invariant Lévy processes. In this case,
the Fourier expansion is given in terms of irreducible characters, a more manageable form of
Fourier series. Some examples are computed explicitly in the last section.
The results of this chapter are stated in terms of convolution semigroups. They may also
be regarded as a study of infinitely divisible distributions on compact Lie groups because
such measures may be embedded in continuous convolution semigroups (see 3.3). The convergence to the uniform distribution on a compact Lie group studied here is clearly related
to the convergence of the convolution powers on a compact group. We briefly mention this
intensively studied area. For the weak convergence, there is Kawada-Itô’s theory of 1940
([35, Theorem 2.1.4]). See Major-Shlosman [66] for the uniform convergence at exponential
rate on compact connected groups. Diaconis’ influential lecture notes [17] demonstrated how
the group representation can be used effectively to determine the rate of convergence to the
uniform distribution for many concrete problems that can be modelled as random walks in
groups. Rosenthal [80] performed detailed study of the rate of convergence on SO(n). The
reader is also referred to Applebaum [5] for a more extensive study of probability measures
on compact Lie groups using Fourier analytic methods.
97
4.1
Fourier analysis on compact Lie groups
This section is devoted to a brief discussion of Fourier theory on a compact Lie group G
based on the Peter-Weyl theorem. See Bröcker and Dieck [12] for more details on the related
representation theory. We note that Peter-Weyl Theorem holds on compact groups, but we
will only consider compact Lie groups here.
Let V be a complex vector space of complex dimension dimC V = n. The set GL(V ) of
all complex linear bijections: V → V is a Lie group. With a basis {v1 , . . . , vn } of V , GL(V )
may be identified with the complex general linear group GL(n, C) of n×n invertible complex
matrices via the Lie group isomorphism
GL(V ) ∋ f 7→ {fij } ∈ GL(n, C),
where f (vj ) =
n
∑
fij vi .
i=1
The unitary group U (n) is the closed subgroup of GL(n, C) consisting of the unitary
′
matrices, that is, the set of n × n complex matrices X satisfying X −1 = X ∗ , where X ∗ = X ,
X is the complex conjugation of X and the prime denotes the matrix transpose. Its Lie
algebra u(n) is the space of all n × n skew-Hermitian matrices, that is, the set of matrices
X satisfying X ∗ = −X. The special unitary group SU (n) is the closed subgroup of U (n)
consisting of unitary matrices of determinant 1 and its Lie algebra su(n) is the space of
traceless skew-Hermitian matrices.
Let G be a Lie group and let V be a complex vector space. A Lie group homomorphism
F : G → GL(V ) is called a representation of G on V . It is called faithful if F is injective.
A representation F of G on V may be regarded as a (complex) linear action of G on
V , given by (g, v) 7→ F (g)v for g ∈ G and v ∈ V . It is called irreducible if the only
subspaces of V invariant under F are {0} and V . Two representations F1 : G → GL(V1 )
and F2 : G → GL(V2 ) are called equivalent if there is a linear bijection f : V1 → V2 such
that f ◦ [F1 (g)] = F2 (g) ◦ f for any g ∈ G. Assume V is equipped with a Hermitian inner
product ⟨·, ·⟩. The representation F is called unitary if it leaves the Hermitian inner product
invariant, that is, if ⟨F (v1 ), F (v2 )⟩ = ⟨v1 , v2 ⟩ for any v1 , v2 ∈ V . If G is compact, then any
representation of G on a finite dimensional complex space V may be regarded as unitary by
properly choosing a Hermitian inner product on V .
Let U be a unitary representation of G on a complex vector space V of complex dimension n = dimC (V ) equipped with a Hermitian inner product. Given an orthonormal basis {v1 , v2 , . . . , vn } of V , U may be regarded as an unitary matrix valued function
∑
U (g) = {Uij (g)} given by U (g)vj = ni=1 vi Uij (g) for g ∈ G. Let Ĝ denote the set of all the
equivalence classes of irreducible unitary representations. For δ ∈ Ĝ, let U δ be a unitary
representation belonging to the class δ and let d(δ) be its dimension. We will denote by Ĝ+
the set Ĝ excluding the trivial one-dimensional representation given by U δ = 1.
98
For a compact Lie group G, the normalized Haar measure on G will be denoted by ρG ,
or simply by dg. Let L2 (G) be the space of functions f on G with finite L2 -norm
∥f ∥2 = [ρG (|f |2 )]1/2 = [
∫
|f (g)|2 dg]1/2 ,
identifying functions that are equal almost everywhere under ρG . We recall that ρG is
invariant under left and right translations, and the inverse map on G.
By the Peter-Weyl Theorem (see II.4 and III.3 in Bröcker and Dieck [12]), the family
d(δ)1/2 Uijδ ,
i, j = 1, 2, . . . , d(δ) and δ ∈ Ĝ,
(4.1)
is a complete orthonormal system on L2 (G). Moreover, any continuous function on G can be
approximated arbitrarily close in supremum norm by a linear combination of finitely many
functions of the form Uijδ . We note that because L2 (G) is separable, Ĝ is countable.
The Fourier series of a function f ∈ L2 (G) with respect to the orthonormal system (4.1)
may be written as
∑
f = ρG (f ) +
d(δ) Tr[ρG (f U δ ∗ ) U δ ]
(4.2)
δ∈Ĝ+
in L2 sense, that is, the series converges to f in L2 (G).
The L2 -convergence of the series in (4.2) is equivalent to the convergence of the series of
nonnegative numbers in the following Parseval identity:
∥f ∥22 = |ρG (f )|2 +
∑
d(δ)Tr[ρG (f U δ ∗ )ρG (f U δ )],
(4.3)
δ∈Ĝ+
noting that Tr[ρG (f U δ ∗ )ρG (f U δ )] is the Euclidean norm of the matrix ρG (f U δ ).
The character of δ ∈ Ĝ is
χδ = Tr(U δ ),
(4.4)
which is independent of the choice of the unitary matrix U δ in the class δ. By [12, II.Theorem
(4.12)], a representation is uniquely determined by its character up to equivalence. The
normalized character is
ψδ = χδ /d(δ).
(4.5)
Proposition 4.1 The character χδ is positive definite in the sense that
k
∑
χδ (gi gj−1 )ξi ξj ≥ 0
i,j=1
for any finite set of gi ∈ G and complex numbers ξi . In particular, |ψδ | ≤ ψδ (e) = 1.
Moreover, for any u, v ∈ G,
∫
ψδ (gug −1 v)dg = ψδ (u)ψδ (v).
99
(4.6)
Proof Let U be a unitary representation in the class δ on a complex vector space V . Then
k
∑
i,j=1
χδ (gi gj−1 )ξi ξj = Tr[
k
∑
∑
U (gi gj−1 )ξi ξj ] = Tr{[
i,j=1
U (gi )ξi ] [
i
∑
U (gj )ξj ]∗ } ≥ 0.
j
The inequality involving ψδ follows easily from the positive definiteness of ψδ . To prove (4.6),
∫
let A(u) = dg U (gug −1 ) for any u ∈ G. Then A(u) is a linear map: V → V . The invariance
of dg implies that A(u) commutes with U (v) for any v ∈ G, that is,
U (v)A(u)U (v −1 ) =
∫
dg U (vgug −1 v −1 ) =
∫
dg U (gug −1 ) = A(u).
By Schur’s lemma (see [12, I.Theorem (1.10)]), this implies that A(u) is a multiple of the
∫
identity map idV on V . Taking the trace on both sides of dg U (gug −1 ) = A(u), we see that
∫
dg U (gug −1 ) = ψδ (u) idV . Multiplying by U (v) on the right and taking the trace again
proves (4.6). 2
A function f on G is called conjugate invariant if f (hgh−1 ) = f (g) for any g, h ∈ G.
Such a function is also called a class function or a central function in the literature. Let
L2ci (G) denote the closed subspace of L2 (G) consisting of conjugate invariant functions. The
set of irreducible characters, {χδ ; δ ∈ Ĝ}, is an orthonormal basis of L2ci (G), see [12, II.4 and
III.3]. Therefore, for f ∈ L2ci (G),
f = ρG (f ) +
∑
d(δ)ρG (f ψδ )χδ
(4.7)
δ∈Ĝ+
in L2 sense. Moreover, any continuous function f on G, that is conjugate invariant, may be
approximated arbitrarily close in supremum norm by a linear combination of finitely many
characters.
4.2
Lévy processes in compact Lie groups
Let G be a d-dim compact Lie group with Lie algebra g, and let gt be a Lévy process in G.
By Hunt’s Theorem 2.1, the domain D(L) of the generator L of gt contains C ∞ (G) (because
G is compact) and for f ∈ C ∞ (G), Lf is given by (2.3) with drift vector ξ0 , covariance
matrix aij and Lévy measure η. Recall that the triple (ξ0 , aij , η) determine the distribution
of the Lévy process gt given g0 = e, and are called the characteristics of gt . However, ξ0
depends on the choice of a basis {ξ1 , . . . , ξd } of g and a set of associated coordinate functions
ϕ1 , . . . , ϕd , but aij only depends on {ξ1 , . . . , ξd }.
We will assume the Lévy process gt starts at the identity element e of G, that is, g0 = e.
As mentioned in §3.2, the study of the Lévy process gt in distribution is equivalent to the
100
study of the associated convolution semigroup µt , and the generator L and the characteristics
(ξ0 , aij , η) of gt are also called the generator and the characteristics of µt .
The density of a measure on G will always mean the density function with respect to
the normalized Haar measure ρG unless explicitly stated otherwise. If a measure µ on G has
density p so that µ(dg) = p(g)ρG (dg), then µ may be written as p.ρG . For any two functions
p and q in B+ (G), their convolution is the function p ∗ q ∈ B+ (G) on G defined by
p ∗ q(g) =
∫
p(gh−1 )q(h)ρG (dh) =
∫
p(h)q(h−1 g)ρG (dh)
(4.8)
for g ∈ G, where the second equality holds because of the invariance of the Haar measure
dh. It is easy to show that if µ = p.ρG and ν = q.ρG , then µ ∗ ν = (p ∗ q).ρG .
Lemma 4.2 Let µ and ν be two probability measures on G such that one of them has a
density p. Then µ ∗ ν has a density q with ∥q∥2 ≤ ∥p∥2 .
Proof We will only consider the case when p is the density of µ. The other case can
be treated by a similar argument. For any f ∈ C(G), by the translation invariance of
dg = ρG (dg),
µ ∗ ν(f ) =
∫∫
∫∫
f (gh)p(g) dg ν(dh) =
f (g)p(gh−1 ) dg ν(dh) =
∫
∫
f (g)[
p(gh−1 )ν(dh)]dg.
∫
Hence, q(g) = p(gh−1 )ν(dh) is the density of µ ∗ ν. It is easy to see, by the Schwartz
inequality and the translation invariance of dg, that ∥q∥2 ≤ ∥p∥2 . 2
Remark 4.3 Using the Hölder inequality instead of the Schwartz inequality, we can prove
the same conclusion in Lemma 4.2 with ∥ · ∥2 replaced by the Lr -norm ∥ · ∥r for 1 ≤ r ≤ ∞,
where ∥f ∥∞ is the essential supremum of |f | on G for any Borel function f on G.
Note that if µt is a convolution semigroup and it has a density pt for each t > 0, then by
Lemma 4.2, ∥pt ∥2 ≤ ∥ps ∥2 for 0 < s < t.
Proposition 4.4 Let µt be a continuous convolution semigroup on G with µ0 = δe , generator
L and characteristics (ξ0 , aij , η). Then
µt (U δ ) = exp[tL(U δ )(e)]
= exp{
(4.9)
d
1 ∑
aij ξi ξj U δ (e) + ξ0 U δ (e) +
2 i,j=1
∫
G
[U δ (h) − I −
for δ ∈ Ĝ, where I is the d(δ) × d(δ) identity matrix.
101
d
∑
i=1
ϕi (h)ξi U δ (e)]η(dh)}
Note: The above expression, which resembles the classical Lévy -Khinchin formula on Rd ,
provides a complete characterization for a continuous convolution semigroup µt on a compact
Lie group G, as any finite measure µ on G is completely determined by µ(U δ ), δ ∈ Ĝ.
Proof of Proposition 4.4: Note that µ0 (U δ ) = I, and
∫
δ
µt+s (U ) =
∫
δ
µt (dg)µs (dh)U (gh) =
µt (dg)µs (dh)U δ (g)U δ (g) = µt (U δ )µs (U δ ).
It follows that µt (U δ ) = etY for some matrix Y . Because (d/dt)µt (U δ ) |t=0 = L(U δ )(e), we
obtain Y = L(U δ )(e) and prove (4.9). 2
For a square complex matrix A, let
A = Q diag[B1 (λ1 ), B2 (λ2 ), . . . , Br (λr )] Q−1
(4.10)
be the Jordan decomposition of A, where Q is an invertible matrix and Bi (λi ) is a Jordan
block of the following form





B(λ) = 




λ
0
0
···
0
1
λ
0
···
0
0
1
λ
···
0
0
0
1
···
0
···
···
···
···
···
0
0
0
···
λ





.




(4.11)
We note that if A is a Hermitian matrix, that is, if A∗ = A, then Q is unitary and all
Bi (λi ) = λi are real.
√
√∑
2
A
=
Recall that the Euclidean norm of a matrix A is ∥A∥ =
Tr(AA∗ ).
ij ij
Proposition 4.5 Let A be a square complex matrix. If all its eigenvalues λi have negative
real parts Re(λi ), then etA → 0 exponentially as t → ∞ in the sense that for any λ > 0
satisfying maxi Re(λi ) < −λ < 0, there is a constant K > 0 such that
∀t ∈ R+ ,
∥etA ∥ ≤ Ke−λt .
(4.12)
Proof Let the matrix A have the Jordan decomposition (4.10) with Jordan blocks Bi (λi )
given by (4.11). A direct computation shows that

etB(λ)




=




eλt
0
0
···
0
teλt
eλt
0
···
0
t2 eλt /2!
teλt
eλt
···
0
t3 eλt /3!
t2 eλt /2!
teλt
···
0
102
···
···
···
···
···
tk−1 eλt /(k − 1)!
tk−2 eλt /(k − 2)!
tk−3 eλt /(k − 3)!
···
eλt





.




Let bij (t) be the element of the matrix etA = Qdiag[et B1 (λ1 ) , et B2 (λ2 ) , . . . , et Br (λr ) ]Q−1 at
∑
place (i, j). From the expression for etB(λ) , it is easy to see that bij (t) = rm=1 pijm (t)eλm t ,
∑
∑
where pijm (t) are polynomials in t. Then Tr[etA (etA )∗ ] = i,j | m pijm (t)eλm t |2 and from
this (4.12) follows. 2
Recall the diffusion part of the generator L of the Lévy process gt or the convolution
∑
semigroup µt is the operator (1/2) dj,k=1 ajk ξj ξk given in (2.6). It will be called nondegenerate if the matrix a = {aij } is positive definite. This definition clearly does not depend on
the choice for the basis {ξ1 , . . . , ξd } of g and the associated coordinate functions ϕi .
In general, let m (≤ d) be the rank of a. Let
L = span{
d
∑
aij ξj , 1 ≤ i ≤ d}.
(4.13)
j=1
Then dim(L) = m. By the proposition below, the subspace L of g does not depend on
the choice for the basis of g and the associated coordinate functions. The Lévy process gt
or the convolution semigroup µt is said to have a hypoelliptic diffusion part if Lie(L), the
Lie algebra generated by L, is equal to g. It is clear that a nondegenerate diffusion part is
hypoelliptic.
Proposition 4.6 The subspace L of g does not depend on the choice for the basis {ξ1 , . . . , ξd }
of g and the associated coordinate functions ϕi . Moreover,
L = span{ζ1 , ζ2 , . . . , ζm },
where
ζi =
d
∑
(4.14)
σij ξj for 1 ≤ i ≤ m,
(4.15)
j=1
and σ is the m×d matrix in Theorem 2.13 such that a = σ ′ σ (noting a = σ = 0 when m = 0).
∑
Furthermore, the diffusion part of the generator L given by (2.3), (1/2) di,j=1 aij ξi ξj , may
∑
be written as (1/2) m
i=1 ζi ζi .
∑
Proof: By Proposition 2.4, the operator di,j=1 aij ξi ξj is independent of {ξ1 , . . . , ξd }. Let
{ξ1′ , . . . , ξd′ } be another basis of g with associated covariance matrix a′ij , and let L′ be the sub∑
∑
space defined by (4.13) with ξi replaced by ξi′ . Then ξj = dp=1 bjp ξp′ and a′pq = di,j=1 aij bip bjq
for some invertible matrix b = {bij }. We have, for 1 ≤ p ≤ d,
d
∑
q=1
a′pq ξq′ =
d
∑
aij bip bjq ξq′ =
d
∑
aij ξj bip .
i,j=1
i,j,q=1
∑
∑
∑
This shows L′ ⊂ L, and hence L′ = L. Because dj=1 aij ξj = dj=1 m
p=1 σpi σqj ξj =
∑m
2
p=1 σpi ζp , L ⊂ span{ζ1 , . . . , ζm }. However, as dim(L) = m, L = span{ζ1 , . . . , ζm }.
103
A continuous Lévy process with a hypoelliptic diffusion part is an example of hypoelliptic diffusion processes. It is well known (see for example Theoreme 5.1 on page 27 in [6,
chapitre 2]) that such a process has a smooth transition density for t > 0. In this case, µt
has a smooth density pt for t > 0. The following result provides a simple condition for the
distribution of a more general Lévy process to have an L2 density. More such conditions will
be given later. We note that a necessary and sufficient condition for a probability measure
µ on a compact Lie group to have an L2 density, in terms of µ(U δ ), can be found in [5,
Theorem 4.5.1].
Theorem 4.7 Let µt be a continuous convolution semigroup on a compact Lie group G with
µ0 = δe , a finite Lévy measure and a nondegenerate diffusion part. Then µt has a density
pt ∈ L2 (G) for t > 0.
Proof Because the Lévy measure η is finite, by the discussion in Section 1.3, the associated
Lévy process gt may be constructed from a continuous Lévy process xt by interlacing jumps
at exponentially spaced time intervals. More precisely, let xt be a continuous Lévy process in
G whose generator is given by (2.8) with η = 0. Assume g0 = e. Let {τn } be a sequence of
exponential random variables with a common rate λ = η(G) and let {σn } be a sequence of
G-valued random variables with a common distribution η(·)/η(G) such that all these objects
are independent. Let Tn = τ1 + τ2 + · · · + τn for n ≥ 1 and set T0 = 0. Let gt0 = xt , gt1 = gt0
for 0 ≤ t < T1 and gt1 = g 0 (T1 )σ1 x(T1 )−1 x(t) for t ≥ T1 , and inductively, let gtn = gtn−1 for
t < Tn and gtn = g n−1 (Tn )σn x(Tn )−1 x(t) for t ≥ Tn . Define gt = gtn for Tn ≤ t < Tn+1 . Then
gt is a Lévy process in G with generator given by (2.8).
Note that Tn has a Gamma distribution with density rn (t) = λn tn−1 e−λt /(n − 1)! with
respect to the Lebesgue measure on R+ . Let qt denote the smooth density of the distribution
of xt for t > 0. For f ∈ C(G) and t > 0, using the independence, we have
µt (f ) = E[f (xt ); t < T1 ] +
= E[f (xt )]P (T1 > t) +
∞ ∫
∑
∞
∑
E[f (gt ); Tn ≤ t < Tn + τn+1 ]
n=1
t
n=1 0
rn (s) ds E[f (gsn−1 σn x−1
s xt )] P (τn+1 > t − s). (4.16)
We now show that for n ≥ 1 and 0 ≤ s < t,
E[f (gsn−1 σn x−1
s xt )]
∫
=
f (g)ps,t,n (g)dg
for some ps,t,n with ∥ps,t,n ∥2 ≤ ∥qt/2n ∥2 . (4.17)
To prove (4.17) for n = 1, first assume s ≥ t/2. We have
−1
E[f (gs0 σ1 x−1
s xt )] = E[f (xs σ1 xs xt )] = µ ∗ ν(f ),
where µ and ν are respectively the distributions of xs and σ1 x−1
s xt . By Lemma 4.2, µ ∗ ν
has a density ps,t,1 with ∥ps,t,1 ∥2 ≤ ∥qs ∥2 ≤ ∥qt/2 ∥2 . If s ≤ t/2, then we may take µ and ν to
104
be the distributions of xs σ1 and x−1
s xt respectively, and still obtain a density ps,t,1 of µ ∗ ν
with ∥ps,t,1 ∥2 ≤ ∥qt/2 ∥2 . This proves (4.17) for n = 1. Now using induction, assume (4.17) is
proved for n = 1, 2, . . . , k for some integer k > 0. Because
∫
E[f (gtk )] = E[f (gtk−1 )]P (Tk > t) +
t
0
E[f (gsk−1 σk x−1
s xt )]P (Tk ∈ ds),
this implies in particular that the distribution of gtk has a density pkt with ∥pkt ∥2 ≤ ∥qt/2k ∥2 .
Consider E[f (gsk σk+1 x−1
s xt )] = µ ∗ ν(f ), where µ are ν are taken to be the distributions of
k
−1
gs and σk+1 xs xt respectively if s ≥ t/2, and those of gsk σk+1 and x−1
s xt if s ≤ t/2. By
Lemma 4.2, we can show that µ ∗ ν has a density whose L2 -norm is bounded by
∥pks ∥2 ≤ ∥qs/2k ∥2 ≤ ∥qt/2k+1 ∥2
if s ≥ t/2, and bounded by ∥qt/2 ∥2 if s ≤ t/2. In either case, the L2 -norm of the density of
µ ∗ ν is bounded by ∥qt/2k+1 ∥2 . This proves (4.17) for any n ≥ 1.
∫
By (4.16) and the fact that P (τn > t) = e−λt for t > 0, we see that µt (f ) = f (g)pt (g)dg
with
∞ ∫
pt = qt e−λt +
∑
t
n=1 0
rn (s) ds e−λ(t−s) ps,t,n
and
∥pt ∥2 ≤ ∥qt ∥2 e
−λt
+
≤ ∥qt ∥2 e−λt +
∞ ∫
∑
t
n=1 0
∞ ∫ t
∑
n=1 0
rn (s) ds e−λ(t−s) ∥ps,t,n ∥2
rn (s) ds e−λ(t−s) ∥qt/2n ∥2 .
(4.18)
It is well known that the density of a nondegenerate diffusion process xt on a d-dimensional
compact manifold is bounded above by ct−d/2 for small t > 0, where c is a constant independent of t. See for example [6, Chapitre 9]. Therefore, |qt | ≤ ct−d/2 and ∥qt/2n ∥2 ≤ c(2n /t)d/2 .
∫
Since 0t rn (s)ds ≤ (λt)n /n!, it is easy to see that the series in (4.18) converges. This proves
pt ∈ L2 (G). 2
A subspace a of g and a subset A of G, either may be empty, generate a closed subgroup
H of G, which is the smallest closed subgroup of G containing exp(a) = {eξ ; ξ ∈ a} and A.
In this case, exp(Lie(a)) ⊂ H.
Theorem 4.8 Let µt be a continuous convolution semigroup on a compact Lie group G with
µ0 = δe , and let L be its generator. Assume µt has a density pt ∈ L2 (G) for t > 0.
(a) For t > 0, g ∈ G and δ ∈ Ĝ, µt (U δ ) = exp[t L(U δ )(e)], and
pt (g) = 1 +
∑
d(δ) Tr{exp[t L(U δ ∗ )(e)]U δ (g)},
δ∈Ĝ+
105
(4.19)
where the series converges absolutely and uniformly for (t, g) ∈ [r, ∞) × G for any fixed
r > 0. Consequently, (t, g) 7→ pt (g) is continuous on (0, ∞) × G.
(b) Assume Lie(ζ1 , . . . , ζm ) and supp(η) generate G, where ζi are defined in (4.15) and η is
the Lévy measure. Then all the eigenvalues of L(U δ ∗ )(e), δ ∈ Ĝ+ , have negative real parts.
Consequently, pt → 1 uniformly on G as t → ∞.
Proof For f = pt , the series in (4.19) is just the Fourier series in (4.2). Let Aδ (t) =
ρG (pt U δ ∗ ) = µt (U δ ∗ ). We have Aδ (t) = exp[tL(U δ ∗ )(e)] by (4.9).
We now prove the absolute and uniform convergence of series in (4.19). Note that by
∑
∑
the Parseval identity, ∥pt ∥22 = 1 + δ d(δ)Tr[Aδ (t)Aδ (t)∗ ], where the summation δ is taken over δ ∈ Ĝ+ . For any r > 0 and ε > 0, there is a finite subset Γ of Ĝ+ such that
∑
∗
2
δ
δ∈Γc d(δ)Tr[Aδ (r/2)Aδ (r/2) ] ≤ ε . By the Schwartz inequality and the fact that U is a
unitary matrix, for any finite Γ′ ⊃ Γ and t > r,
∑
d(δ) |Tr[Aδ (t) U δ ]| =
δ∈Γ′ −Γ
∑
≤
∑
d(δ) |Tr[Aδ (r/2) Aδ (t − r/2) U δ ]|
δ∈Γ′ −Γ
d(δ) {Tr[Aδ (r/2) Aδ (r/2)∗ ]}1/2 {Tr[(Aδ (t − r/2)U δ )(Aδ (t − r/2)U δ )∗ ]}1/2
δ∈Γ′ −Γ
∑
=
d(δ) {Tr[Aδ (r/2) Aδ (r/2)∗ ]}1/2 {Tr[Aδ (t − r/2) Aδ (t − r/2)∗ ]}1/2
δ∈Γ′ −Γ
≤ {
∑
∑
d(δ) Tr[Aδ (r/2) Aδ (r/2)∗ ]}1/2 {
δ∈Γ′ −Γ
d(δ) Tr[Aδ (t − r/2) Aδ (t − r/2)∗ ]}1/2
δ∈Γ′ −Γ
≤ ε ∥pt−r/2 ∥2 ≤ ε ∥pr/2 ∥2 ,
where the last inequality follows from Lemma 4.2. This proves the absolute and uniform
convergence stated in part (a).
To complete the proof, we will show that the real parts of all the eigenvalues of the
matrix L(U δ ∗ )(e) are ≤ 0, and if Lie(ζ1 , . . . , ζm ) and supp(η) generate G, then they are < 0
for δ ∈ Ĝ+ . This would imply that Aδ (t) → 0 exponentially and, combined with the uniform
convergence of the series in (4.19), the uniform convergence of pt to 1 as t → ∞ follows.
Write U = U δ and n = d(δ) for δ ∈ Ĝ+ . Consider the quadratic form Q(z) =
z ∗ [L(U ∗ )(e)]z for z = (z1 , . . . , zn )′ , a column vector in Cn . Since the eigenvalues of L(U ∗ )(e)
are the values of Q(z) with |z| = 1, it suffices to show that Re[Q(z)] ≤ 0 for all z ∈ Cn , and
that Re[Q(z)] < 0 for all nonzero z ∈ Cn if under the condition stated above. For ξ ∈ g, let
˜ and hence ξ˜∗ is a skew-Hermitian matrix. Moreover,
ξ˜ = (ξU ∗ )(e). Then U (etξ )∗ = exp(tξ),
d
d
(ξU ∗ )(g) = U (getξ )∗ |t=0 = U (etξ )∗ U (g)∗ |t=0 = ξ˜ U (g)∗ .
dt
dt
∗
∗
˜
˜
Therefore, (ζξU )(e) = ζ(ξU )(e) = ξ ζ̃ for ζ ∈ g, and if Z = [ξ, ζ] (Lie bracket), then
∑m ∗
∑m
∗
˜ We have ∑d
Z̃ = [ζ̃, ξ].
i=1 ζ̃i ζ̃i and by (2.3),
i=1 ζ̃i ζ̃i = −
i,j=1 aij ξi ξj U (e) =
∫
m
1∑
∗
L(U )(e) = −
ζ̃i ζ̃i + ζ̃V −
[I − U (g)∗ ]η(dg) + rV ,
c
2 i=1
V
∗
106
(4.20)
∫ ∑
where V is a neighborhood of e, ζ̃V = ξ˜0 − V c di=1 ϕi (g)ξ˜i η(dg) and
∫
rV =
∗
V
[U (g) − I −
d
∑
ϕi (g)ξ˜i ]η(dg) → 0
as V ↓ {e}.
i=1
We may write
Q(z) = −
∫
m
1∑
|ζ̃i z|2 + z ∗ ζ̃V z −
[|z|2 − z ∗ U (g)∗ z]η(dg) + z ∗ rV z.
2 i=1
Vc
(4.21)
Since U (g)∗ is unitary, |z|2 ≥ |z ∗ U (g)∗ z|. Because Re(z ∗ W z) = 0 for any skew-Hermitian
matrix W , it follows that Re[Q(z)] ≤ 0. If Re[Q(z)] = 0 for some nonzero z ∈ Cn , then
ζ̃i z = 0 for 1 ≤ i ≤ m and |z|2 = z ∗ U (g)∗ z for z ∈ supp(η). For ζ = [ζi , ζj ], we have
ζ̃z = [ζ̃j , ζ̃i ]z = ζ̃j ζ̃i z − ζ̃i ζ̃j z = 0. Then ζ̃z = 0 for any ζ ∈ L =Lie(ζ1 , . . . , ζm ). Because
U (etζ )∗ = exp(tζ̃), U (etζ ) = exp(−tζ̃), and hence, U (g)z = z for all g contained the subgroup
of G generated by exp(L). Next, |z|2 = z ∗ U (g)∗ z implies U (g)z = z, and hence for g ∈
supp(η), U (g)z = z. Because L and supp(η) generate G, U (g)z = z for all g ∈ G. This
would contradict the irreducibility of the representation U unless n = 1. When n = 1,
U (g)z = z would imply that U is the trivial representation, which is impossible as δ ∈ Ĝ+ .
Therefore, Re[Q(z)] < 0 for nonzero z ∈ Cn . 2
The total variation norm of a signed measure ν on G is defined by ∥ν∥tv = sup |ν(f )|
with f ranging over all Borel functions on G with |f | ≤ 1. The following corollary follows
easily from the uniform convergence of pt to 1 and the Schwartz inequality.
Corollary 4.9 Under the assumption in Theorem 4.8 (b), µt converges to the normalized
Haar measure ρG under the total variation norm, that is,
∥µt − ρG ∥tv → 0
as t → ∞.
Remark 4.10 If G is connected, then the uniform convergence pt → 1 in Theorem 4.8 (b)
can be strengthened to have an exponential rate, even without the other assumptions. That
is, if G is connected and pt ∈ L2 (G) for t > 0, then for any r > 0, there are constants a > 0
and λ > 0 such that |pt − 1| ≤ ae−λt for any t > r on G. This follows from Major-Shlosman
[66] which says that if µ is a probability measure on a compact connected group G with
an L2 density p, then there are constants b > 0 and q ∈ (0, 1) such that |pn − 1| ≤ bq n
on G for any integer n > 1, where pn is the density of µ∗n . Let pt be the density of µt
in Theorem 4.8, and for t > r, let n be the largest integer (≥ 2) such that (r/2)n ≤ t.
Because p ∗ 1 = 1 ∗ p = 1 for any density p, pt − 1 = (pnr/2 − 1) ∗ (pt−nr/2 − 1). By MajorShlosman’s result with p = pr/2 , |pt − 1| ≤ 2bq n = 2bhnr/2 , where h = q 2/r ∈ (0, 1). Then
|pt − 1| ≤ 2bht (1/h)t−nr/2 ≤ (2b/h)ht = ae−λt with a = 2b/h and λ = ln(1/h).
107
4.3
Symmetric Lévy processes
Let G be a d-dim compact Lie group, and let J: G → G be the inverse map J(g) = g −1 . A
Lévy process gt in G, or the associated convolution semigroup µt , will be called symmetric if
it is J-invariant, that is, if Jµt = µt for all t ≥ 0. Recall µt is the distribution of gte = g0−1 gt .
Let (ξ0 , aij , η) be the characteristics of gt or µt . Let the coordinate functions ϕi be
∑
exponential, that is, g = exp[ di=1 ϕi (g)ξi ] for g near e. Then ϕi (g −1 ) = −ϕi (g) for g near
e. Replacing ϕi by ϕϕi for a J-invariant function ϕ ∈ C ∞ (G) with ϕ = 1 near e and ϕ = 0
outside a small neighborhood of e, one may assume
∀g ∈ G,
ϕi (g −1 ) = −ϕi (g),
1 ≤ i ≤ d.
(4.22)
Some simple characterizations of the symmetry of a Lévy process gt are summarized in
the following proposition.
Proposition 4.11 Let µt be a continuous convolution semigroup on a compact Lie group G
with µ0 = δe , generator L and characteristics (ξ0 , aij , η). Then µt is symmetric if and only
if (a) below holds, and this is also equivalent to (b) below if the coordinate functions ϕi are
chosen to satisfy (4.22).
(a) L(U δ )(e) is a Hermitian matrix for all δ ∈ Ĝ.
(b) ξ0 = 0 and η is J-invariant.
Moreover, (b) implies that for f ∈ C ∞ (G) and g ∈ G,
Lf (g) =
d
1∫
1 ∑
aij ξi ξj f (g) +
[f (gh) − 2f (g) + f (gh−1 )]η(dh).
2 i,j=1
2 G
(4.23)
Note: It will be clear from the proof that the integral in (4.23) is absolutely integrable.
Proof: We note that L(U δ )(e) is a Hermitian matrix for all δ ∈ Ĝ if and only if Aδ (t) =
exp[t L(U δ ∗ )(e)] is a Hermitian matrix for all δ ∈ Ĝ and all t > 0. Since Aδ (t)∗ = µt (U δ ) =
µt (U δ ∗ ◦ J), and Uijδ span a dense subset of C(G) under the supremum norm, we see that this
is also equivalent to the J-invariance of µt for all t > 0. This shows that the J-invariance of
µt is equivalent to (a).
For f ∈ C ∞ (G), Lf (e) = (d/dt)µt (f ) |t=0 , and hence, if µt is J-invariant, then L(f ◦
J)(e) = Lf (e). By the generator formula (2.3), and (4.22), a simple computation yields
d
1 ∑
L(f ◦ J)(e) =
aij ξi ξj f (e) − ξ0 f (e) +
2 i,j=1
=
∫
G
[f (h−1 ) − f (e) +
d
∑
ϕi (h)ξi f (e)]η(dh)
i=1
∫
d
d
∑
1 ∑
aij ξi ξj f (e) − ξ0 f (e) + [f (h) − f (e) −
ϕi (h)ξi f (e)]Jη(dh)
2 i,j=1
G
i=1
108
Comparing this with (2.3), for any f ∈ C ∞ (G) vanishing near e, we obtain η(f ) = Jη(f ),
and hence η is J-invariant. Then comparing with (2.3) again shows ξ0 = 0. This proves
(b) under J-invariance of gt . Now suppose (b) holds. By the J-invariance of η and (4.22),
∫
V c ϕi (h)η(dh) = 0 for any J-invariant neighborhood V of e. Then by (4.20) in the proof
∫
∑
∗
∗
of Theorem 4.8, for U = U δ , L(U ∗ )(e) = −(1/2) m
i=1 ζ̃i ζ̃i − lim V c [I − U (g) ]η(dg), where
the limit is taken as a J-invariant open set V shrinks to {e}. By the J-invariance of η, this
integral is a Hermitian matrix. Then L(U δ )(e) is Hermitian for any δ ∈ Ĝ. By (a), this
shows that µt is J-invariant.
∫
Because ξ0 = 0 and V c ϕi dη = 0, by (2.3),
Lf (g) =
∫
d
d
∑
1 ∑
aij ξi ξj f (g) + [f (gh) − f (g) −
ϕi (h)ξi f (g)]η(dh).
2 i,j=1
G
i=1
By the J-invariance of η and (4.22), the above may be written as
Lf (g) =
∫
d
d
∑
1 ∑
aij ξi ξj f (g) + [f (gh−1 ) − f (g) −
ϕi (h−1 )ξi f (g)]η(dh)
2 i,j=1
G
i=1
∫
d
d
∑
1 ∑
=
aij ξi ξj f (g) + [f (gh−1 ) − f (g) +
ϕi (h)ξi f (g)]η(dh).
2 i,j=1
G
i=1
Averaging these two different forms of Lf (g) yields (4.23). 2
Let µt be a continuous symmetric convolution semigroup on G with µ0 = δe . By Proposition 4.11 (a), L(U δ )(e) is Hermitian. Then
L(U δ ∗ )(e) = L(U δ )(e) = Qδ diag[exp(λδ1 t), . . . , exp(λδd(δ) t)] Q∗δ ,
where Qδ is a unitary matrix, and λδ1 ≤ λδ2 ≤ · · · ≤ λδd(δ) are the eigenvalues of L(U δ )(e). If
µt has a density pt ∈ L2 (G), then by Theorem 4.8, its Fourier series is
pt (g) = 1 +
∑
d(δ) Tr{Qδ diag[exp(λδ1 t), . . . , exp(λδd(δ) t)] Q∗δ U δ (g)}.
(4.24)
δ∈Ĝ+
Because Qδ is unitary, the associated Parseval identity is
∥pt − 1∥22 =
d(δ)
∑
d(δ)
δ∈Ĝ+
∑
exp(2λδi t).
(4.25)
i=1
By the proof of Theorem 4.8, the convergence of the series in (4.25) implies that the Fourier
series (4.24) converges absolutely and uniformly for (t, g) ∈ [ε, ∞) × G for any fixed ε > 0.
Let ϕ and ψ be two nonnegative functions, possibly only defined near a point x0 . They
will be called asymptotically equal at x0 , denoted as ϕ ≍ ψ at x0 , if there are constants
109
c2 ≥ c1 > 0 such that c1 ϕ ≤ ψ ≤ c2 ϕ near x0 . Similarly, two measures µ and ν are
called asymptotically equal at x0 , denoted as µ ≍ ν, if c1 µ ≤ ν ≤ c2 µ when both measures
are restricted to a neighborhood of x0 . This is equivalent to the asymptotic equality of
their densities under a reference measure when such exists. We will say µ asymptotically
dominates ν at x0 if µ dominates a measure that is asymptotically equal to ν at x0 .
Fix a Riemannian metric on G, and let r = r(g) denote the Riemannian distance from
g ∈ G to e. Note that for different choices of Riemannian metrics on G, the associated
distance functions r are asymptotically equal at e. We are now ready to state the main
result of this section.
Theorem 4.12 Let µt be a continuous symmetric convolution semigroup on a compact Lie
group G of dimension d with µ0 = δe . Assume either it has a hypoelliptic diffusion part,
or its Lévy measure η asymptotically dominates r−β .ρG at e for some β > d, where r is the
Riemannian distance to e (this condition does not depend on the choice of a Riemannian
metric on G).
(a) For t > 0, µt has a density pt ∈ L2 (G) such that (t, g) 7→ pt (g) is continuous on (0, ∞)×
G. Moreover, the Fourier series of pt , given in (4.19) and (4.24), converges absolutely and
uniformly for (t, g) ∈ [ε, ∞) × G for any fixed ε > 0.
(b) If G is connected, or more generally if the assumption of Theorem 4.8 (b) holds, then
all eigenvalues λδi of L(U δ )(e), δ ∈ Ĝ+ , are negative and they have a maximum −λ < 0.
Moreover, for any ε > 0, there are constants 0 < c1 < c2 such that for t > ε,
∥pt − 1∥∞ ≤ c2 e−λt ,
c1 e−λt ≤ ∥pt − 1∥2 ≤ c2 e−λt ,
c1 e−λt ≤ ∥µt − ρG ∥tv ≤ c2 e−λt .
Proof By Proposition 4.11, and using the notation in the proof of Theorem 4.8, with
∫
∑
˜∗
U = U δ for δ ∈ Ĝ+ and ξ˜ = ξU ∗ (e) for ξ ∈ g, L(U )(e) = L(U ∗ )(e) = − 21 m
i=1 ζi ζ̃i − G [I −
U (h)∗ ]η(dh). Let
∫
m
1∑
2
Q(z) = z [L(U )(e)]z = −
|ζ̃i z| − [|z|2 − z ∗ U ∗ (h)z]η(dh)
2 i=1
G
∗
(4.26)
for z ∈ Cd regarded as a column vector.
It is known that the eigenvalues λ1 ≤ λ2 ≤ · · · ≤ λd of any d × d Hermitian matrix H
possess the following min-max representation:
λi = min
Vi
max z ∗ Hz
z∈Vi , |z|=1
for 1 ≤ i ≤ n,
(4.27)
where Vi ranges over all i-dimensional subspaces of Cd (see for example [14, Theorem 1.9.1]).
Assume µt has a hypoelliptic diffusion part. If η = 0, then the associated Lévy process
gt is a hypoelliptic diffusion process, and µt has a smooth density for t > 0. If we add a Jinvariant Lévy measure η, then because |z|2 ≥ |z ∗ U (h)∗ z|, Q(z) can only be smaller. By the
110
min-max representation (4.27), λδi can only be smaller, so the series in (4.25) still converges.
This implies that the series in (4.24) defines a function pt ∈ L2 (G), which may also be written
∑
as pt = 1 + δ d(δ) Tr[µt (U δ ∗ ) U δ ]. Then for δ ∈ Ĝ, ρG (pt U δ ∗ ) = µt (U δ ∗ ). This shows that pt
is the density of µt . As mentioned after (4.25), its Fourier series (4.24) converges absolutely
and uniformly. We have proved (a) under the assumption of a hypoelliptic diffusion part.
Suppose the series (4.25) converges. If the Lévy measure η is modified outside a neighborhood of e, then the change in Q(z) is controlled by a constant term, that is, if Q′ (z)
is Q(z) after the change, then there is constant c > 0, independent of δ ∈ Ĝ+ , such that
Q(z) − c ≤ Q′ (z) ≤ Q(z) + c for all z ∈ Cd(δ) with |z| = 1. By the min-max representation
(4.27), the changes in the eigenvalues λδi , δ ∈ Ĝ, are controlled by the same constant, and
hence the series (4.25) still converges. It also clear from (4.27) that if η is replaced by a larger Lévy measure, the series (4.25) will still converge. Because a change of λδi by a constant
factor may be absorbed by t, so will not affect the convergence of (4.25), it is then easy to
show from (4.27) that a change in η by a constant factor will not affect the convergence of
(4.25). Therefore, if the series (4.25) converges for some choice of a Lévy measure η, then it
will still converge for any Lévy measure that asymptotically dominates η at e. An example
of such an η is the Lévy measure of a subordinated Brownian motion in G to be discussed
below.
Let ht (x) the density of the convolution semigroup µB
t of the Riemannian Brownian
motion in G with generator ∆G (time changed by factor 1/2 for simplicity). Then for δ ∈ Ĝ,
δ
δ
∗
µB
t (U ) = exp[t∆G U (e)] = Qδ diag{exp(λ1 t), . . . , exp(λd(δ) t)}Qδ
(4.28)
for some unitary matrix Qδ , where λ1 ≤ λ2 ≤ · · · ≤ λd(δ) are the eigenvalues of the Hermitian
matrix ∆G U δ (e). By [13, Theorem 4], for any finite T > 0,
∀t ∈ [0, T ] and x ∈ G,
ht (x) ≤ at−d/2 exp[−r2 /(bt)],
(4.29)
where a > 0 and b > 0 are some constants, and r =dist(e, x) (Riemannian distance). This
upper bound holds on any complete Riemannian manifold with bounded curvature with e
being a fixed reference point.
Fix α ∈ (0, 1). Let νtα be the convolution semigroup on [0, ∞) associated to the α-stable
subordinator (see [82]). This is a nondecreasing Lévy process in R with Laplace transform
∫
0
∞
e−su νtα (du) = exp(−tsα ) = exp[−t
∫
0
∞
(1 − e−su )ξ α (du)]
(4.30)
for s > 0, and Lévy measure ξ α (du) = αu−1−α du/Γ(1 − α) on (0, ∞).
Let bt be a Riemannian Brownian motion in G with b0 = e and let τt be an independent
α-stable subordinator with τ0 = 0. The distribution of the subordinated process bαt = bτt is
111
given by
∫
µαt =
∞
0
∫
α
µB
u (·)νt (du)
with density
hαt (x) =
∞
0
hu (x)νtα (du).
(4.31)
A direct computation shows that µαt is a continuous convolution semigroup on G. This fact
holds on any Lie group G and homogeneous space G/K with K compact.
By (4.15) in [65], the density qtα of νtα at time t = 1 satisfies
q1α (u) ≍ Au−a exp(−bu−c ) at u = 0
(4.32)
for some constants A, a, b, c > 0 depending only on α. The above estimate holds also for qtα
at any time t > 0 if we allow the constants depend on α and t. Then it is easy to see from
(4.29) and (4.31) that hαt ∈ L2 (G) for t > 0.
By (4.28) and (4.30),
∫
µαt (U δ ) =
∞
0
∫
δ α
µB
u (U )νt (du) = Qδ exp{−t
∫
= exp{−t
∞
∫0
= exp{−t
G
0
∞
[I − diag(eλ1 u , . . . , eλd(δ) u )]ξ α (du)}Q∗δ .
[I − Qδ diag(eλ1 u , . . . , eλd(δ) u )Q∗δ ]ξ α (du)}
[I − U δ (x)]η α (dx)},
∫
(4.33)
∫
where η α (dx) = [ 0∞ hu (x)ξ α (du)]dx. Because 01 uξ α (du) < ∞, and for any f ∈ Cc∞ (G) with
α 2
f (e) = 0, (1/u)µB
u (f ) → ∆G f (e) as u → 0, it is then easy to show η (r 1[r≤1] ) < ∞, and
hence η α is a Lévy measure on G. Let µ̃t be a continuous convolution semigroup on G with
∫
µ̃0 = δe and characteristics (0, 0, η α ). Then µ̃t (U δ ) = exp[−t G (I − U δ )dη α ]. Therefore,
µαt (U δ ) = µ̃t (U δ ) for all δ ∈ Ĝ. This shows that µαt = µ̃t , and hence η α is the Lévy measure
of µαt .
By (4.29),
∫
ηα ≍ [
0
T
u−d/2 e−r
2 /(bu)
u−1−α du]ρG (dx) ≍ r−d−2α .ρG
at x = e.
(4.34)
Because µαt has an L2 density hαt , if Lévy measure η asymptotically dominates r−d−2α dx at
x = e, 0 < α < 1, then as mentioned earlier, µt will have a density pt ∈ L2 (G). The
conclusions in (a) then follows from Theorem 4.8 (a).
We note that if gt has a hypoelliptic diffusion part, then with ζi defined by (4.15),
Lie(ζ1 , . . . , ζm ) = g, and hence G is generated by Lie(ζ1 , . . . , ζm ) when G is connected. On
the other hand, the asymptotic condition on η stated in the present theorem implies that
supp(η) contains a neighborhood of e, so it generates G when G is connected. This shows
that if G is connected, then the assumption in Theorem 4.8 (b) holds, and hence λδi < 0
for δ ∈ Ĝ+ . From the convergence of the series in (4.25), it is easy to see that λδi should
converge to −∞ as δ leaves any finite subset of Ĝ+ . This implies that there is a largest
112
number, denoted by −λ, in the set of negative numbers λδi for δ ∈ Ĝ+ and 1 ≤ i ≤ d(δ).
Using Schwartz inequality, as for proving the absolute and uniform convergence of the series
(4.19) in the proof of Theorem 4.8, we can show that for t > ε > 0,
|pt − 1| ≤
∑
d(δ) |Tr[Aδ (t) U δ ]| (where Aδ (t) = exp[tLU δ ∗ (e)])
δ∈Ĝ+
≤ ∥pε/2 ∥2 {
∑
d(δ) Tr[Aδ (t − ε/2)Aδ (t − ε/2)∗ ]}1/2
δ∈Ĝ+
= ∥pε/2 ∥2 {
∑
d(δ)
∑
d(δ)
i=1
δ∈Ĝ+
≤ ∥pε/2 ∥2 {e−2λ(t−ε)
exp[2λδi (t − ε/2)]}1/2
∑
d(δ)
d(δ)
δ∈Ĝ+
∑
exp(2λδi (ε/2))}1/2
i=1
≤ e−λt eλε ∥pε/2 ∥2 ∥pε/2 − 1∥2 ,
where the last inequality above follows from (4.25). This proves the inequality for ∥pt − 1∥∞
in (b). By this inequality, ∥pt − 1∥2 ≤ c2 e−λt for t > ε. However, by (4.25), ∥pt − 1∥22 ≥
d(δ) exp(2λδi t) for any δ ∈ Ĝ+ and 1 ≤ i ≤ d(δ). This proves the inequalities for ∥pt − 1∥2 .
By ∥pt − 1∥2 ≤ c2 e−λt and the Schwartz inequality, ∥µt − ρG ∥tv ≤ c2 e−λt . However, since
|Uiiδ | ≤ 1 and ρG (Uiiδ ) = 0 for δ ∈ Ĝ+ ,
d(δ)
∥µt − ρG ∥tv ≥
|µt (Uiiδ )|
= |Aδ (t)ii | =
∑
δ
|(Qδ )ij |2 eλj t .
j=1
For any j, (Qδ )ij ̸= 0 for some i, this completes the proof of (b). 2
Remark 4.13 The range of exponent β in Theorem 4.12 is β > d. It is not possible to allow
∫
β fall below d, because if η ≍ r−β .ρG at e for some β < d, then η(G) ≍ 01 r−β rd−1 dr < ∞.
In this case, if there is no diffusion part, then the Lévy process is a pure jump process that
jumps at a sequence of stopping times τn ↑ ∞, and hence cannot have a distribution density.
We note also that it is not possible to have a Lévy measure η on G that asymptotically
dominates r−β .ρG at e with β ≥ d + 2. Indeed, if η ≍ r−β dx at x = e for β ≥ d + 2, then
∫
∫ 1 2 −β d−1
2
r dr = ∞, and hence η cannot be a Lévy measure.
r≤1 r dη ≍ 0 r r
4.4
bi-invariant Lévy processes
A Lévy process gt in a Lie group G is called bi-invariant if as a Markov process, its transition
semigroup Pt is bi-invariant, that is, Pt (f ◦ lg ) = (Pt f ) ◦ lg and Pt (f ◦ rg ) = (Pt f ) ◦ rg for any
t ∈ R+ , f ∈ B+ (G) and g ∈ G, where lg and rg : G → G are, respectively, the left and right
113
translations x 7→ gx and x 7→ xg. A bi-invariant Lévy process in G may also be defined as a
bi-invariant Markov process with rcll paths and an infinite life time.
Because as a Lévy process, its transition semigroup Pt is already assumed to be left
invariant, so a bi-invariant Lévy process is the same thing as a right invariant Lévy process.
It is also the same as a conjugate invariant Lévy process in the sense that Pt (f ◦cg ) = (Pt f )◦cg ,
where cg : G → G is the conjugation map x 7→ gxg −1 .
Let µt be the convolution semigroup associated to the Lévy process gt . Because Pt f (x) =
µ(f ◦ lx ), it is easy to see that gt is bi-invariant if and only if µt is conjugate invariant, that
d
is, cg µt = µt for any g ∈ G and t ≥ 0. When g0 = e, this is also equivalent to ggt g −1 = gt
(equal in distribution) for any g ∈ G and t ≥ 0.
Let ϕ1 , . . . , ϕd be exponential coordinate functions associated to the basis {ξ1 , . . . , ξd } of
∑
the Lie algebra g of G, that is, x = exp[ di=1 ϕi (x)ξi ] for x near e. Then for any g ∈ G,
∑
gxg −1 = exp[ di=1 ϕi (x)Ad(g)ξi ]. It follows that if G is a compact Lie group, then for
∑
∑
any g ∈ G, di=1 ϕi (gxg −1 )ξi = di=1 ϕi (x)Ad(g)ξi for x near e. Replacing ϕi by ϕϕi for a
conjugate invariant ϕ ∈ Cc∞ (G) such that ϕ = 1 near e and ϕ = 0 outside a sufficiently small
neighborhood of e, we may assume that on a compact Lie group G,
∀g, x ∈ G,
d
∑
ϕi (gxg −1 )ξi =
i=1
d
∑
ϕi (x)Ad(g)ξi .
(4.35)
i=1
In fact, by choosing the above ϕ to be invariant also under the inverse map, we may assume
both (4.35) and (4.22) hold on a compact Lie group G.
A d × d symmetric matrix bij is called Ad(G)-invariant if for any g ∈ G,
d
∑
bpq [Ad(g)]ip [Ad(g)]jq = bij
p,q=1
where [Ad(g)] is the matrix representing the map Ad(g): g → g under the basis {ξ1 , . . . , ξd },
∑
that is, Ad(g)ξj = di=1 [Ad(g)]ij ξi .
An element ξ ∈ g is called Ad(G)-invariant if Ad(g)ξ = ξ for any g ∈ G. Such a ξ
belongs to the center of g, and hence if G has no center, or has a discrete center, then g
has no Ad(G)-invariant element. The following result provides a simple criterion for the
bi-invariance of a Lévy process gt .
Proposition 4.14 Let µt be a continuous convolution semigroup on a compact Lie group G
with µ0 = δe and characteristics (ξ0 , aij , η). Assume the coordinate functions ϕi are chosen to
satisfy (4.35). Then µt is conjugate invariant if and only if ξ0 and aij are Ad(G)-invariant,
and η is conjugate invariant.
Note: Because aij and η do not depend on the choice of coordinate functions ϕi , without
assuming (4.35), the conjugate invariance of µt implies the Ad(G)-invariance of aij and the
conjugate invariance of η.
114
Proof of Proposition 4.14: Suppose µt is conjugate invariant. Then for f ∈ Cc∞ (G), and
g ∈ G, µt (f ◦ cg ) = µt f . Let L be the generator of µt . Then L(f ◦ cg )(e) = (d/dt)µt (f ◦
cg ) |t=0 = (d/dt)µt (f ) |t=0 = Lf (e). For ξ ∈ g, because ξ(f ◦ cg )(e) = (d/dt)f (getξ g −1 ) |t=0 =
[Ad(g)ξ]f (e), by (2.3), we can write L(f ◦ cg )(e) as follows:
L(f ◦ cg )(e) =
∫
+
d
1 ∑
aij [Ad(g)ξi ][Ad(g)ξj ]f (e) + [Ad(g)ξ0 ]f (e)
2 i,j=1
{f (σ) − f (e) −
d
∑
ϕi (g −1 σg)[Ad(g)ξi ]f (e)}(cg η)(dσ).
(4.36)
i=1
This is equal to Lf (e) given by (2.3). If f vanishes near e, then cg η(f ) = L(f ◦ cg )(e) =
∑
∑
Lf (e) = η(f ), and hence cg η = η. Because of (4.35), di=1 ϕi (g −1 σg)[Ad(g)ξi ] = di=1 ϕi (σ)ξi ,
it follows from L(f ◦ cg )(e) = Lf (e) that
d
d
1 ∑
1 ∑
aij [Ad(g)ξi ][Ad(g)ξj ]f (e) + [Ad(g)ξ0 ]f (e) =
aij ξi ξj f (e) + ξ0 f (e).
2 i,j=1
2 i,j=1
This being true for any f ∈ Cc∞ (G) implies the Ad(G)-invariance of aij and ξ0 .
Next assume ξ0 and aij are Ad(G)-invariant, and η is conjugate invariant. Then for g ∈ G
and f ∈ Cc∞ (G), L(f ◦cg )(e) = Lf (e). To show µt is conjugate invariant, let µ′t = cg µt . Then
µ′t is a continuous convolution semigroup with generator L′ given by L′ f = [L(f ◦ cg )] ◦ c−1
g .
The invariance assumption on (ξ0 , aij , η) implies L′ f (e) = Lf (e). Because both operators L
and L′ are left invariant, it follows L′ f = Lf on G. By Theorem 2.1, µ′t = µt . 2
Proposition 4.15 Let µt be a conjugate invariant continuous convolution semigroup on a
compact Lie group G with µ0 = δ0 . If it has a hypo-ellitic diffusion part (see §4.2), then it
has a nondegenerate diffusion part.
Proof: Let m be the rank of a = {aij }, and let ζ1 , . . . , ζm ∈ g be given in (4.15). They
may be extended to form a basis of g. Under this basis, the covariance matrix a is given
by aij = δij for i, j ≤ m and aij = 0 for all other (i, j). By the Ad(G)-invariance of a,
∑
∑
p [Ad(g)]ip [Ad(g)]jp = δij for i ≤ m and
p [Ad(g)]ip [Ad(g)]jp = 0 for i > m. Because the
matrix [Ad(g)] is invertible, this implies m = d. 2
Recall that ψδ = χδ /d(δ) is the normalized character. Because Re(ψδ ) takes the maximum
value ψδ (e) = 1 at e, all its first order derivatives vanish at e. It follows that
| Re(ψδ ) − 1| = O(|x|2 ).
∫
Therefore, by the condition (2.4) on the Lévy measure η, the integral (1 − Re ψδ )dη in the
following proposition exists. Because |ψδ | ≤ 1, this integral is in fact non-negative.
115
Proposition 4.16 Let µt be a conjugate invariant continuous convolution semigroup on a
compact Lie group G with µ0 = δe and generator L. Then for δ ∈ Ĝ,
µt (ψδ ) = etL(ψδ )(e) .
(4.37)
Moreover, letting (ξ0 , aij , η) be the characteristics of µt ,
|etL(ψδ )(e) | = etRe[L(ψδ )(e)] = exp{t[λδ −
∫
G
(1 − Re ψδ )dη]},
(4.38)
∑
where λδ = (1/2) di,j=1 aij ξi ξj ψδ (e) is real and is ≤ 0. If the assumption in Theorem 4.8 (b)
is also satisfied, then Re[L(ψδ )(e)] < 0 for δ ∈ Ĝ+ .
Proof: By the conjugate invariance of µt and (4.6),
∫
µt+s (ψδ ) =
∫
=
∫
ψδ (uv)µt (du)µs (dv) =
ψδ (gug −1 v)µt (du)µs (dv)dg
ψδ (u)ψδ (v)µt (du)µs (dv) = µt (ψδ )µs (ψδ ).
This combined with µ0 (ψδ ) = ψδ (e) = 1 implies that µt (ψδ ) = ety for some complex number
y. We have y = (d/dt)µt (ψδ ) |t=0 = Lψδ (e) and, hence, µt (ψδ ) = etL(ψδ )(e) . This proves (4.37).
For ξ ∈ g, let ξ δ = ξU δ (e). Like ξ˜ = ξU δ ∗ (e) in the proof of Theorem 4.8, ξ δ is a
skew Hermitian matrix. Because ξ(ψδ )(e) = Tr(ξ δ )/d(δ), ξ(Re ψδ )(e) = 0. Let ζi be defined
∑d
∑
∑
by (4.15) so that di,j=1 aij ξi ξj = m
i,j=1 aij ξi ξj ψδ (e) =
i=1 ζi ζi . It follows that 2λδ =
∑m δ δ
∑m δ∗ δ
Tr[ i=1 ζi ζi ]/d(δ) = −Tr[ i=1 ζi ζi ]/d(δ) ≤ 0. Now (4.38) follows from the formula (2.3)
for the generator L.
Because U δ (eξ ) = exp(ξ δ ), if ξ δ = 0, then U δ (eξ ) = I, the identity matrix. Note that
∫
Re(ψδ ) ≤ 1, and Re(ψδ ) = 1 only when U δ = I. Because ReL(ψδ )(e) = λδ − (1 − Re ψδ )dη,
it follows that if Re[L(ψδ )(e)] = 0, then U δ (g) = I for g contained in the closed subgroup
generated by exp(ζi ), 1 ≤ i ≤ m, and supp(η). If the assumption in Theorem 4.8 (b) holds,
then U δ (g) = I for any g ∈ G. This is impossible for δ ∈ Ĝ+ . 2
Theorem 4.17 Let G be a compact Lie group of dimension d, and let µt be a conjugate
invariant continuous convolution semigroup on G with µ0 = δe and generator L. Assume
either µt has a nondegenerate diffusion part, or its Lévy measure η asymptotically dominates
r−β .ρG at e for some β > d (defined in §4.3), where r is the Riemannian distance to e (this
condition does not depend on the choice of a Riemannian metric on G).
(a) For t > 0, µt has a density pt ∈ L2 (G) and for g ∈ G,
pt (g) = 1 +
∑
µt (χ̄δ )χδ (g) = 1 +
δ∈Ĝ+
∑
d(δ)etL(ψδ )(e) χδ (g),
(4.39)
δ∈Ĝ+
where the series converges absolutely and uniformly for (t, g) ∈ [ε, ∞) × G for any fixed
ε > 0.
116
(b) If G is connected, or more generally if the assumption of Theorem 4.8 (b) holds, then
L[Re(ψδ )](e), δ ∈ Ĝ+ , are all negative and they have a maximum −λ < 0. Moreover, for
any ε > 0, there are constants 0 < c1 < c2 such that for t > ε,
∥pt − 1∥∞ ≤ c2 e−λt ,
c1 e−λt ≤ ∥pt − 1∥2 ≤ c2 e−λt ,
c1 e−λt ≤ ∥µt − ρG ∥tv ≤ c2 e−λt .
Note: It will be shown in Remark 5.23 that pt (g) in (a) is in fact smooth in (t, g).
Proof of Theorem 4.17: Suppose µt has an L2 density pt for t > 0. Then pt is conjugate
invariant and hence, by (4.7) and (4.37), may be expanded into a Fourier series in terms of
irreducible characters as in (4.39) in L2 -sense.
By (4.38), the Parseval identity associated to the Fourier series in (4.39) is
∥pt ∥22 = 1 +
∑
d(δ)2 e2tRe[L(ψδ )(e)] = 1 +
δ∈Ĝ+
∑
d(δ)2 e−2t[|λδ |+
∫
(1−Reψδ )dη]
.
(4.40)
δ∈Ĝ+
The nonnegative series in (4.40) converges if and only if the Fourier series in (4.39) converges
in L2 (G). Because |χδ | ≤ d(δ) and |etL(ψδ )(e) | = etRe[L(ψδ )(e)] , the converges of the series in
(4.40) implies the absolute and uniform convergence of the series in (4.39).
Assume µt has a nondegenerate diffusion part. If η = 0, then the associated Lévy process,
as a nondegenerate diffusion process, has a distribution density pt ∈ L2 (G) for t > 0. Thus,
(4.39) holds, and hence the series in (4.40) converges. When η is nonzero, the terms of series
in (4.40) can only be smaller, so the series still converges. Then the series in (4.39) defines
a function pt such that ρG (pt χ̄δ ) = µt (χ̄δ ) for any δ ∈ Ĝ, and hence pt is the density of µt .
Suppose the series in (4.40) converges for a given Lévy measure η without a diffusion part.
If η is modified outside a neighborhood U of e, then, because η(U c ) < ∞, the change in
Re[L(ψδ )(e)] is controlled by a fixed constant term, so the convergence of the series in (4.40)
is not affected. Adding a diffusion part or replacing η by a larger Lévy measure will only
make Re[L(ψδ )(e)] smaller, so the series in (4.40) will still converge. The same ∫holds when
η is multiplied by a constant as the constant can be absorbed by t in e−2t[λδ + (1−Reψδ )dη] .
Therefore, if η is replaced by a Lévy measure that asymptotically dominates η, then the
series in (4.40) will still converge.
α
α
As in the proof of Theorem 4.12, for α ∈ (0, 1), let µB
t , νt and µt be, respectively,
the continuous convolution semigroups associated to the Riemannian Brownian motion in
G, the α-stable subordinator in R and the subordinated Brownian motion in G, such that
∫
∫
∫ ∞ −su α
α
νt (du) = e−ts = exp[−t 0∞ (1 − e−su )ξ α (du)] for s > 0, and µαt = 0∞ µB
u (·)νt (du),
0 e
where ξ α (du) = αu−1−α du/Γ(1−α) is the Lévy measure of νtα . Let ht and hαt be, respectively,
2
α
α
the densities of µB
t and µt for t > 0. Then ht ∈ L (G), and this holds on any complete
α
Riemannian manifold. Note that both µB
t and µt are conjugate invariant. The parseval
117
identity associated to the Fourier series of hαt is ∥hαt ∥22 = 1 +
γδ = λδ (≤ 0) in Proposition 4.16 when µt = µB
t . Then
∫
µαt (ψδ ) =
∫
∞
= exp[−t
0
∞
0
∫
α
µB
u (ψδ )νt (du) =
−u|γδ |
(1 − e
∞
δ∈Ĝ+
d(δ)2 |µαt (ψδ )|2 . Let
e−u|γδ | ν α (du)
0
∫
α
)ξ (du)] = exp[−t
G
∫
∑
(1 − ψδ )dη α ],
∑
∫
where η α (dx) = [ 0∞ hu (x)ξ α (du)]dx. It follows that ∥hαt ∥22 = 1+ δ∈Ĝ+ d(δ)2 e−2t (1−Re ψδ )dη .
Comparing this with (4.40) shows that if η asymptotically dominates η α at e, then the series
in (4.40) converges. By (4.34), η α ≍ r−n−2α .ρG at e. Therefore, if η asymptotically dominates
r−d−2α .ρG at e, then (4.40) converges. This proves (a).
As in the proof of Theorem 4.12, it can be shown that if G is connected, then the assumtion of Theorem 4.8 (b) holds. By Proposition 4.16, under this assumption, Re[L(ψδ )(e)] < 0
for δ ∈ Ĝ+ . The convergence of the series in (4.40) implies that Re[L(ψδ )(e)] → −∞ as δ
leaves any finite subset of Ĝ+ . In particular, this implies that the set of negative numbers
Re[L(ψδ )(e)], δ ∈ Ĝ+ , has a largest number −λ < 0. For t > ε > 0,
|pt − 1| ≤ e−λ(t−ε)
≤ e
−λ(t−ε)
∑
d(δ) |e
2
∑
α
d(δ)|eεL(ψδ )(e) χδ |
δ
εL(ψδ )(e)
| ≤ e−λ(t−ε) ∥pε/2 ∥22 ,
δ
this proves the inequality for ∥pt −1∥∞ in (b), and from which the upper bounds for ∥pt −1∥2
and ∥µt − ρG ∥tv follow. The lower bounds follow from ∥pt − 1∥22 ≥ d(δ)2 |etL(ψδ )(e) |2 and
∥µt − ρG ∥tv ≥ |µt (ψδ )| = |etL(ψδ )(e) | for δ ∈ Ĝ+ . Part (b) is proved. 2
A Lie algebra g with dim(g) > 1 is called simple if it does not contain any ideal except
{0} and g. It is called semisimple if it does not contain any abelian ideal except {0}. Here,
an ideal i of g is called abelian if [i, i] = {0}. A Lie group G is called simple or semisimple if
its Lie algebra g is so.
We note that the adjoint action of a Lie group G on its Lie algebra g, given by (g, ξ) 7→
Ad(g)ξ, has kernel A = {g ∈ G; Ad(g)ζ = ζ for any ζ ∈ g}, which is a closed subgroup of G
with Lie algebra a = {ξ ∈ g; ad(ξ)ζ = 0 for any ζ ∈ g}. It is clear that a is an abelian ideal
of g. It follows that if G is semisimple, then A is a discrete subgroup. Therefore, a connected
semisimple Lie group is a class C Lie group (see the discussion preceding Theorem 3.18).
Proposition 4.18 Let G be a compact semisimple Lie group, and let µt be a conjugate invariant continuous convolution semigroup on G with µ0 = δe , generator L and characteristics
(ξ0 , aij , η). Then
µt (ψδ ) = exp[tL(ψδ )(e)] = exp{t[
∫
d
1 ∑
aij ξi ξj ψδ (e) + (ψδ − 1)dη]}.
2 i,j=1
G
Moreover, if G is simple, then aij = aδij for some constant a ≥ 0.
118
(4.41)
Note 1: (4.41) may be regarded as a type of Lévy -Khinchin formula for conjugate invariant
Lévy processes in a semisimple compact Lie group G. Note that any finite and conjugate
invariant measure µ on G is completely determined by µ(ψδ ), as δ ranges over Ĝ, because µ is
determined by µ(f ) for conjugate invariant f ∈ C(G), and such an f may be approximated
by linear combinations of ψδ uniformly on G.
Note 2: The basis {ξ1 , . . . , ξd } of g determines a left invariant Riemannian metric on G
under which this basis is orthonormal (see §3.4). When G is simple, the diffusion part of the
∑
generator L is (a/2)∆G , where ∆G = di=1 ξi ξi is the Laplace-Betami operator on G under
the above metric (see Proposition 3.21).
Proof of Proposition 4.18: Note that a semisimple Lie algebra has a zero center because
the center is an abelian ideal. If G is semisimple, then its Lie algebra g has no nonzero Ad(G)invariant element, because Ad(g)ζ = ζ for some ζ ∈ g and any g ∈ G implies [ξ, ζ] = 0 for
any ξ ∈ g, so ζ belongs to the center of g, which is zero. Because ψδ is conjugate invariant,
∫
∫
for ξ ∈ g and g ∈ G, ξψδ (e) = [Ad(g)ξ]ψδ (e) = G dgAd(g)ξψδ (e). Because G dgAd(g)ξ is
conjugate invariant, ξψδ (e) = 0. It follows that 1 − ψδ is absolutely integrable with respect
to the Lévy measure η. By the expression (2.3) for the generator L, this proves (a).
Because any subspace of g invariant under the adjoint action Ad(G) of G on g is an
ideal, if G is simple, then Ad(G) is irreducible. Then by Lemma 3.6, any Ad(G)-invariant
symmetric matrix is a multiple of the identity matrix. This shows aij = aδij for some
constant a ≥ 0 2
4.5
Examples
Example 4.19 (SU (2)): For n ≥ 2, let SU (n) be the connected compact group of unitary
matrices of determinant 1. This is a closed subgroup of the unitary group U (n) and is called
the special unitary group. Its Lie algebra su(n) is the space of n×n skew-Hermitian matrices
of trace 0. In particular, G = SU (2) is the group of matrices of the form


α
β 

,
−β̄ ᾱ
α, β ∈ C with |α|2 + |β|2 = 1.
(4.42)
√
Note that any g ∈ G is conjugate to a = diag(eθi , e−θi ) for a unique θ ∈ [0, π] where i = −1,
that is, g = kak −1 for some k ∈ G. It follows that if f is a conjugate-invariant function on
G, then f (g) = f (a). We may write f (θ) for f (g) when g is conjugate to a = diag(eθi , e−θi ),
0 ≤ θ ≤ π.
We now describe the irreducible representations of G = SU (2). Let Vn be the linear
space of homogeneous polynomials of degree n in two complex variables z1 and z2 . Let G
119
act trivially on V0 = C, and for n ≥ 1, let G act on Vn by the map
(G × Vn ) ∋ (g, P ) 7→ gP ∈ Vn ,
(4.43)
where gP (z1 , z2 ) = P (z1 g11 + z2 g21 , z1 g12 + z2 g22 ) for g = {gpq } ∈ G. The G-action is a
representation of G on Vn of dimension n + 1, which will be denoted by Un . It can be shown
(see [12, II.5]) that all irreducible representations of G, up to equivalence, are given by Un
for n = 0, 1, 2, 3, . . ..
To compute the character χn of Un , take the basis {z1k z2n−k ; 0 ≤ k ≤ n} of Vn . Since χn
is conjugate-invariant, it suffices to determine χn (θ). Because Un (θ)(z1k z2n−k ) = ekθi e−(n−k)θi
(z1k z2n−k ),
n
∑
sin((n + 1)θ)
χn (θ) = Tr[Un (θ)] =
e2kθi e−nθi =
(4.44)
sin θ
k=0
for 0 ≤ θ ≤ π. Note that χn (0) = (n + 1) and χn (π) = (−1)n (n + 1), obtained as limit of
the above expression as θ → 0 and π.
The matrices




1 i 0 
,
ξ1 = 
2 0 −i
1 
1 0
ξ2 = 
,
2 −1 0


1 0 i 
ξ3 = 
.
2 i 0
(4.45)
form a basis of g = su(2), the Lie algebra of G = SU (2), which is orthonormal under the
Ad(G)-invariant inner product ⟨ξ, η⟩ = 2Tr(ξη ∗ ). Let G be equipped with the bi-invariant
Riemannian metric induced by this inner product. By §3.4, the Laplace-Beltrami operator
on G = SU (2) is given by
∆SU (2) = ξ1 ξ1 + ξ2 ξ2 + ξ3 ξ3 .
Because the adjoint action Ad(G) is transitive on unit sphere in g = su(2), ξ1 ξ1 χn (e) =
ξ2 ξ2 χn (e) = ξ3 ξ3 χn (e). Note that ξ1 ξ1 χn (e) = (1/4)(d2 /dθ2 )χn (θ) |θ=0 = χ′′n (0). An elementary computation shows that χ′′n (0) = −(1/12)n(n + 1)(n + 2), and hence
∆SU (2) χn (e) = −
n(n + 1)(n + 2)
.
4
(4.46)
Let F : G = SU (2) → [0, π] be the map g 7→ θ if g is conjugate to diag(eθi , e−θi ). For
any conjugate invariant measure η on G, let η ′ = F η be its projection to [0, π]. Then η is a
∫
Lévy measure on G if and only if η ′ ({0}) = 0 and 0π θ2 η ′ (dθ) < ∞.
∑
Because G = SU (2) is simple, aij = aδij in Theorem 4.17. By (4.46), (1/2) 3i=1 ξi ξi ψn (e) =
−n(n + 2)/8, and hence the formula (4.39) takes the following form on G = SU (2):
pt (θ) = 1 +
∞
∑
∫π
(n + 1)e−t{an(n+2)/8+
0
[1−χn (β)]η ′ (dβ)}
n=1
120
χn (θ),
χn (θ) =
sin((n + 1)θ)
, (4.47)
sin θ
for 0 ≤ θ ≤ π and t > 0. Moreover,
−λ ≤ −
∫ π
3a
− sup [1 − χn (β)]η ′ (dβ)
8
n≥1 0
in Theorem 4.17 (b).
Normalized Haar measure on SU (2) Note that pt (θ) given in (4.47) is the density of µt
with respect to the normalized Haar measure ρSU (2) on SU (2). To do an actual computation,
one may need an expression for ρSU (2) . For this purpose, we will describe an identification
of SU (2) with the unit sphere S 3 embedded in the quaternion algebra, which is just R4 as a
vector space with basis {1, I, J, K} obeying the multiplication rule:
II = JJ = KK = −1, IJ = −JI = K, JK = −KJ = I, KI = −IK = J.
The Euclidean norm is compatible with the quaternion product in the sense that |xy| = |x| |y|
for x, y ∈ R4 . Any nonzero quaternion x = (a + bI + cJ + dK) has a multiplicative inverse
x−1 = (a − bI − cJ − dK)/|x|2 . To each g ∈ SU (2) given by the matrix in (4.42) with
α = x0 + x1 i and β = x2 + x3 i in C, associate the quaternion
ĝ = x0 + x1 I + x2 J + x3 K
(4.48)
of unit norm. The map: g 7→ ĝ provides is a bijection from SU (2) onto the unit sphere S 3
in R4 which preserves the products on SU (2) and on R4 (the quaternion algebra).
We note that the normalized Haar measure ρS 3 on S 3 is the uniform distribution on
S 3 . This is because the left translations on S 3 preserves the Euclidean norm and so are
orthogonal transformations which preserves the uniform distribution.
The spherical polar coordinates θ, ψ, ϕ on S 3 are defined by
x0 = cos θ,
x1 = sin θ cos ψ,
x2 = sin θ sin ψ cos ϕ,
x3 = sin θ sin ψ sin ϕ,
(4.49)
with 0 ≤ θ ≤ π, 0 ≤ ψ ≤ π and 0 ≤ ϕ ≤ 2π. Under the identification of SU (2) and
S 3 , diag(eiθ , e−iθ ) in SU (2) corresponds to (cos θ, sin θ, 0, 0) in S 3 , and the normalized Haar
measure ρSU (2) on SU (2) is the uniform distribution ρS 3 on S 3 given by
ρS 3 (dθ, dψ, dϕ) =
1
sin2 θ sin ψ dθ dψ dϕ.
2π 2
(4.50)
Thus for a conjugate invariant function f (g) on SU (2), written as f (θ),
µt (f ) =
2∫π
f (θ)pt (θ) sin2 θ dθ.
π 0
(4.51)
A covering map from SU (2) onto SO(3) Before presenting the next example for G =
SO(3), we will now describe a covering map p from SU (2) onto SO(3). Let R3 be identified
121
with the subspace of R4 spanned by I, J, K, the subspace of pure quaternions. For g ∈ SU (2),
the map: R4 → R4 given by x 7→ ĝxĝ −1 , where ĝ = x0 + x1 I + x2 J + x3 K is given in (4.48),
preserves both the norm and the pure quaternion subspace R3 , and hence restrict to a norm
preserving map: R3 → R3 . Because SU (2) is connected, this map is an element of SO(3).
We can now define
p : SU (2) → SO(3)
given by
p(g)x = ĝxĝ −1 for x ∈ R3 .
(4.52)
This is a Lie group homomorphism from SU (2) onto SO(3).
We now show that p is a double covering map from SU (2) onto SO(3) with kernel Ker(p)
containing only two elements: I2 (2 × 2 identity matrix) and −I2 . It is clear that p(±I2 ) is
the identity element of SO(3). Because |ĝ| = 1 for g ∈ SU (2), a simple computation shows
that the coefficient of I in p(g)I is x21 + x20 − x23 − x22 , and the coefficient of J in p(g)J is
x20 +x22 −x23 −x21 . If g ∈Ker(p), then p(g)I = I and p(g)J = J, and hence x21 +x20 −x23 −x22 = 1
and x20 + x22 − x23 − x21 = 1. Adding these two equations yields x20 − x23 = 1. This implies
x20 = 1, and x1 = x2 = x3 = 0, and hence g = ±I2 . This proves Ker(p) = {I2 , −I2 }. Because
both SU (2) and SO(3) are connected Lie groups of dim 3, as a Lie group homomorphism
with a kernel of two elements, p is necessarily onto and is a double covering map.
Example 4.20 (SO(3)): Let G = SO(3). Via the covering map p, an irreducible representation Ũ of G becomes an irreducible representation U = Ũ ◦ p of SU (2), and for an
irreducible representation U of SU (2), there is a representation Ũ of G such that U = Ũ ◦ p
if and only if U (−I2 ) is an identity matrix. In Example 4.19, all irreducible representations
of SU (2) are determined to be Un , given by (4.43), for n = 0, 1, 2, 3, . . . with dimension n + 1.
It is easy to see that only for even n, Un (−I2 ) is an identity matrix, therefore, the irreducible
representations of G are Ũn with U2n = Ũn ◦ p and dimensions 2n + 1 for n = 0, 1, 2, 3, . . ..
We will identify SU (2) with the unit sphere S 3 in R4 (quaternion algebra) via the map
g 7→ ĝ in (4.48). Then p becomes a covering map from S 3 onto G = SO(3). Regard
R3 as the space of pure quaternions with basis {I, J, K}. For H ∈ S 2 = S 3 ∩ R3 , let
uH (θ) = cos θ + H sin θ ∈ S 3 and let rH (θ) ∈ SO(3) be the rotation of angle θ in R3 around
H. In particular, uI (θ), uJ (θ), uK (θ) in S 3 are identified with


eθi 0
,

0 e−θi


cos θ
sin θ 

,
− sin θ cos θ


cos θ i sin θ 

i sin θ cos θ
in SU (2) respectively, and






1 0
0
cos θ
0 sin θ
cos θ − sin θ 0











rI (θ) =  0 cos θ − sin θ  , rJ (θ) =  0
1 0
0 
 , rK (θ) =  sin θ cos θ
.
0 sin θ cos θ
− sin θ 0 cos θ
0
0
1
122
A direct computation shows that p(uI (θ)) = rI (2θ). Using conjugation, it can be shown that
∀H ∈ S 2 ,
p(uH (θ)) = rH (2θ).
(4.53)
Because any g ∈ G = SO(3) is conjugate to rI (θ) for a unique θ ∈ [0, π], any conjugate
invariant function f (g) on G, such as a character of G, may be regarded as a function
of θ ∈ [0, π], and may be written as f (θ). Because the character χ̃n of the irreducible
representation Ũn of G is determined by χ2n = χ̃n ◦ p, by (4.44),
χ̃n (θ) =
sin((2n + 1)θ/2)
.
sin(θ/2)
(4.54)
Note that under the identification of SU (2) and S 3 , the basis vectors ξ1 , ξ2 , ξ3 of su(2)
given by (4.45), which are orthonormal under the inner product ⟨ξ, η⟩ = 2Tr(ξη ∗ ), are tangent
vectors of curves θ 7→ uI (θ), θ 7→ uJ (θ), θ 7→ uK (θ) respectively at θ = 0. Let ξ˜j = Dp(ξj )
for j = 1, 2, 3. Then


0 0 0



ξ˜1 = 
 0 0 −1  ,
0 1 0


0
0 1


ξ˜2 = 
0 0 
 0
,
−1 0 0


0 −1 0


ξ˜3 = 
0 
 1 0

0 0
0
form a basis of g = o(3), the Lie algebra of G = SO(3), which are orthonormal under
the Ad(G)-invariant inner product ⟨ξ, η⟩ = (1/2)Tr(ξη ∗ ) on o(3). The Laplace-Beltrami
operator determined by the induced bi-invariant Riemannian metric on G = SO(3), given
by ∆SO(3) = ξ˜1 ξ˜1 + ξ˜2 ξ˜2 + ξ˜3 ξ˜3 , is p-related to ∆SU (2) , and hence by (4.46),
∆SO(3) χ̃n (e) = −2n(2n + 1)(2n + 2)/4 = −n(n + 1)(2n + 1)
(4.55)
for n = 0, 1, 2, 3, . . ..
Let F : G = SO(3) → [0, π] be the map g 7→ θ if g is conjugate to rI (θ). For any
conjugate invariant measure η on G, let η ′ = F η be its projection on [0, π]. Then η is a
∫
Lévy measure on G if and only if η ′ ({0}) = 0 and 0π θ2 η ′ (dθ) < ∞.
Because SO(3) is simple, aij = aδij in Theorem 4.17 for some constant a ≥ 0, and (4.39)
takes the following form on G = SO(3).
pt (θ) = 1 +
∞
∑
∫π
(2n + 1)e−t{an(n+1)/2+
0
[1−χ̃n (β)]η ′ (dβ)}
χ̃n (θ),
n=1
for θ ∈ [0, π] and t > 0. Moreover,
−λ ≤ −a − sup
∫
n≥1
in Theorem 4.17 (b).
123
π
0
(1 − χ̃n )dη ′
χ̃n (θ) =
sin((2n + 1)θ/2)
,
sin(θ/2)
(4.56)
Note that pt is the density with respect to the normalized Haar measure ρSO(3) . Let
(θ, ψ, ϕ) denote the rotation of angle θ around the point on the unit sphere S 2 in R3 with
cartesian coordinates (cos ψ, sin ψ cos ϕ, sin ψ cos ϕ) for 0 ≤ θ ≤ π, 0 ≤ ψ ≤ π and 0 ≤ ϕ ≤
2π. Because the covering map p preserves the Haar measures, by (4.50) and (4.53),
θ
1
sin2 ( ) sin ψ dθ dψ dϕ.
2
2π
2
Thus for a conjugate invariant function f (g) on SO(3), written as f (θ),
ρSO(3) (dθ, dψ, dϕ) =
(4.57)
2∫π
θ
f (θ)pt (θ) sin2 ( )dθ.
(4.58)
π 0
2
Example 4.21 (O(3)): The othogonal group G = O(3) has two connected components:
G0 = SO(3) and (−I3 )G0 , where I3 is the 3 × 3 identity matrix. Note that
µt (f ) =
O(3) = SO(3) × (Z/2)
is a Lie group product, where Z/2 = {0, 1} is the additive group of integers modulo 2. By
[12, (4.15)], if G1 and G2 are two compact Lie groups, then the map (U, U ′ ) 7→ U ⊗U ′ (tensor
product) is a bijection from Ĝ1 × Ĝ2 onto (G1 × G2 )ˆ, where U and U ′ are (equivalence classes
of) irreducible unitary representations of G1 and G2 respectively. Then the irreducible
unitary representations of G = O(3) may be obtained from those of SO(3), discussed in
Example 4.20, as follows:
For n ≥ 0, each representation Ũn of G0 = SO(3) corresponds to two irreducible representations of G = O(3), denoted as Ũn+ and Ũn− respectively, both are just Ũn when restricted
to G0 , with Ũn+ (−I3 ) = idn and Ũn− (−I3 ) = −idn , where idn is the identity map on the
representation space of Ũn .
−
+
−
Let χ̃+
n and χ̃n be the characters of the representations Ũn and Ũn respectively. Then
for n = 0, 1, 2, 3, . . .,
∀g ∈ G0 = SO(3),
sin((2n + 1)θ/2)
,
sin(θ/2)
−
χ̃−
n (g) = χ̃n (g) and χ̃n (−I3 g) = −χ̃n (g).
+
χ̃+
n (g) = χ̃n (−I3 g) = χ̃n (g) =
−
−
In particular, χ̃+
0 ≡ 1 is the trivial character, and for g ∈ G0 , χ̃0 (g) = 1 and χ̃0 (−I3 g) = −1.
The equality (4.39) in Theorem 4.17 (a) takes the following form on G = O(3): For g ∈ G,
pt (g) = 1 +
∞
∑
∫
(2n + 1)e−t[an(n+1)/2+
n=1
+e−2tη(−I3 G0 ) χ̃−
0 (g)
+
∞
∑
G
(1−χ̃n )dη]
χ̃n (g)
∫
−t[an(n+1)/2+
(2n + 1)e
G
(1−χ̃−
n )dη] −
χ̃n (g).
(4.59)
n=1
Note that when η(−I3 G0 ) = 0, µt is concentrated on G0 = SO(3), and pt given in (4.59) is
0 on −I3 G0 and is twice of the expression in (4.56) on G0 . This is because pt is the density
with respect to the normalized Haar measure on O(3) which has only 1/2 charge on SO(3).
124
Chapter 5
Spherical transform and
Lévy -Khinchin formula
The Fourier transform on Euclidean space is a powerful tool for probability analysis and
the celebrated Lévy -Khinchin formula is the Fourier transform of a convolution semigroup,
representing the distribution of a Lévy process. On a symmetric space, a special type of
homogeneous spaces, the spherical transform plays a similar role, and leads naturally to
a Lévy -Khinchin type formula. The purpose of this chapter is to obtain a spherical Lévy Khinchin formula, and use this formula together with the inverse spherical transform to prove
the existence of a smooth density for a convolution semigroup on a symmetric space, and to
obtain a representation for this density in terms of spherical functions. As an application,
the exponential convergence to the uniform distribution is obtained in the compact case.
The results will be stated in terms of convolution semigroups, and established under a
nondegenerate diffusion part or an asymptotic condition on the Lévy measure.
A spherical Lévy -Khinchin formula on symmetric spaces was first obtained in Gangolli
[29] for infinitely divisible distributions. The same formula was obtained in Applebaum [2]
and Liao-Wang [58] for convolution semigroups by a much simpler method based on Hunt’s
generator formula. Because an infinitely divisible distribution on a symmetric space can
be embedded in a convolution semigroup (see §3.3), the more difficult result in [29] can be
derived by the easier method in [2, 58].
The content of this chapter is an improved and extended version of [58]. The reader is
referred to Heyer [36] for an overview of spherical transforms and convolution semigroups,
including central limit problems. A generalized Gangolli-Lévy -khinchin formula on semisimple Lie groups without K-bi-invariance assumption was obtained in Applebaum-Dooley [4].
We would also like to draw reader’s attention to Picard-Savona [74] for the smoothness of
densities of more general type of Markov processes with jumps, and to Klyachko [49] for
using spherical functions to study random walks on symmetric spaces.
125
5.1
Spherical functions
In this chapter, let G be a connected Lie group with identity element e, and let X = G/K
be a homogeneous space with a compact subgroup K. As before, π: G → X is the natural
projection and o = eK ∈ X.
Recall that for g ∈ G, lg and rg are respectively the left and right translations on G.
A function f on G is called left (resp. right) invariant if f ◦ lg = f (resp. f ◦ rg = f ) for
any g ∈ G. It is called bi-invariant if it is both left and right invariant. It is called K-left
(resp. K-right, resp. K-bi-) invariant if the above holds for g ∈ K. A function f on X
is called G-invariant (resp. K-invariant) if f ◦ g = f for g ∈ G (resp. g ∈ K). Note that
a G-invariant function on X is necessarily a constant. Also recall that an operator T on
G is called left invariant if for f ∈ Dom(T ), f ◦ lg ∈ Dom(T ) and T (f ◦ lg ) = (T f ) ◦ lg for
any g ∈ G. The right invariant, K-left invariant and K-right invariant operators on G, and
G-invariant operators on X are defined similarly. The domain of a differential operator is
taken to be the space of smooth functions unless when explicitly stated otherwise.
Let D(G) denote the space of left invariant differential operators on G, let DK (G) be the
subspace of D(G) consisting of those which are also K-right invariant, and let D(X) be the
space of G-invariant differential operators on X = G/K.
A complex valued smooth function ϕ on G is called a spherical function if ϕ(e) = 1, if it
is K-bi-invariant and if it is a common eigenfunction of the operators in DK (G), that is,
∀T ∈ DK (G),
T ϕ = β(T, ϕ) ϕ
for some constant β(T, ϕ).
(5.1)
We may write dk for the normalized Haar measure ρK (dk) on K. By Proposition 2.2 in
[34, chapter IV], a nonzero complex valued continuous function ϕ on G is spherical if and
only if
∫
ϕ(xky)dk = ϕ(x)ϕ(y).
(5.2)
K
Because of its K-right invariance, a spherical function ϕ on G may be naturally regarded
as a function on X = G/K. Thus, a function ϕ on X will be called spherical if ϕ ◦ π
is a spherical function on G. Equivalently, a complex valued smooth function ϕ on X is
spherical if ϕ(o) = 1, if it is K-invariant and if it is a common eigenfunction of the Ginvariant differential operators on X, that is, if (5.1) holds with DK (G) replaced by D(X).
From its definition, it is clear that if ϕ is a spherical function on G or on X, then so is
its complex conjugate ϕ̄.
Let {ξ1 , . . . , ξd } be a basis of the Lie algebra g of G. For any integer m ≥ 0, an open
subset U of G and f ∈ C m (G), define
(m)
∥f ∥U
= sup |f (x)| +
x∈U
m
∑
∑
sup |ξi1 ξi2 · · · ξij f (x)|.
j=1 i1 ,i2 ,...,ij x∈U
126
(m)
Write ∥f ∥(m) for ∥f ∥G . Let Cbm (G) = {f ∈ C m (G); ∥f ∥(m) < ∞}.
Lemma 5.1 If ϕ is a bounded spherical function on G, then ϕ ∈ Cbm (G) for any m > 0.
Proof Because K is compact, G may be equipped with a left invariant Riemannian metric
that is also K-right invariant. Let ∆ be the Laplace operator on G under this metric. Then
∆ϕ = λϕ for some constant λ. Let U and V be open neighborhoods of e such that V
contains the closure U of U and is a diffeomorphic image of a neighborhood of 0 in g under
the exponential map. Let L be a strictly elliptic second order differential operator on V with
smooth coefficients. By the Schauder estimate ([30, corollary 6.3]), there is a constant c such
(2)
(0)
(0)
that if u and f are smooth functions on V such that Lu = f , then ∥u∥U ≤ c(∥u∥V +∥f ∥V ).
(2)
(0)
Applying the Schauder estimate to the equation ∆ϕ = λϕ, we obtain ∥ϕ∥U ≤ c(1+|λ|)∥ϕ∥V
for some constant c = c(U, V ). Using the left translation of U and V , and the left invariance
of ∆, it can be shown that ∥ϕ∥(2) ≤ c(1 + |λ|)∥ϕ∥(0) .
For any ξ ∈ g, F = ∆ξ − ξ∆ is a left invariant differential operator on G of second order.
Let W be open neighborhood of e such that U ⊂ W and W ⊂ V . Apply the Schauder
estimate to the equation ∆(ξϕ) = λ(ξϕ) + F ϕ, we have
(2)
∥ξϕ∥U
≤ c′ (∥ξϕ∥W + ∥λξϕ + F ϕ∥W ) ≤ c′ (1 + |λ|)∥ξϕ∥W + c′ ∥F ϕ∥W
(0)
(0)
(0)
(0)
≤ c′ (1 + |λ|)∥ϕ∥W + c′ ∥ϕ∥W ≤ c′′ (1 + |λ|)2 ∥ϕ∥(0)
(2)
(2)
for some constants c′ and c′′ . By the invariance as before, ∥ϕ∥(3) ≤ c3 (1 + |λ|)2 ∥ϕ∥(0) .
Inductively it can be shown that
∥ϕ∥(m) ≤ cm (1 + |λ|)m−1 ∥ϕ∥(0)
(5.3)
for some constant cm independent of ϕ. 2
For a finite K-invariant measure µ on X = G/K, the map
ϕ 7→ µ(ϕ)
(5.4)
from the set of bounded spherical functions ϕ on X (or some other set of spherical functions)
into the set C of complex numbers is called the spherical transform of µ.
Recall that the convolution of two K-invariant measures µ and ν on X is given by
∫
µ ∗ ν(f ) = f (xy)µ(dx)ν(dy) for f ∈ Bb (X), which is independent of the choice for the
section map S to represent xy = S(x)y and is a K-invariant measure on X.
Proposition 5.2 For any bounded spherical function ϕ on X, and two finite K-invariant
measures µ and ν on X,
µ ∗ ν(ϕ) = µ(ϕ)ν(ϕ).
(5.5)
127
Proof: Let ψ = ϕ◦π, which is a bounded spherical function on G. Then by the K-invariance
of µ and ν, and (5.2), with the choice of a section map S on X,
µ ∗ ν(ϕ) =
∫ ∫
∫
=
∫
∫
∫
∫
∫
ϕ(S(x)y)µ(dx)ν(dy) =
X
ϕ(S(x)ky)µ(dx)ν(dy)dk
X
∫K ∫X ∫X
ϕ(S(x)kπ(g))dk µ(dx)Sν(dg) =
∫G ∫X
=
K
∫
∫
ψ(S(x))ψ(g)µ(dx)Sν(dg) =
G
X
X
X
ψ(S(x)kg)dk µ(dx)Sν(dg)
G
X
K
ϕ(x)ϕ(y)µ(dx)ν(dy) = µ(ϕ)ν(ϕ). 2
Let p be an Ad(K)-invariant subspace of g complementary to the Lie algebra k of K, and
let a basis {ξ1 , . . . , ξd } of g be chosen so that ξ1 , . . . , ξn form a basis of p and ξn+1 , . . . , ξd
form a basis of k. A set of exponential coordinate functions on X, ϕ1 , . . . , ϕn ∈ Cc∞ (X), are
also chosen so (3.9) holds near o and (3.10) holds on X. By Proposition 3.2, any second
order G-invariant linear differential operator T on X with T 1 = 0 is given at point o by
T f (o) =
n
1 ∑
aij ξi ξj (f ◦ π)(e) + ξ0 (f ◦ π)(e)
2 i,j=1
(5.6)
for f ∈ Cc∞ (X), where {aij } is an Ad(K)-invariant symmetric matrix and ξ0 is an Ad(K)invariant vector in p. Because of the G-invariance, T is completely determined by its action
at o given by (5.6).
5.2
Lévy -Khinchin formula on homogeneous spaces
Let xt be a Lévy process in X = G/K with x0 = o, and let µt be the associated convolution
semigroup. Recall that each µt is K-invariant and is the distribution of xt . The generator
L of the Lévy process xt , or of the convolution semigroup µt , when restricted to Cc∞ (X), is
given by (3.11) in Theorem 3.3. Because of the G-invariance, L is completely determined by
its action at point o given by the following simpler expression: For f ∈ Cc∞ (X),
∫
Lf (o) = T f (o) +
X
[f (y) − f (o) −
n
∑
ϕi (y)ξi f (o)]η(dy),
(5.7)
i=1
∑
where T = (1/2) ni,j=1 aij ξi ξj + ξ0 is a G-invariant differential operator given by (5.6), and
η is the Lévy measure which is an K-invariant measure on X satisfying (3.12).
A bounded spherical function ϕ on X may not belong to the domain Dom(L) of the
generator L of a Lévy process in X, but Lϕ(o) given by (5.7) still makes sense. Moreover,
by Lemma 5.1 and (3.23) in §3.1,
Lϕ(o) =
d
µt (ϕ)|t=0 .
dt
128
(5.8)
Theorem 5.3 Let µt be a continuous convolution semigroup on X = G/K with µ0 = δo ,
and let L be its generator, given by (5.7) at o with T being the differential operator part of
L.
(a) For any bounded spherical function ϕ on X,
∫
µt (ϕ) = etLϕ(o) = exp{t[β(T, ϕ) +
X
(ϕ(x) − 1 −
n
∑
ϕi (x)ξi ϕ(o))η(dx)]},
(5.9)
i=1
where β(T, ϕ) is the eigenvalue of T for eigenfunction ϕ as given in (5.1).
(b) Assume there is no nonzero Ad(K)-invariant vector in p (that is, if ξ ∈ p with Ad(k)ξ =
ξ for any k ∈ K, then ξ = 0). Then ξϕ(o) = 0 for any ξ ∈ p, and hence the formula (5.9)
takes the following simpler form:
∫
µt (ϕ) = exp{t[β(T, ϕ) +
X
(ϕ(x) − 1)η(dx)]}.
(5.10)
Proof By (5.5), µs+t (ϕ) = µs (ϕ)µt (ϕ) for any s, t ≥ 0. Because µt (ϕ) is continuous in t, it
follows that µt (ϕ) = ety and by (5.8), y = (d/dt)µt (ϕ)|t=0 = Lϕ(o). This proves (a).
Because ϕ is K-invariant, for ξ ∈ p, ξϕ(o) = (d/dt)ϕ(etξ o) |t=0 = (d/dt)ϕ(ketξ o) |t=0 =
∫
(d/dt)ϕ(eAd(k)ξ o) |t=0 = [Ad(k)ξ]ϕ(o) for any k ∈ K. Then ξϕ(o) = [ K dkAd(k)ξ]ϕ(o). Note
∫
that K dkAd(k)ξ is an Ad(K)-invariant vector in p, which is zero by assumption. Then
ξϕ(o) = 0. This proves (b). 2
Example 5.4 A classical Lévy process in X = Rn is a translation invariant Markov process.
Then G is the group of translations x 7→ x+a, a ∈ Rn , which may be naturally identified with
Rn , and K = {0}. In this case, D(X) is the space of translation invariant linear differential
operators on Rn , which are just linear differential operators with constant coefficients. It
follows that the spherical functions are the exponentials
ϕλ (x) = ei
∑
j
λj xj
√
for x = (x1 , . . . , xn ) ∈ Rn and λ = (λ1 , . . . , λn ) ∈ Cn , where i = −1. The bounded
spherical functions are given by ϕλ (x) with λ ∈ Rn . The spherical transform λ → µ(ϕλ ) is
the usual Fourier transform of µ.
One may take ξj = ∂/∂xj and choose exponential coordinates ϕj so that ϕj (x) = xj near
∑
the origin 0. Then ξ0 = j bj (∂/∂xj ) for some constants bj and
β(T, ϕλ ) = −(1/2)
∑
ajk λj λk + i
j,k
∑
b j λj .
j
The formula (5.9) becomes the classical Lévy-Khinchin formula, and hence it may be regarded
as a Lévy-Khinchin type formula on a general homogeneous space X = G/K.
129
Recall from §3.1 that the space X = G/K is called irreducible if the K-action on To X
is irreducible, that is, if it has no nontrivial invariant subspace of To X. This is equivalent
to the irreducibility of the adjoint action Ad(K) on p. When dim(X) > 1, the irreducibility
of X is stronger than the assumption in Theorem 5.3 (b), which means that the K-action
on To X does not fix any nonzero vector. For an example of a non-irreducible G/K with
K-action not fixing any nonzero vector in To X, one may take two irreducible G1 /K1 and
G2 /K2 of dimension > 1, and let G = G1 × G2 and K = K1 × K2 .
The space X = G/K is called isotropic if the K-action on To M is transitive on the rays
through the origin of To M , that is, for any two vectors u and v in To M with v ̸= 0, there
is k ∈ K such that u = aDk(v) for some constant a. This is equivalent to the transitivity
of the adjoint action Ad(K) on the rays through the origin of p and is stronger than the
irreducibility of G/K. An example of an irreducible G/K that is not isotropic will be given
later (see Example 5.15).
Let ⟨·, ·⟩ be an Ad(K)-invariant inner product on p. Let the basis {ξ1 , . . . , ξn } of p be
chosen so that it is an orthonormal. The inner product induces a G-invariant Riemannian
metric on X = G/K, and by Proposition 3.21, if g is unimodular, then the associated
∑
Laplace-Beltrami operator ∆X is given by ∆X = ni=1 ξi ξi .
Proposition 5.5 Assume X = G/K is irreducible with dim(X) > 1. Let T be a G-invariant
second order linear differential operator on X with T 1 = 0. Then
T = a∆X
for some constant a. In particular for T in Theorem 5.3, T = a∆X with a ≥ 0.
Proof Recall that any T as above is given by (3.3) with an Ad(K)-invariant symmetric
matrix {aij } and an Ad(K)-invariant ξ0 ∈ p. The irreducibility of the action Ad(K) on p with
dim(X) > 1 implies ξ0 = 0. We may assume that the basis {ξ1 , . . . , ξn } of p is chosen to be
orthonormal under an Ad(K)-invariant inner product on p. Then the matrix representation
[Ad(k)] of Ad(k) restricted to p belongs to O(n) for k ∈ K. By the Ad(K)-invariance of
{aij }, [Ad(k)]{aij }[Ad(k)]′ = {aij } for any k ∈ K. Because Ad(K) is irreducible on p,
[Ad(k)] for k ∈ K form a subgroup of O(n) whose action on p is irreducible. By Lemma 3.6,
aij = aδij for some constant a. This shows that all G-invariant second order differential
operators on X are proportional to each other. Because ∆X is such an operator, T = a∆X
for some constant a. For T in Theorem 5.3, {aij } is nonnegative definite and hence a ≥ 0.
2
Proposition 5.6 Assume X = G/K is isotropic. Then any G-invariant differential operator on X is a polynomial in ∆X . Therefore, any smooth K-invariant function ϕ on X with
ϕ(o) = 1 is a spherical function if and only if it is an eigenfunction of ∆X .
130
Proof With the identification of p and To X under the natural projection G → G/K, the
basis {ξ1 , . . . , ξn } of p may be regarded as an orthonormal basis of T0 X under a K-invariant
inner product. For k ∈ K, let {kpq } be the matrix representing Dk under this basis and
∑
for t = (t1 , . . . , tn ) ∈ Rn , kt ∈ Rn is the usual matrix product, that is, (kt)p = q kpq tq .
A polynomial P (t) = P (t1 , . . . , tn ) is called K-invariant if P (kt) = P (t) for k ∈ K. Any
G-invariant differential operator T on X is identified with a K-invariant polynomial P (t) in
the sense that
∑
∂
∂
T f (o) = P (
,...,
)f (e p tp ξp o) |t=0
∂t1
∂tn
for f ∈ Cc∞ (X). In the proof of Proposition 5.5, it is shown that ∆X is identified with the
polynomial Q(t) = t21 +· · ·+t2n up to a multiplicative constant. It is therefore enough to show
that any K-invariant polynomial P (t) is a polynomial in Q. Let Pj be the sum of jth degree
monomials in P (t). Then Pj (t) is also K-invariant. Because G/K is isotropic, the K-action
is transitive on the unit sphere S in To X. In particular, for any t ∈ Rn , there is k ∈ K such
that kt = −t. This implies P (−t) = P (t) and consequently Pj (t) = 0 for odd j. For an even
j, Pj (t) is a constant aj for t ∈ S. Because Q(t) = n for n ∈ S, Pj (t) = aj Q(t)j/2 /nj/2 for
t ∈ S. This implies that Pj (t) = aj Q(t)j/2 /nj/2 for all t ∈ Rn , and hence proves that P (t) is
a polynomial in Q. 2
The transformation group on Rn generated by translations and orthogonal transformations is called the Euclidean motion group on Rn and is denoted by M (n). It contains
the translation group, naturally identified with Rn , and the orthogonal group O(n) as closed
subgroups. Both O(n) and M (n) have two connected components. The identity component
of O(n) is the special orthogonal group SO(n) formed by orthogonal transformations of determinant 1. The transformations in SO(n) are also called rotations on Rn . The identity
component of M (n) is the subgroup M0 (n) generated by translations and rotations.
Both M (n) and M0 (n) act transitively on Rn with O(n) and SO(n) respectively as the
isotropy subgroups at the origin 0. This leads to two homogeneous space structures on Rn ,
M (n)/O(n) and M0 (n)/SO(n). However, for n ≥ 2, the Lévy processes in Rn defined by
these two homogeneous space structures are the same. To show this, note that a Lévy process
in G/K is determined in distribution by a continuous convolution semigroup of K-invariant
probability measures. For n ≥ 2, an SO(n)-invariant measure on Rn is determined by its
projection to a ray through origin, so it is also O(n)-invariant, and vice versa.
Example 5.7 Let xt be a classical Lévy process in Rn with n ≥ 2. Assume it is rotationinvariant, that is, SO(n)-invariant. Then it is a Lévy process in the homogeneous space
Rn = M0 (Rn )/SO(n). It is easy to see that this space is isotropic, thus a smooth SO(n)invariant function ϕ on Rn with ϕ(0) = 1 is a spherical function if and only if it is an
131
eigenfunction of the Laplace operator ∆ on Rn . Moreover, T in Theorem 5.3 is given by
T = a∆ for some constant a ≥ 0, and the simplified Lévy -Khinchin formula (5.10) holds.
We will now identify all bounded spherical functions ϕ on Rn = M0 (Rn )/SO(n) which
appear in (5.10). Let σ be the uniform distribution on the unit sphere S n−1 in Rn . Then for
any complex number λ, it is easy to show that
∫
ϕλ (x) =
S n−1
exp(iλ
n
∑
xp yp )σ(dy)
(5.11)
p=1
is SO(n)-invariant with ϕ(0) = 1 and is an eigenfunction of ∆ with eigenvalue −λ2 , and
hence is a spherical function on Rn = M0 (Rn )/SO(n). Then ϕλ (x) with λ ∈ R is a bounded
spherical function for which the simplified form of the Lévy -Khinchin formula (5.10) holds
with β(T, ϕλ ) = −aλ2 . Note that by the reflection-invariance of σ, ϕ−λ = ϕλ .
We now show that any spherical function ϕ is equal to ϕλ for some λ ∈ C based on the fact
that a spherical function is analytic (see the proof of Proposition 2.2 in [34, Chapter IV]).
The SO(n)-invariance implies that ϕ is a radial function, that is, ϕ(x) depends only on
∑
r = |x| = ( p x2p )1/2 . Writing ϕ(r) for ϕ(x) with r = |x| and expressing ∆ under the
spherical polar coordinates, we see that ϕ satisfies the ordinary differential equation
ϕ′′ (r) +
n−1 ′
ϕ (r) = βϕ(r)
r
(5.12)
for some complex number β (eigenvalue). Using the usual power series method to solve the
differential equation (5.12) shows that for any β, up to a multiplicative constant, it has only
one solution that is analytic at r = 0, which must be proportional to ϕ because a spherical
function is analytic. Since ϕλ is also such a solution if λ is a square root of −β, it follows
that ϕ = ϕλ . This shows that ϕλ with λ ∈ C form the complete set of spherical functions on
Rn = M0 (Rn )/SO(n), then all bounded spherical functions are given by ϕλ with λ ∈ R.
Recall the well known Bessel differential equation:
r2 ϕ′′ (r) + rϕ′ (r) + (r2 − α2 )ϕ(r) = 0.
(5.13)
Its solutions that are analytic at 0, up to a multiplicative constant, are given by the following
Bessel functions of the first kind
Jα (r) =
∞
∑
(−1)m
x
( )2m+α
m=0 m!Γ(m + α + 1) 2
(5.14)
J(n/2)−1 (λ|x|)
.
(λ|x|)(n/2)−1
(5.15)
for α ≥ 0. We have, for n ≥ 2,
ϕλ (x) = 2(n/2)−1 Γ(n/2)
To prove (5.15), one just needs to use (5.13) and (5.14) to verify that the right hand side of
(5.15) satisfies (5.12) and has value 1 at r = 0. In particular, ϕλ (x) = J0 (λ|x|) for n = 2.
132
We return now to a general homogeneous space X = G/K. A continuous complex-valued
function ϕ on G is called positive definite if
m
∑
ϕ(xi x−1
j )ξi ξj ≥ 0
(5.16)
i,j=1
for any finite set of xi ∈ G and ξi ∈ C. It is easy to show that a positive definite function ϕ
on G satisfies
ϕ(e) ≥ 0,
ϕ(x−1 ) = ϕ(x),
|ϕ| ≤ ϕ(e).
A function ϕ on X is called positive definite if ϕ ◦ π is so on G. Such a function is always
bounded with |ϕ| ≤ ϕ(o).
Proposition 5.8 In Theorem 5.3(b), if ϕ is a positive definite spherical function on X =
G/K, then β(T, ϕ) is real and ≤ 0.
Proof The map ξ 7→ eξ o is a diffeomorphism from an Ad(K)-invariant neighborhood W
of 0 in p onto a K-invariant neighborhood U of o in X. By the K-invariance of spherical
functions, for x = eξ o ∈ U , ϕ(x) = ϕ(keξ o) = ϕ(eAd(k)ξ o) = ϕ(erAd(k)ζ ) for any k ∈ K, where
r = ∥ξ∥ is the norm of ξ associated to an Ad(K)-invariant inner product on p and ζ = ξ/r.
Taking the Taylor expansion of ϕ(x) regarded as a function of r at r = 0,
1
ϕ(x) = 1 + r[Ad(k)ζ]ϕ(o) + r2 [Ad(k)ζ]2 ϕ(esAd(k)ζ )
2
Because there is no nozero Ad(K)-invariant vector in p,
(for some s = s(x, k) in [0, r]).
∫
K
dk[Ad(k)ξ] = 0 and hence
1 ∫
ϕ(x) = 1 + r2 { dk[Ad(k)ζ]2 ϕ(esAd(k)ζ )}
2
K
1 2 ∫
= 1 + r { dk[Ad(k)ζ]2 ϕ(o)} + O(r3 ).
2
K
(5.17)
Because ϕ is positive definite, ϕ̄(x) = ϕ(x−1 ) = ϕ(e−rAd(k)ζ ) has the same Taylor expansion
as ϕ(x) up to r2 -term (with r-term being 0), and hence
Im[ϕ(x)] =
1
[ϕ(x) − ϕ̄(x)] = O(r3 ).
2i
With the basis {ξ1 , . . . , ξn } of p suitably chosen, the coefficient matrix aij in T may be
diagonalized with nonnegative eigenvalues λi . By (5.17), ξϕ(o) = 0 for ξ ∈ p, and hence
∑
T ϕ(o) = ni=1 λi ξi ξi ϕ(o). Because Re(ϕ) is maximized at o and Im(ϕ) = O(r3 ), it follows
that ξξϕ(o) ≤ 0 for ξ ∈ p and β(T, ϕ) = T ϕ(o) is real and ≤ 0. 2
133
5.3
Symmetric spaces
On a special class of homogeneous spaces, called symmetric spaces, the spherical transform:
ϕ 7→ µ(ϕ) of a K-invariant measure µ, introduced in the last section, determines µ, and hence
the Lévy -Khinchin formula (5.9), or its simplified form (5.10), provides a complete characterization for a continuous K-invariant convolution semigroup µt . Moreover, the spherical
transform on a symmetric space may be inverted, and thus provides a means to prove the
existence of a smooth density for µt and to obtain its representation as a series or an integral
of spherical functions. The present section collects some basic facts on symmetric spaces.
Let G be a connected Lie group with Lie algebra g, and let K be a compact subgroup
with Lie algebra k. Assume dim(K) < dim(G) to avoid triviality. A non-trivial Lie group
automorphism Θ on G with Θ2 = idG is called a Cartan involution on G. The pair (G, K)
together with a Cartan involution Θ is called a (Riemannian) symmetric pair if
Θ
GΘ
0 ⊂ K ⊂ G ,
(5.18)
where GΘ is the fixed point set of Θ (a closed subgroup of G) and GΘ
0 is its identity component. The space X = G/K is called a symmetric space which plays an important role
in differential geometry. See [33] for a comprehensive treatment of symmetric spaces. A
short and self-contained introduction to this subject may be found in [55, Chapter 5]. The
definition of symmetric spaces given here is slightly less general than that given in [33].
In the rest of this section, we will assume X = G/K is a symmetric space. The differential
DΘ of Θ at e, denoted by θ, is a nontrivial Lie algebra automorphism on g with θ2 = idg
and eigenvalues ±1. By (5.18), k is the eigenspace associated to eigenvalue 1. Let p be the
eigenspace of θ associated to eigenvalue −1. Because of θ2 = idg , the matrix representation
of θ is completely diagonalizable (considering its Jordan form) and hence g = p ⊕ k (a
direct sum). Because Θ fixes points in K, it commutes with K-conjugation on G, and
hence θ commutes with Ad(k) on g for k ∈ K. It follows that p is Ad(K)-invariant. The
commutativity of Ad(k) and θ also means that there is an Ad(K)-invariant inner product
⟨·, ·⟩ on g that is also invariant under θ. Fix such an inner product on g. Then p is the
orthogonal complement of k in g, and it is easy to show that
[k, k] ⊂ k,
[k, p] ⊂ p,
[p, p] ⊂ k.
(5.19)
The restriction of ⟨·, ·⟩ to p induces a G-invariant Riemannian metric on X = G/K. By
(5.19), the condition (3.30) in §3.4 is satisfied, and hence t 7→ etξ o is a geodesic in X from
o = eK for ξ ∈ p. By the general theory of differential geometry, the Riemannian metric on
X is complete and any point in X is connected to o by such a geodesic. It follows that any
g ∈ G can be written as
g = eξ k, ξ ∈ p and k ∈ K.
(5.20)
134
Let π: G → X be the natural projection as before. By Theorem 3.1 in [34, chapter IV],
G is unimodular, so left and right Haar measures on G are the same, and they are invariant
under the inverse map on G. Fix a nonzero Haar measure ρG on G and let ρX = πρG . Then
by Proposition 1.10 (b), ρX is a G-invariant Radon measure on X, unique up to a constant
factor. For simplicity, we will write dg and dx respectively for ρG (dg) and ρX (dx), and write
dk for the normalized Haar measure ρK (dk) on K.
Let Lp (G) and Lp (X) denote respectively the spaces of Lp -functions on G and X under
the measures dg and dx for p ≥ 1. When we say a measure µ on G has a density p we
mean that µ is absolutely continuous with respect to dg with Radon-Nikodym derivative
∫
p = dµ/dg, that is, µ(f ) = f (g)p(g)dg for f ∈ B+ (G). A density of a measure on X has a
similar meaning with respect to dx.
The convolution of two functions p and q in L1 (G) is the function p ∗ q ∈ L1 (G) defined
by
∫
p ∗ q(x) =
p(y)q(y −1 x)dy
(5.21)
G
for x ∈ G. Note that if µ and ν are finite measures on G with densities p and q respectively,
then µ ∗ ν has density p ∗ q.
For two K-invariant functions p and q in L1 (X), their convolution is the function p ∗ q
∫
∫
in L1 (X) defined by (5.21) for x ∈ X and with G replaced by X , which is independent of
the choice of a section map S to represent y −1 x = S(y)−1 x. A simple computation using the
K-invariance of q shows that p ∗ q is also K-invariant. As for the convolution on G, if µ and
ν are K-invariant finite measures on X with densities p and q respectively, which may be
chosen to be K-invariant, then µ ∗ ν has density p ∗ q.
Proposition 5.9 Let µ and ν be two finite K-invariant measures on a symmetric space
X = G/K. Then µ ∗ ν = ν ∗ µ.
Proof By Proposition 1.9, the map γ 7→ πγ is a bijection from K-bi-invariant measures
γ on G onto K-invariant measures on X, which also preserves the convolution. Therefore,
it suffices to prove µ ∗ ν = ν ∗ µ for two finite K-bi-invariant measures µ and ν on G.
By Theorem 3.1 in [34, chapter IV], µ ∗ ν = ν ∗ µ if µ and ν are compactly supported
with continuous densities. Suppose µ and ν are just compactly supported. Let ϕn ≥ 0 be
∫
continuous functions on G such that G ϕn (g)dg = 1 and supp(ϕn ) ↓ {e}, and let γn be the
measure on G with density ϕn . Here as usual, for any continuous function f , supp(f ) denotes
its support, which is the smallest closed set F such that f = 0 on F c . Then µn = γn ∗ µ
∫
and νn = γn ∗ ν are compactly supported with continuous densities ϕn (gh−1 µ(dh) and
∫
ϕn (gh−1 )ν(dh) respectively, and hence µn ∗ νn = νn ∗ µn . Because µn → µ and νn → ν
weakly, µ ∗ ν = ν ∗ µ holds for compactly supported µ and ν. Without assuming the compact
supports of µ and ν, let Un be K-bi-invariant open subsets of G such that Un are relatively
135
compact and Un ↑ G, and let µn and νn be respectively the restrictions of µ and ν to Un .
Then µn ∗ νn = νn ∗ µn . The weak convergence of µn → µ and νn → ν shows µ ∗ ν = ν ∗ µ.
2
By the following result, the Lévy-Khinchin formula (5.9) or its simplified form (5.10)
provides a complete characterization of continuous K-invariant convolution semigroups µt
on a symmetric space X = G/K.
Theorem 5.10 A finite K-invariant measure µ on a symmetric space X = G/K is completely determined by its spherical transform µ(ϕ) for ϕ ranging over bounded spherical functions on X.
Proof By Proposition 3.8 in [34, chapter IV], any K-invariant function f ∈ L1 (X) is
∫
completely determined by its spherical transform f˜(ϕ) = f (x)ϕ(x)dx with ϕ ranging over
bounded spherical functions on X. Thus, if µ has a density, then it is determined by
its spherical transform. For a general µ, let pn ≥ 0 be K-invariant functions on X with
∫
compact supports supp(pn ) ↓ {o} and pn (x)dx = 1. Then µn (dx) = pn (x)dx → δo and
∫
µ ∗ µn → µ ∗ δo = µ weakly. Because µ ∗ µn has density qn (x) = µ(dy)pn (y −1 x), it is
determined by µ ∗ µn (ϕ) = µ(ϕ)µn (ϕ). It follows that µ as the weak limit of µ ∗ µn is
determined by µ(ϕ). 2
In §4.1, a representation of a Lie group G is introduced as a linear action of G on a finite
dim complex vector space. In general, an action of G is a (continuous) linear action F on a
topological complex vector space V of possibly infinite dim. It is called irreducible if V has
no closed subspace, except {0} and V , that is invariant under F . Two representations F1 on
V1 and F2 on V2 are called equivalent if there is a continuous linear bijections f : V1 → V2
such that F2 (g) ◦ f = f ◦ F1 (g) for any g ∈ G. A representation F on V is called unitary if
V is a complex Hilbert space equipped with a Hermitian inner product ⟨·, ·⟩ that is invariant
under F .
Let F be a unitary representation of G on a Hilbert space V . Then for any v ∈ V ,
ϕ(x) = ⟨v, F (x)v⟩ is a positive definite function on G (see (5.16)). Conversely, if ϕ is a
positive definite function on G, then there is a unitary representation Fϕ of G on a Hilbert
space Vϕ such that ϕ(x) = ⟨v, Fϕ (x)v⟩ for some v ∈ Vϕ . In fact, Vϕ is the closure of the set Vϕ′
∑
of functions of the form f (x) = ki=1 αi ϕ(gi x), αi ∈ C and gi ∈ G, under the inner product
⟨f1 , f2 ⟩ =
∑
αi β̄j ϕ(gi gj−1 )
i,j
∑
∑
for f1 (x) = i αi ϕ(gi x) and f2 (x) = j βj ϕ(gj x), and the action of Fϕ on Vϕ is extended
from its action on Vϕ′ by Fϕ (g)f (x) = f (g −1 x) for f ∈ Vϕ′ . Note that ϕ(x) = ⟨v, Fϕ (x)v⟩ for
v = ϕ. See [34, IV§1.1] for more details.
136
Let K be a compact subgroup of G. A representation F of G on V is called spherical if
V has a nonzero vector v that is fixed by F (k) for all k ∈ K. Then the set V K = {v ∈ V ;
F (k)v = v for k ∈ K} is a closed nonzero subspace of V . The following result, proved in [34,
IV.§3] (see Theorems 3.4 and 3.7, and Lemma 3.6 in [34, IV.§3]), deals with an interesting
relation between spherical functions and spherical representations.
Theorem 5.11 Let G/K be a symmetric space. Then for any positive definite spherical
function ϕ, Fϕ is an irreducible spherical unitary representation with VϕK being the 1-dim
subspace spanned by ϕ. Moreover, the map: ϕ 7→ {Fϕ } is a bijection from the set of positive
definite spherical functions onto the set of equivalence classes of irreducible unitary spherical
representations, where {Fϕ } is the class containing Fϕ . Furthermore, for any irreducible
unitary spherical representation F on V , the associated spherical function ϕ is give by ϕ(x) =
⟨v, F (x)v⟩ for any v ∈ V K with ⟨v, v⟩ = 1.
Recall ad(ξ) is the linear map: g → g given by ad(ξ)η = [ξ, η] (Lie bracket). The Killing
form of the Lie algebra g is defined by
B(ξ, η) = Tr[ad(ξ)ad(η)]
(5.22)
for ξ, η ∈ g. This is a symmetric bilinear form on g. Recall the Lie algebra g with dim(g) > 1
is called semisimple if it has no abelian ideal. By Cartan’s criterion (see Theorem 1 in I.6 [11,
I.6]), g is semisimple if and only if its Killing form B is nondegenerate, that is, if B(ξ, ·) ̸= 0
for nonzero ξ ∈ g.
The Killing form B is invariant under any Lie algebra automorphism on g, in particular
it is invariant under the adjoint action Ad(G), that is,
∀ξ, η ∈ g and g ∈ G,
B(Ad(g)ξ, Ad(g)η) = B(ξ, η).
(5.23)
Setting g = etζ in (5.23) and taking (d/dt) at t = 0 yields
∀ξ, η, ζ ∈ g,
B([ζ, ξ], η) + B(ξ, [ζ, η]) = 0.
(5.24)
Note that the Killing form of an ideal h of g is equal to the restriction of the Killing form
of g to h, but this is not true if h is merely a Lie subalgebra of g.
Proposition 5.12 In a symmetric space X = G/K, if G is semisimple, then there is no
nonzero Ad(K)-invariant vector in p, that is, the assumption in Theorem 5.3(b) is satisfied,
and consequently the simplified Lévy -Khinchin formula (5.10) holds.
Proof Because G is semisimple, B is nondegenerate. The Ad(K)-invariance of ξ ∈ p
implies [ξ, ζ] = 0 for any ζ ∈ k. Then for any η ∈ p, B([ξ, η], ζ) = −B(η, [ξ, ζ]) = 0. Because
[ξ, η] ∈ k, we must have [ξ, η] = 0 and hence ad(ξ) = 0. By the semisimplicity, ξ = 0. 2
137
The symmetric space X = G/K is said to be of compact type if the Killing form B of
g is negative definite, it is said to be of noncompact type if B is negative definite on k and
positive definite on p, and it is said to be of Euclidean type if p is an abelian ideal of g. For
a compact or noncompact type symmetric space X = G/K, B is nondegenerate and so G is
semisimple. If X is of compact type, then G is compact (see Proposition 6.6, Corollary 6.7
and Theorem 6.9 in [33, Chapter II]), and hence so is X. If X is of noncompact type, then G
is noncompact and X is diffeomorphic to Rn (see Theorem 1.1 (iii) in [33, Chapter VI]). In
general, a symmetric space G/K with G simply connected is a direct product of symmetric
spaces belonging to these basic types (see Proposition 4.2 in [33, Chapter V]). We note that a
compact or noncompact symmetric space is not necessarily of compact or noncompact type.
5.4
Examples of symmetric spaces
Example 5.13 (Euclidean space) A Euclidean space Rn is a trivial example of a Euclidean type symmetric space with G = Rn (additive group) and K = {0}, and the Cartan
involution Θ: g 7→ −g. A less trivial example is given by Rn = M0 (Rn )/SO(n), where
M0 (Rn ) and SO(n) are respectively the connected Euclidean motion group and the special
orthogonal group introduced in Example 5.7. The group M0 (Rn ) consists of transformations
on Rn of the form (A, b): x 7→ Ax + b for A ∈ SO(n) and b ∈ Rn . One may regard SO(n)
and Rn as subgroups of M0 (Rn ). Then Rn is a normal abelian subgroup and hence its Lie
algebra is an abelian ideal of the Lie algebra of M0 (Rn ). Note that (M0 (Rn ), SO(n)) is
a symmetric pair with Cartan involution Θ: (A, b) 7→ (A, −b). The associated symmetric
space Rn = M0 (Rn )/SO(n) is of Euclidean type.
To present other examples, let us first compute the Killing forms for some matrix groups.
Let GL(n, R) be the group of real invertible n×n matrices, called the general linear group on
Rn , see Appendix A.1 for more details. Note that the Lie algebra g of any real matrix group
G, as a subgroup of GL(n, R), is a space of matrices with Lie bracket [ξ, η] = ξη−ηξ. Because
ad(ξ)η = [ξ, η], by choosing a convenient basis of g, the Killing form B(ξ, η) = Tr[ad(ξ)ad(η)]
may be computed. We note that by the discussion in §2.4, for a matrix group G, the action
of Ad(G) on g is given by a matrix conjugation, that is, Ad(g)ξ = gξg −1 for g ∈ G and
ξ ∈ g.
General linear group For the general linear group G = GL(n, R), the Lie algebra g is the
space gl(n, R) of all n × n real matrices. A convenient basis of g is formed by matrices Eij
for i, j = 1, 2, . . . , n, where Eij is the matrix that has 1 at place (i, j) and 0 elsewhere. For
X, Y ∈ gl(n, R),
ad(X)ad(Y )Eij = [X, [Y, Eij ]] = XY Eij − XEij Y − Y Eij X + Eij Y X.
138
(5.25)
Then denoting Xij for the (i, j)-element of matrix X,
[ad(X)ad(Y )Eij ]ij =
∑
Xik Yki − Xii Yjj − Yii Xjj +
k
∑
Yjk Xkj .
(5.26)
k
Summing over (i, j), we obtain the Killing form of gl(n, R):
B(X, Y ) = 2n Tr(XY ) − 2(Tr X)(Tr Y ).
(5.27)
Taking X to be the identity matrix I yields B(I, Y ) = 0 for all Y . It follows that the Killing
form is degenerate and hence GL(n, R) is not semisimple.
Special linear group The special linear group SL(n, R) is formed by n × n real matrices of
determinant 1 and is a closed connected subgroup of GL(n, R). Its Lie algebra is the space
sl(n, R) of n × n real matrices of trace 0, an ideal of gl(n, R). The Killing form of sl(n, R)
may be obtained by restricting the Killing form of gl(n, R) to sl(n, R). Since Tr(X) = 0 for
X ∈ sl(d, R), we obtain the Killing form of sl(n, R):
B(X, Y ) = 2n Tr(XY ).
Because B(X, X ′ ) = 2n Tr(XX ′ ) = 2n
SL(n, R) is semisimple for n ≥ 2.
∑
i,j
(5.28)
Xij2 , it follows that B is nondegenerate and hence
Orthogonal group The common Lie algebra of the orthogonal group O(n) and its identity component SO(n), the special orthogonal group, is the space o(n) of skew-symmetric
matrices in gl(n, R), that is, o(n) = {A ∈ gl(n, R); A′ = −A}. The Killing form of o(n) is
B(X, Y ) = (n − 2) Tr(XY ).
(5.29)
To prove this, note that the matrices ηij = Eij − Eji , 1 ≤ i < j ≤ n, form a basis of o(n).
By (5.25), for i < j,
[ad(X)ad(Y )Eji ]ij = −2Xij Yij .
Subtracting from (5.26) and using the skew-symmetry of X, Y ∈ o(n),
[ad(X)ad(Y )ηij ]ij =
∑
Xik Yki +
k
∑
Xjk Ykj + 2Xij Yij .
k
∑
Note that for each index p between 1 and n, the sum k Xpk Ykp appears exactly n − 1
∑
∑
times in the expressions k Xik Yki + k Xjk Ykj when they are summed as (i, j) varies over
∑
all possible choices with i < j. Because ad(X)ad(Y )ηij = p<q [ad(X)ad(Y )ηij ]pq ηpq and
Yij = −Yji ,
Tr[ad(X)ad(Y )] =
∑
[ad(X)ad(Y )ηij ]ij = (n − 1)Tr(XY ) − Tr(XY ).
i<j
139
This proves (5.29).
∑
By (5.29) and the skew-symmetry, B(X, X) = −(n − 2) i<j Xij2 . It follows that for
n ≥ 3, B is negative definite and hence SO(n) is semisimple.
We are now ready to present some symmetric spaces of compact or noncompact types.
Example 5.14 (Sphere) Let S n be the n-dimensional unit sphere in Rn+1 . Because
G = SO(n + 1) acts transitively on S n and the closed subgroup fixing the point (1, 0, . . . , 0)
is K = diag{1, SO(n)}, S n = G/K may be identified with SO(n + 1)/SO(n). Let J =
diag(1, −In ), where In is the n × n identity matrix. Then the map Θ: A 7→ JAJ is a Cartan
involution on G with the fixed point set GΘ = K ∪ diag{−1, O(n) − SO(n)}, and so under
Θ, (G, K) is a symmetric pair. Because the killing form of g = o(n + 1) is negative definite,
the sphere S n = SO(n + 1)/SO(n) is a symmetric space of compact type. Note that it is an
isotropic homogenous space.
Example 5.15 (Space of symmetric matrices) Now consider the Lie group G =
SL(n, R) with Lie algebra g = sl(n, R) and the compact subgroup K = SO(n) with Lie
algebra k = o(n). The map Θ: g 7→ (g ′ )−1 is a Cartan involution on G with GΘ = K, and
hence under which (G, K) is a symmetric pair. The differential of Θ at e = In is θ: g → g
given by A 7→ −A′ , and hence its eigenspace of eigenvalue −1 is precisely the subspace p
of g consisting of all symmetric matrices. By (5.28), the Killing form B is negative definite
on k and positive definite on p, and hence the symmetric space G/K = SL(n, R)/SO(n) is
of noncompact type. It may be identified with the space X of real n × n positive definite
symmetric matrices of unit determinant via the map g 7→ gg ′ from G onto X with kernel K.
Note that the space SL(n, R)/SO(n) is not isotropic for n > 2, but by [55, Proposition 7.5], it is irreducible.
Duality Let X = G/K be a symmetric space with associated Cartan involution Θ: G → G
and decomposition g = k ⊕ p of the Lie algebra g of G introduced in §5.3. Let gC be the
complexification of g, which by definition is the complex linear space spanned by g with Lie
bracket extended by complex linearity. Regard gC as a real Lie algebra and note that k ⊕ (ip)
√
is a Lie subalgebra of gC , called the dual algebra of g, where i = −1. Extend θ = DΘ on
g = k ⊕ p to k ⊕ (ip) by setting
θ(ξ + iζ) = ξ − iζ,
for ξ ∈ k and ζ ∈ p.
By complex linear extension, the Killing form B on g becomes the Killing form on gC , which
will also be denoted by B.
Let X ∗ = G∗ /K be a symmetric space with the same compact subgroup K. Assume
there is a Lie algebra isomorphism ϕ from k ⊕ (ip) onto the Lie algebra g∗ of G∗ that is an
140
identity on k, and is compatible with Cartan involutions in the sense that ϕ ◦ θ = θ∗ ◦ ϕ,
where θ∗ = DΘ∗ is the differential of the Cartan involution Θ∗ associated to X ∗ . In this case
the two symmetric spaces X and X ∗ are said to be in duality. Because B(iξ, iζ) = −B(ξ, ζ)
for ξ, ζ ∈ p, it is easy to show (see also Proposition 2.1 in [33, Chapter V]) that if X = G/K
is of compact type (resp. noncompact type), then X ∗ = G∗ /K is of noncompact type (resp.
compact type).
For example, consider X = SL(n, R)/SO(n), the space of symmetric matrices. Then
g = sl(n, R) has complexification gC = sl(n, C), and k = o(n). Because p is the space
of traceless real symmetric matrices, k ⊕ (ip) is naturally identified with su(n). The map
g 7→ g ′−1 is the Cartan involution on G = SL(n, R), and is also a Cartan involution on
G∗ = SU (n). In both cases, it fixes the same compact subgroup K = SO(n). Then
X ∗ = SU (n)/SO(n) is a symmetric space. The natural identification of k ⊕ (ip) with su(n)
is compatible with Cartan involutions. It follows that the symmetric space SL(n, R)/SO(n),
of noncompact type, is in duality with the symmetric space X ∗ = SU (n)/SO(n), which is
necessarily of compact type.
Example 5.16 (Lorentz group and hyperbolic space) The Lorentz group O(1, n) is
the group of real (n + 1) × (n + 1) matrices A satisfying A′ JA = J, where J = diag(1, −In ).
′
Differentiating etξ Jetξ = J at t = 0, we obtain its Lie algebra o(1, n) = {ξ ∈ gl(n + 1, R);
ξ ′ J + Jξ = 0}. For u ∈ R, x, y ∈ Rn regarded as row vectors and an n × n real matrix A, let


u x 
.
ξ(u, x, y, A) =  ′
y A
(5.30)
Then o(1, n) is formed by matrices ξ(0, x, x, A) with x ∈ Rn and A ∈ o(n). It is known (see
[55, §5.4]) that the Lorentz group O(1, n) has four connected components and its identity
component SO(1, n)+ is a closed subgroup of SL(n+1, R) formed by the matrices ξ(u, x, y, A)
with u > 0, x, y ∈ Rn and A ∈ SO(n) such that
u2 = 1 + xx′ ,
uy − xA = 0,
A′ A = In + y ′ y.
We may identify o(n) with the subspace of o(1, n) formed by matrices ξ(0, 0, 0, A) with
A ∈ o(n), and SO(n) with the subgroup of SO(1, n)+ formed by matrices ξ(1, 0, 0, A) with
A ∈ SO(n). Then (SO(1, n)+ , SO(n)) is a symmetric pair with Cartan involution Θ: ξ 7→
ξ ′−1 . The symmetric space H n = SO(1, n)+ /SO(n) is called a hyperbolic space.
The space p is formed by matrices ξ(0, x, x, 0) with x ∈ Rn and is naturally identified
with Rn . A direct computation shows that the Ad(K)-action on p corresponds to the usual
SO(n)-action on Rn .
Note that the Lie algebra o(n + 1) of SO(n + 1) is formed by matrices ξ(0, x, −x, A) with
x ∈ Rn and A ∈ o(n). It is easy to show that the map
ξ(0, 0, 0, A) + iξ(0, x, −x, 0) 7→ ξ(0, 0, 0, A) + ξ(0, x, x, 0),
141
√
where i = −1, is a Lie algebra isomorphism from the dual algebra of o(n + 1) onto o(1, n)
and is the identity map on o(n). Moreover, it is compatible with Cartan involutions. Then
S n = SO(n + 1)/SO(n) and H n = SO(1, n)+ /SO(n) are in duality, and hence H n is of
noncompact type.
5.5
Compact symmetric spaces
In this section, let X = G/K is a symmetric space with G compact, but not necessarily
of compact type. Recall that the set Ĝ of equivalence classes of irreducible representations
of G is countable. Now let ĜK be the subset of equivalence classes of irreducible spherical
representations of G (defined in §5.3), and let ĜK+ be ĜK with the trivial representation
removed. As in §4.1, χδ is the character of δ ∈ Ĝ with dimension d(δ).
Let ρG be the normalized Haar measure on G, and let ρX = πρG be the unique Ginvariant probability measure on X, where π: G → X is the natural projection. Let L2 (X)
be the space of L2 -functions on X under ρX , and let L2K (X) be the subspace of K-invariant
functions in L2 (X). We may write dg, dk and dx for ρG (dg), ρK (dk) and ρX (dx) respectively.
Theorem 5.17 (a) The spherical functions on G, with respect to the symmetric space
X = G/K, are positive definite and are precisely the functions ψδ given below,
∫
ψδ (g) =
K
χδ (kg)dk
for g ∈ G,
(5.31)
with δ varying over ĜK . Moreover, if δ ∈ Ĝ is not spherical, then ψδ ≡ 0.
(b) Let ϕδ be the spherical function on X associated to ψδ (that is, ψδ = ϕδ ◦ π). Then
{d(δ)1/2 ϕδ ; δ ∈ ĜK } is a complete orthonormal system in L2K (X).
Proof By [34, IV.Theorem 4.2], the spherical functions on G are positive definite and are
precisely the functions in (5.31), noting that ψδ in (5.31) is the complex conjugate of the
spherical function in [34]. We note that for δ ∈ Ĝ, χ̄δ is the character of δ̄ ∈ Ĝ, where δ̄ is
the conjugate representation of δ (defined in Remark 5.24, see also [12, II.3 and II.4]).
∫
If δ ∈ Ĝ is not spherical, then the linear map H = K dkU δ (k) on the representation
space is a zero map because its image is fixed by U δ (k) for k ∈ K. It follows that ψδ (g) =
∫
∫
δ
δ
K χδ (kg)dk = Tr K dkU (k)U (g) = 0. This proves (a).
∫
To prove (b), for ϕ ∈ C(G) and f ∈ C(X), define ϕ ∗ f (x) = G ϕ(g)f (g −1 x)dg, x ∈ X.
By [34, V.Theorem 3.5], if f ∈ C(X), then
f=
∑
d(δ)χ̄δ ∗ f
δ∈ĜK
142
(5.32)
is a convergent series with orthogonal terms in L2 (X). Moreover, if f ∈ C ∞ (X), then the
series converges absolutely and uniformly. For a K-invariant f ∈ C(X) and x ∈ X with
x = ho for some h ∈ G,
χ̄δ ∗ f (x) =
∫
=
∫G
=
ψδ (gh−1 )f (go)dg
=
K
=
∫G
G
χ̄δ (g)f (g −1 x)dg =
χδ (gh−1 )f (go)dg =
∫G ∫
∫
∫
G
∫
K
G
G
G
χδ (g −1 )f (g −1 x)dg =
χδ (kgh−1 )f (go)dgdk
∫
G
χδ (g)f (gho)dg
(because f is K-invariant)
(by (5.31))
ψδ (gkh−1 )f (go)dgdk
ψδ (g)ψδ (h−1 )f (go)dg
= [
∫
∫
(may replace h by hk −1 )
(by (5.2))
ϕδ (go)f (go)dg]ψ̄δ (h) = [
∫
X
ϕδ (x)f (x)dx]ϕ̄δ (x).
∫
∑
Because f is real valued, by (5.32), f = f¯ = δ∈ĜK d(δ)[ G ϕ̄δ (x)f (x)dx]ϕδ . This implies
that d(δ)1/2 ϕδ , δ ∈ ĜK , form a complete orthonormal system in L2K (X). 2
Recall that the generator L of a Lévy process gt in X = G/K, or a continuous convolution
semigroup µt on X with µ0 = δo , is given by (5.7). It is said to have a nondegenerate diffusion
part if the coefficient matrix {aij } in (5.7) is positive definite. When G/K is irreducible, the
differential operator part of L is a∆X ,
Theorem 5.18 Let X = G/K be a symmetric space with G compact, and let µt be a continuous convolution semigroup on X with µ0 = δo . Assume either its generator L has a
nondegenerate diffusion part or its Lévy measure η asymptotically dominates r−β .ρX at o
for some β > n (defined in §4.3), where r is the distance to o under a Riemannian metric on X (this condition is independent of the choice of a Riemannian metric on X), and
n = dim(X). Then µt has a density pt = dµt /dρX for t > 0 such that pt (x) is continuous in
(t, x) ∈ (0, ∞) × X, and
pt (x) =
∑
d(δ) µt (ϕ̄δ )ϕδ (x) =
δ∈ĜK
∑
d(δ) etLϕ̄δ (o) ϕδ (x),
(5.33)
δ∈ĜK
where the series converges absolutely and uniformly for (t, x) ∈ [ε, ∞) × X for any ε > 0.
Moreover, pt (x) is smooth in (t, x) ∈ (0, ∞) × X.
Proof If µt has an L2 density pt , then by Theorem 5.3 (a), ρX (pt ϕ̄δ ) = µt (ϕ̄δ ) = etLϕ̄δ (o) , and
by Theorem 5.17(b), pt is given by the series in (5.33), which converges in L2 (X). Because
ϕδ is positive definite, its real part Re(ϕδ ) takes the maximum value 1 at o. It follows that
Re[ξϕδ (o)] = 0 for any ξ ∈ p, and
∫
|etLϕ̄δ (o) | = etRe[Lϕ̄δ (o)] = et[Re β(T,ϕ̄δ )−
143
X
(1−Re ϕδ )dη]
,
where T is the differential operator part of L. The Parseval identity corresponding to the
series in (5.33) is
ρX (p2t ) =
∑
∑
d(δ)e2tRe[Lϕδ (o)] =
δ∈ĜK
d(δ)e2t[Re β(T,ϕ̄δ )−
∫
X
(1−Re ϕδ )dη]
,
(5.34)
δ∈ĜK
By the positive definiteness of ϕδ , |ϕδ | ≤ 1, so the convergence of the series in (5.34) for
any t > 0 implies that the series in (5.33) converges absolutely and uniformly for (t, x) ∈
[ε, ∞) × X for any ε > 0.
Assume L has a nondegenerate diffusion part. If η = 0, then the associated Lévy process
xt is a nondegenerate diffusion process in X. It is well known that such a process has a
smooth, and hence bounded, distribution density pt for t > 0, see for example Theoreme 5.1
in [6, chapitre 2 (page 27)]. Because Re(1 − ϕδ ) ≥ 0 and |ϕδ | ≤ 1, if η is nonzero, then the
series in (5.34) can only be smaller in absolute value than when η = 0, and hence it together
with the series in (5.33) will still converge absolutely and uniformly. In this case, let pt be
the series in (5.33). Then ρX (pt ϕ̄δ ) = etLϕ̄δ (o) = µt (ϕ̄δ ). By Theorem 5.10, µt = pt dx.
Suppose the series in 5.33) converges for a given Lévy measure η without a diffusion part.
If η is modified outside a neighborhood U of o, then, because η(U c ) < ∞, the change in
Re[L(ϕδ )(o)] is controlled by a fixed constant term, so the convergence of the series in (5.34)
is not affected. Adding a diffusion part or replacing η by a larger Lévy measure will only
make Re[L(ϕδ )(o)] smaller, so the series in (5.34) will still converge. The same holds when
η is multiplied by a constant as the constant can be absorbed by t in e2tReLϕδ (o) . Therefore,
if η is replaced by a Lévy measure that asymptotically dominates η, then the series in (5.34)
will still converge.
α
α
As in the proofs of Theorems 4.12 and 4.17, for α ∈ (0, 1), let µB
t , νt and µt be,
respectively, the continuous convolution semigroups associated to the Riemannian Brownian
motion in X, the α-stable subordinator in R and the subordinated Brownian motion in
∫
∫
X, such that 0∞ e−su νtα (du) = e−ts = exp[−t 0∞ (1 − e−su )ξ α (du)] for s > 0, and µαt =
∫∞ B
α
α
−1−α
du/Γ(1 − α) is the Lévy measure of νtα . Let ht
0 µu (·)νt (du), where ξ (du) = αu
α
α
2
and hαt be, respectively, the densities of µB
t and µt for t > 0. Then ht ∈ L (G). The
∑
parseval identity associated to the Fourier series of hαt is ρX ((hαt )2 ) = δ∈ĜK d(δ)|µαt (ϕδ )|2 .
Let λδ = β(∆X , ϕδ ) (≤ 0 by Proposition 5.8 and positive definiteness of ϕδ ). Then
∫
µαt (ϕδ )
∞
=
0
∞
∫
= exp[−t
0
∫
α
µB
u (ϕδ )νt (du)
(1 − e
−u|λδ |
∞
=
e−u|λδ | ν α (du)
0
∫
α
)ξ (du)] = exp[−t
G
∫
(1 − ϕδ )dη α ],
∑
∫
where η α (dx) = [ 0∞ hu (x)ξ α (du)]dx. It follows that ρX ((hαt )2 ) = δ∈ĜX d(δ)e−2t (1−Re ϕδ )dη .
Comparing this with (5.34) shows that if η asymptotically dominates η α at e, then the
series in (5.34) converges. By (4.34), η α ≍ r−n−2α .ρG at e. Therefore, if η asymptotically
144
α
dominates r−d−2α .ρG at e, then (5.34) converges. This proves (5.33), and its absolute and
uniform convergence.
It remains to prove the smoothness of pt (x). Let ψδ = ϕδ ◦ π. By (5.3) in the proof
of Lemma 5.1, for any integer m > 0, ∥ψδ ∥(m) ≤ cm (1 + |λδ |)m−1 ∥ψδ ∥(0) = cm (1 + |λδ |)m−1
for some constant cm > 0 independent of δ. By the Taylor expansion and using ∥ψδ ∥(2) ≤
c2 (1 + |λδ |), we obtain
|ϕδ − ϕδ (o) −
n
∑
ϕi ξi ϕδ (o)| ≤ b1 (1 + |λδ |)r2
i=1
in a neighborhood U of o for some constant b1 > 0. Then
|
∫
X
[ϕδ − ϕδ (o) −
n
∑
ϕi ξi ϕδ (o)]dη| ≤ b1 (1 + |λδ |)
i=1
∫
U
r2 dη + b2 η(U c ) ≤ b3 (1 + |λδ |)
for some constants b2 > 0 and b3 > 0. By (5.7), |Lϕ̄δ (o)| ≤ b4 (1 + |λδ |) for some constant
b4 > 0. Let ζ1 , . . . , ζm ∈ g. Applying (d/dt)n ζ1 · · · ζm to the series in (5.33) with ϕδ replaced
by ψδ , the resulting series is dominated in absolute value by
cm
∑
d(δ)[b4 (1 + |λδ |)]n (1 + |λδ |)m−1 etRe[Lϕδ (o)] .
(5.35)
δ
If we can prove
Re[Lϕδ (o)] ≤ −a|λδ |α + b
for some constants a > 0, b ≥ 0 and α > 0, then [b4 (1 + |λδ |)]n (1 + |λδ |)m−1 eεRe[Lϕδ (o)] is
bounded in λδ for any fixed ε > 0. It follows from the convergence of the series in (5.34)
that the series (5.35) converges for t > 0. This implies the smoothness of pt (x) in (t, x).
∑
As in the proof of Proposition 5.8, T ϕδ (o) = ni=1 λi ξi ξi ϕδ (o) with λi ≥ 0 and ξi ξi ϕδ (o) ≤
0. If the diffusion part is nondegenerate, then λi ≥ a for some a > 0. Then Re[Lϕδ (o)] ≤
∑
β(T, ϕδ ) = T ϕδ (o) ≤ a ni=1 ξi ξi ϕδ (o) = a∆X ϕδ (o) = aβ(∆X , ϕδ ) = aλδ = −a|λδ |. This
proves the smoothness of pt under a nondegenerate diffusion part.
Now assume η asymptotically dominates η α ≍ r−n−2α .ρX at o. Let Lα be the generator
of µαt , the convolution semigroup associated to the α-subordinated Brownian motion on X.
Then
α
exp[tRe L ϕδ (o)] =
= |
∫
0
∞
|µαt (ϕδ )|
ν (du) = |
∫
u∆X ϕδ (o) α
e
0
=|
∞
∫
∞
0
α
µB
u (ϕδ )νt (du)|
euλδ νtα (du)
∫
=
0
∞
e−u|λδ | νtα (du) = e−t|λδ | .
α
This shows that Re Lα ϕδ (o) = −|λδ |α . Because η asymptotically dominates the Lévy measure
η α at o, Re Lϕδ (o) ≤ aRe Lα ϕδ (o) + b for some constants a > 0 and b > 0. 2
145
Remark 5.19 If X is of compact type, then G is semisimple. By Proposition 5.12, Lϕδ (o) =
∫
β(T, ϕδ ) − (1 − ϕδ )dη. By Proposition 5.8, β(T, ϕδ ) is real and ≤ 0, it follows that (5.33)
may be written as
∑
pt (x) = 1 +
d(δ) et[β(T,ϕδ )−
∫
(1−ϕ̄δ )dη]
ϕδ (x).
(5.36)
δ∈ĜK+
Remark 5.20 The proof of Theorem 5.18 shows that the derivative of qt may be obtained
by a term-by-term differentiation in the series (5.33), that is, g ∈ G,
(d/dt)n ξi1 · · · ξim (pt ◦ π)(g) =
∑
d(δ)[Lϕ̄δ (o)]n etLϕ̄δ (o) ξi1 · · · ξim (ϕδ ◦ π)(g),
δ∈ĜK
where the series converges absolutely and uniformly for (t, g) ∈ (0, ∞) × G.
∫
Let ∥f ∥2 = [ X f (x)2 dx]1/2 be the L2 -norm of a function f ∈ L2 (X) and let ∥f ∥∞ =
supx∈X |f (x)| for f ∈ Bb (X). The total variation norm of a signed measure µ on X is
defined, as in §4.2, by ∥µ∥tv = sup |µ(f )| with f ranging over f ∈ Bb (X) with |f | ≤ 1. The
corresponding norms on G are defined similarly and will be denoted in the same way. The
following result shows the convergence of µt to the normalized Haar measure ρX on X as
t → ∞ at an exponential rate under the total variation norm, which is stronger than the
weak convergence.
Theorem 5.21 In Theorem 5.18, Re[Lϕδ (o)] < 0 for δ ∈ ĜK+ . Moreover, there is a largest
number −λ among these negative numbers, and for any ε > 0, there are constants c1 > c2 > 0
such that for t > ε,
∥pt − 1∥∞ ≤ c1 e−λt ,
c2 e−λt ≤ ∥pt − 1∥2 ≤ c1 e−λt ,
c2 eλt ≤ ∥µt − ρX ∥tv ≤ c1 eλt .
(5.37)
Proof By Proposition 5.8, β(∆X , ϕδ ) ≤ 0. If β(∆X , ϕδ ) = 0, then ∆X ϕδ = 0, that is, ϕδ
is a harmonic function on X. It is well known that on a compact Riemannian manifold,
only harmonic functions are constants. Then β(∆X , ϕδ ) < 0 for δ ∈ ĜK+ . It is shown in
the proof of Theorem 5.18 that, if µt has a nondegenerate diffusion part, then Re[Lϕδ (o)] ≤
∫
aβ(∆X , ϕδ ) < 0. If the diffusion part is zero, then Re[Lϕδ (o)] = (1 − Re ϕδ )dη. Under the
asymptotic condition on η, supp(η) contains a neighborhood U of o. Then Re[Lϕδ (o)] = 0
would imply ϕδ = 1 on U , and hence ϕδ = 1 on X because ϕδ is analytic. This is impossible
for δ ∈ ĜK+ . We have proved Re[Lϕδ (o)] < 0 for δ ∈ ĜK+ . The convergence of the series in
(5.34) implies Re[Lϕδ (o)] → −∞ as δ leaves any finite subset of ĜK . Therefore, there is a
largest number −λ among Re[Lϕδ (o)] < 0, δ ∈ ĜK+ .
By (5.33), for t > ε,
|pt − 1| ≤ e−λ(t−ε)
∑
d(δ)eεRe[Lϕδ (o)] = [
δ
∑
δ
146
d(δ)eε Re Lϕδ (o) ]eλε e−λt .
Then ∥pt − 1∥∞ ≤ c1 e−λt for some constant c1 > 0. This proves the inequality for ∥pt − 1∥∞
in (5.37), from which the upper bounds for ∥pt − 1∥2 and ∥µt − ρG ∥tv follow. The lower
bounds follow from ∥pt − 1∥22 ≥ d(δ)2 e2tRe[Lϕδ (o)] and ∥µt − ρX ∥tv ≥ |µt (ϕδ )| = etRe[Lϕδ (o)] for
any δ ∈ Ĝ+ . 2
Example 5.22 (Sphere S 2 ): Let G = SO(3) and K = diag{1, SO(2)}. Then X = G/K
may be identified with the unit sphere S 2 in R3 , a symmetric space of compact type. We
may identify K with SO(2). In Example 4.20, the irreducible representations of G = SO(3)
are determined to be Ũn with dimension 2n + 1 and character χ̃n for n = 0, 1, 2, 3, . . .. Note
that the inverse image of K under the covering map p: SU (2) → SO(3), defined in §4.5, is
p−1 (K) = {diag(eθi , e−θi ); 0 ≤ θ ≤ π}, and the vector z1n z2n in the representation space V2n
of U2n = Ũn ◦ p is fixed by p−1 (K). It follows that all the irreducible representation Ũn of
G = SO(3) are spherical.
In Example 4.20, G = SO(3) is equipped with the bi-invariant Riemannian metric on
G = SO(3) induced by the Ad(G)-invariant inner product ⟨ξ, ζ⟩ = (1/2)Tr(ξζ ∗ ) on g = o(3).
Let S 2 be equipped with the G-invariant Riemannian metric induced by this inner product
via the transitive action of G on S 2 as defined in §3.4. As mentioned in Example 3.29,
this metric on S 2 is just the usual Riemannian metric on S 2 as the unit sphere in R3 . By
Theorem 3.28, the Laplace-Beltrami operator ∆SO(3) on SO(3) is π-related to the LaplaceBeltrami operator ∆S 2 on S 2 , where π: SO(3) → S 2 is the natural projection G → G/K.
By Theorem 5.17, the spherical functions on G = SO(3) with respect to the symmetric
∫
space X = G/K is given by ψn (g) = K χ̃n (kg)dk for g ∈ G, n = 0, 1, 2, 3, . . .. Let ϕn be
the associated spherical functions on X = S 2 , determined by ϕn ◦ π = ψn . Because χ̃n is an
eigenfunction of ∆SO(3) with eigenvalue −n(n + 1) and ∆SO(3) is bi-invariant,
∫
∆SO(3) ψn (g) =
K
(∆SO(3) χ̃n )(kg)dk = −n(n + 1)
∫
K
χ̃n (kg)dk = −n(n + 1)ψn (g).
This shows that ψn is an eigenfunction of ∆SO(3) of eigenvalue −n(n + 1). Because ∆SO(3) is
π-related to ∆S 2 , it follows that ϕn is an eigenfunction of ∆S 2 with the same eigenvalue.
We will now obtain an expression for the spherical function ϕn on S 2 in terms of Legendre
polynomials. Let (x1 , x2 , x3 ) be the standard coordinates on R3 and let (θ, α) be the spherical
coordinates on the unit sphere S 2 , where θ ∈ [0, π] is the angle from x1 -axis and α ∈ [0, 2π] is
the angle in (x2 , x3 )-plane. It is well known that the spherical Laplacian ∆S 2 under spherical
coordinates takes the following form:
∂2
∂
1 ∂2
+
cot
θ
+
.
∂θ2
∂θ sin2 θ ∂α2
As a spherical function, ϕn is K-invariant, and hence it is a function of θ alone and may be
written as ϕn (θ). Then
∆S 2 =
ϕ′′n (θ) + cot θ ϕ′n (θ) = −n(n + 1)ϕn (θ).
147
(5.38)
The well known Legendre’s differential equation
(1 − x2 )P ′′ (x) − 2xP ′ (x) + n(n + 1)P (x) = 0
(5.39)
has a polynomial solution
1 dn 2
(x − 1)n
(5.40)
2n n! dxn
normalized by Pn (1) = 1 for each integer n = 0, 1, 2, 3, . . .. A simple computation shows
that Pn (cos θ) satisfies equation (5.38), and hence Pn (cos θ) is a K-invariant eigenfunction of
∆S 2 of eigenvalue −n(n + 1). Because the sphere S 2 as a homogeneous space G/K is clearly
isotropic, by Proposition 5.6, Pn (cos θ) is a spherical function. This implies
Pn (x) =
ϕn (θ) = Pn (cos θ).
(5.41)
Let F : X = S 2 → [0, π] be the map x → θ if x has spherical coordinates (θ, α). For
any K-invariant measure η on X, let η ′ = JF be its projection on [0, π]. Then η is a
∫
Lévy measure on X if and only if η ′ ({0}) = 0 and 0π θ2 η ′ (dθ) < ∞.
Because S 2 = SO(3)/SO(2) is of compact type, by Remark 5.19, the formula (5.33) in
Theorem 5.18 takes the following form on X = S 2 when T = a∆S 2 for some constant a > 0.
qt (θ) = 1 +
∞
∑
∫π
(2n + 1)e−an(n+1)t−
0
[1−Pn (cos β)]η ′ (dβ)
Pn (cos θ)
(5.42)
n=1
for θ ∈ [0, π] and t > 0. Moreover, λ = −2a in (5.37). Note that qt is the density with
respect to the uniform distribution ρS 2 on S 2 given by ρS 2 (dθ, dα) = sin θ dθ dα.
Remark 5.23 Let G be a connected compact Lie group. The product group G × G acts
transitively on G via the map
(G × G) × G ∋ ((g, h), x) 7→ gxh−1 ∈ G,
(5.43)
and the isotropy subgroup at the identity element e of G is Γ = {(g, g); g ∈ G}, the diagonal
of G × G. Thus, G may be regarded as the homogeneous space (G × G)/Γ with the natural
projection π: (G × G) → G given by (g, h) 7→ gh−1 . In fact, G = (G × G)/Γ is a symmetric
space with Cartan involution Θ: (g, h) 7→ (h, g). Under the above identification of (G×G)/Γ
and G, the Γ-invariance on (G × G)/Γ corresponds to the conjugate invariance on G. By the
discussion at the end of [34, IV.3] (page 407), the spherical functions on G = (G × G)/Γ are
precisely the normalized characters ϕδ = χδ /d(δ), δ ∈ Ĝ, which are denoted as ψδ in §4.1. It
follows that when G is connected, Theorems 5.18 and 5.21 correspond precisely to (a) and
(b) of Theorem 4.17, with the additional conclusion that pt is smooth in (t, x) ∈ (0, ∞) × G.
Note that a Lévy process in the Lie group G is more general than a Lévy process in the
symmetric space G = (G × G)/Γ, because the latter is invariant under the action of G × G
148
given by (5.43), and hence is a bi-invariant Markov process in G, or equivalently, a conjugate
invariant Lévy process. The convolution semigroup µt associated to the latter consists of
conjugative invariant probability measures on G.
Remark 5.24 A comparison between (5.33) and (4.39) suggests that the normalized character ϕδ as a spherical function on the symmetric space G = (G × G)/Γ is associated to
a spherical representation of G × G of dimension d(δ)2 as in Theorem 5.17. This spherical
representation of G × G is identified as follows.
We will need the notion of a conjugate representation. For any complex linear space
V , its conjugate V̄ is the same additive space but has the scalar multiplication defined
by (a, v) 7→ āv for (a, v) ∈ C × V . Any representation U of G on V induces naturally a
representation Ū on V̄ , called the representation conjugate to U (see [12, section II.3]), which
is irreducible if so is U . Let U = U δ and V = Cd(δ) . The irreducible representation U of G
on V induces naturally a representation U ⊗ Ū of G × G on the tensor product V ⊗ V̄ of dim
d(δ)2 , via the map ((g, h), (u ⊗ v)) 7→ (U (g)u) ⊗ (Ū (h)v) for g, h ∈ G, u ∈ V and v ∈ V̄ , that
is clearly irreducible with character χδ (x)χ̄δ (y) for (x, y) ∈ G × G. Suppose it is spherical.
Then the spherical function on G × G associated to U ⊗ Ū in Theorem 5.17 is
∫
G
=
∫
χδ (gx)χ̄δ (gy)dg =
∑∫
χδ (g)χ̄δ (gx−1 y)dg =
Ūii (g −1 )Upj (g −1 )Ūpj (x−1 y)dg =
i,j,p
∑∫
∑∫
Uii (g)Ūjp (g)Ūpj (x−1 y)dg
i,j,p
Ūii (g)Upj (g)Ūpj (x−1 y)dg.
i,j,p
∑
By Peter-Weyl Theorem, the above is i Ūii (x−1 y)/d(δ) = χ̄δ (x−1 y)/d(δ) = ϕ̄δ (x−1 y) =
ϕδ (y −1 x) = ϕδ (xy −1 ) (because ϕδ is conjugate invariant). Because it is not identically 0,
by Theorem 5.17(a), U ⊗ Ū must be spherical. It follows that the spherical function ϕδ =
χδ /d(δ) on the symmetric space G = (G × G)/Γ is associated to the irreducible spherical
representation U ⊗ Ū of G × G, of dimension d(δ)2 , as in Theorem 5.17.
5.6
Noncompact and Euclidean types
Now assume X = G/K is a symmetric space of noncompact type. Recall the Lie algebra
g of G is a direct sum of the Lie algebra k of K and an Ad(K)-invariant subspace p, and
the Killing form B is negative definite on k and positive definite on p. Let ⟨·, ·⟩ be the inner
product on g induced by the Killing form B, defined by setting it equal to B on p and −B on
k with k and p orthogonal. Then ⟨·, ·⟩ is Ad(K)-invariant. We will state some basic facts on
noncompact type symmetric spaces, see [33, Chapter VI] or [55, Chapter 5] for more details.
By Theorem 1.1 in [33, Chapter VI], G is diffeomorphic to p×K via the map (ξ, k) 7→ eξ k
from p × K onto G. Therefore, X = G/K may be identified with p via the diffeomorphism
149
X → p, given by eξ K 7→ ξ for ξ ∈ p. Under this identification, the K-action on G/K
corresponds to the Ad(K)-action on p.
Let a be a maximal abelian subspace of p. A real linear functional α on a is called a root
if the linear subspace
gα = {ξ ∈ g; ad(H)ξ = α(H)ξ for H ∈ a},
(5.44)
called the roof space of α, is nonzero. There are finitely many roots with α ≡ 0 being one of
them, and
g = g0 ⊕
∑
gα
(a direct sum).
(5.45)
α̸=0
The hyperplanes α = 0 for nonzero roots α divide a into several convex conical open sets,
called Weyl chambers. Fix a Weyl chamber a+ . A nonzero root α is called positive if α > 0
on a+ , otherwise it is negative, that is, −α is positive.
Let M and M ′ be respectively the centralizer and the normalizer of a in K, that is,
M = {k ∈ K; Ad(k)ξ = ξ for ξ ∈ a} and M ′ = {k ∈ K; Ad(k)a ⊂ a}. Then M is a
normal subgroup of M ′ and the quotient group W = M ′ /M is finite. We may regard W as
a group of linear transformations on a, called the Weyl group. Note that the m = k ∩ g0 is
the common Lie algebra of M and M ′ .
∑
Let n = α>0 gα , the subspace of g spanned by the root spaces of positive roots. Then
n is a Lie algebra, which is in fact a nilpotent Lie algebra, that is, there is an integer j such
that all j-fold Lie brackets formed by the elements of n are 0. Let A and N be respectively
the connected subgroups of G with Lie algebras a and n respectively. Then the exponential
map of G is a diffeomorphic bijections from a to A and from n to N . Note that for a ∈ A,
mam−1 = a for m ∈ M and m′ am′−1 ∈ A for m′ ∈ M ′ .
Let A+ = exp(a+ ). Then A+ = exp(a+ ), where the overline denotes the closure.
The Cartan decomposition (Theorem 1.1 in [33, Chapter IX] or Theorem 5.2 in [55])
G = KA+ K
(5.46)
says that any g ∈ G may be written as g = k1 ak2 for a unique a ∈ A+ and some k1 , k2 ∈ K.
Moreover, if a ∈ A+ , then all possible choices for (k1 , k2 ) are given by (k1 m, m−1 k2 ) with
m ∈ M . Let π: G → X = G/K be the natural projection, and let X ′ = π(A+ K), called
the set of regular points on X. Then X ′ is an open subset of X whose complement is lower
dimensional, that is, X ′ contained in the union of finitely many submanifolds of dimensions
< dim(X). The Cartan decomposition on G induces the polar decomposition on X,
X = KA+ o,
150
(5.47)
where o = π(e), in the sense that any x ∈ X may be written as x = kao for a unique a ∈ A+
and some k ∈ K. Moreover (see [55, Proposition 5.16]),
X ′ = A+ × (K/M ),
(5.48)
in the sense that the map (a, kM ) 7→ kao is a diffeomorphism from A+ × (K/M ) onto X ′ .
The Iwasawa decomposition (Theorem 5.1 in [33, Chapter VI] or Theorem 5.3 in [55])
G = N AK
(5.49)
says that the map N × A × K → G given by (n, a, k) 7→ g = nak is a diffeomorphism.
For g ∈ G, let H(g) ∈ a be defined by the Iwasawa decomposition g = neH(g) k, and for
x ∈ X = G/K, let H(x) = H(g) with x = π(g). This is independent of the choice for g.
Let
1∑
ρ=
mα α,
(5.50)
2 α>0
where mα = dim(gα ), called the multiplicity of root α. Let a∗ be the dual space of a, the
space of linear functionals on a, and let a∗C be the complexification of a∗ .
By Theorem 4.3 in [34, Chapter IV], the spherical functions on X = G/K are given by
∫
ϕλ (x) =
exp[(iλ + ρ)(H(kx))]dk
(5.51)
K
√
for x ∈ X = G/K and λ ∈ a∗C , where i = −1 and dk is the normalized Haar measure on
K. Moreover, ϕλ = ϕµ for λ, µ ∈ a∗C if and only if µ = λ ◦ w for some w ∈ W .
By Exercise B.9 in [34, Chapter IV], if λ ∈ a∗ , then ϕλ is positive definite and hence is
bounded. It is easy to see that ϕλ = ϕ−λ for λ ∈ a∗ .
Let the linear space a, equipped with the inner product ⟨·, ·⟩ (the restriction on a), be
identified with the Euclidean space Rm , and let dH denote the Lebesgue measure on a.
With the natural duality between a and its dual space a∗ , ⟨·, ·⟩ on a induces an inner
product on a∗ by setting ⟨λ, µ⟩ = ⟨Hλ , Hµ ⟩ for λ, µ ∈ a∗ , where Hλ ∈ a is determined by
λ = ⟨·, Hλ ⟩. Let dλ be the induced Lebesgue measure on a∗ .
Because G is unimodular, so a left Haar measure is also invariant under right translations and inverse map, and will be simply called a Haar measure. Let dg be a Haar measure
on G and let dx = π(dg) be the induced G-invariant measure on X = G/K. By Proposition 1.10 (b), a G-invariant measure on X is unique up to a constant factor. We will
∞
(X), the space of K-invariant functions in Cc∞ (X),
normalize dx on X so that for f ∈ Cc,K
∫
∫
f (eH o)
f (x)dx =
X
a+
∏
α>0
(see Exercise C4 in [34, Chapter IV]).
151
(eα(H) − e−α(H) )mα dH
(5.52)
∞
The spherical transform of a function f in Cc,K
(X) is defined by
fˆ(λ) =
∫
X
f (x)ϕ−λ (x)dx
for λ ∈ a∗C .
(5.53)
This spherical transform is similar to the classical Fourier transform on Rd . The action
of the Weyl group W on a induces an action on a∗C , given by (w, λ) 7→ λ ◦ w−1 for w ∈ W and
λ ∈ a∗C . Let HW (a∗C ) be the space of W -invariant entire functions ψ(λ) on a∗C of exponential
type, that is, for some r > 0 and any integer n > 0, there is a constant c = c(r, n) > 0 such
that
∀λ ∈ a∗C , |ψ(λ)| ≤ c(1 + |λ|)−n er|Im(λ)| .
(5.54)
By the Paley-Wiener Theorem (Theorem 7.1 in [34, IV]), f 7→ fˆ is a linear bijection from
∞
the space Cc,K
(X) onto the space HW (a∗C ).
The Harish-Chandra’s c-function c(λ) is a meromorphic function on a∗C , see (43) in [34,
IV.Theorem 6.14]. Its reciprocal on a∗ has a polynomial growth in the sense that
|c(λ)|−1 ≤ c1 + c2 |λ|dim(n)/2
(5.55)
for all λ ∈ a∗ and some constants c1 > 0 and c2 > 0, see IV.Proposition 7.2 in [34].
∞
By Theorem 7.5 in [34, Chapter IV]. for f ∈ Cc,K
(X), we have the inversion formula
∫
f (x) = c0
a∗
fˆ(λ)ϕλ (x)|c(λ)|−2 dλ
with
c0 =
1
,
(2π)m |W |
(5.56)
and the Plancherel formula
∫
|f (x)| dx = c0
∫
2
X
a∗
|fˆ(λ)|2 |c(λ)|−2 dλ,
(5.57)
where m = dim(a) and |W | is the cardinality of the Weyl group W . The normalizing
constant c0 given here is obtained from Exercise C4 in [34, Chapter IV], but note that the
measures dH and dλ in [34] are normalized differently. They are equal to the present dH
and dλ divided by (2π)m/2 , and hence dx and fˆ(λ) are also changed accordingly by a factor
of (2π)m/2 .
∞
(X), fˆ form a dense subset
Moreover, by [34, IV.Theorem 7.5], as f varies over Cc,K
of the space L2W (a∗ ) of W -invariant L2 functions on a∗ under the measure c0 |c(λ)|−2 dλ.
Consequently, by the Plancherel formula (5.57), the spherical transform f 7→ fˆ extends to
the Hilbert space L2K (X) of K-invariant L2 functions on X under measure dx to become a
norm-preserving map from L2K (X) onto L2W (a∗ ). For f ∈ L2K (X), its spherical transform
fˆ ∈ L2W (a∗ ) is defined to be its image under this map.
Proposition 5.25 If f ∈ L2K (X) ∩ L1 (X), then fˆ(λ), λ ∈ a∗ , is given by (5.53). Moreover,
if in addition, fˆ ∈ L1 (a∗ ), then the inversion formula (5.56) holds as well.
152
∞
Proof The first part is proved using a sequence of fn ∈ Cc,K
(X) such that fn → f in
2
1
both L (X) and L (X). To see the existence of such fn , first choose K-invariant, relatively
compact and open sets Xn ↑ X as n ↑ ∞ and let hn = f 1Xn , next for each n, choose
∞
hmn ∈ Cc,K
(Xn ) such that hmn → hn in L2 (X) as m → ∞, and then hmn → hn also
in L1 (X) because Xn has a finite measure. The desired sequence is obtained by setting
fn = hnmn for a suitably chosen subsequence mn .
∞
If fˆ ∈ L1 (a∗ ), then by (5.57), for any g ∈ Cc,K
(X), writing γ(dλ) for c0 |c(λ)|−2 dλ,
∫
∫
f (x)g(x)dx =
∫X ∫
[
=
X
a∗
a∗
¯
fˆ(λ)ĝ(λ)γ(dλ)
=
∫
a∗
fˆ(λ)[
∫
X
g(x)ϕλ (x)dx]γ(dλ)
fˆ(λ)ϕλ (x)γ(dλ)]g(x)dx.
∫
This shows f (x) = a∗ fˆ(λ)ϕλ (x)γ(dλ), which is (5.56). The interchange of integration order
∫ ∫
in the above computation is justified because X a∗ |fˆ(λ)||g(x)|γ(dλ)dx < ∞. 2
Proposition 5.26 A finite K-invariant measure µ on X is completely determined by its
spherical transform µ(ϕλ ) as λ ranges over a∗ .
Note: This is not a direct consequence of Theorem 5.10 because ϕλ , λ ∈ a∗ , do not form
the complete set of bounded spherical functions (see [34, IV.8]).
∞
Proof of proposition For any f ∈ Cc,K
(X),
∫
µ(f ) = µ(c0
a∗
fˆ(λ)ϕλ |c(λ)|−2 dλ) = c0
∫
a∗
fˆ(λ)µ(ϕλ )|c(λ)|−2 dλ,
where the interchange of µ- and dλ- integrations is justified by the fact that fˆ is of exponential
type satisfying (5.54). This implies that µ(ϕλ ) with λ ∈ a∗ determines µ(f ) for any f ∈
∞
Cc,K
(X), and hence also µ. 2
Theorem 5.27 Let X = G/K be a symmetric space of noncompact type, and let µt be a
continuous convolution semigroup on X with µ0 = δo . Assume either µt has a nondegenerate
diffusion part, or its Lévy measure η asymptotically dominates r−β .ρX at o for some β > n
(defined in §4.3), where r is the distance to o under a Riemannian metric on X (this condition
is independent of the choice of a Riemannian metric on X), and n = dim(X). Then µt has
a density pt = dµt /dx for t > 0 such that for t > 0 and x ∈ X,
∫
pt (x) = c0
∫
a∗
et[β(T,ϕλ )−
(1−ϕλ )dη]
ϕλ (x)|c(λ)|−2 dλ,
(5.58)
where the integrand is absolutely and uniformly integrable for (t, x) ∈ [ε, ∞) × X for any
ε > 0. Moreover, qt (x) is smooth in (t, x) ∈ (0, ∞) × X
153
Proof By Propositions 5.12 and 5.8, the simplified Lévy-Khinchin formula (5.10) holds for
ϕ = ϕλ , and β(T, ϕ̄λ ) = β(T, ϕλ ) ≤ 0. Because a nondegenerate diffusion process in X has a
bounded transition density for fixed t > 0 (see for example the second half of Proposition 4.4
on page 143 in [6, chapitre 8]), if L = T is nondegenerate, then pt = dµt /dx is bounded and
hence in L2 . The rest of the proof is essentially the same as that of Theorem 5.18, replacing
the series by integral and the Parveval identity by the Plancherel formula. The expression
for β(∆X , ϕλ ) in Remark 5.28 below is needed for proving the smoothness. 2
Remark 5.28 By (7) in [34, IV.5], β(∆X , ϕλ ) = −(⟨λ, λ⟩ + ⟨ρ, ρ⟩). If X is irreducible, then
T = a∆X for some constant a > 0 and β(T, ϕλ ) = −a(⟨λ, λ⟩ + ⟨ρ, ρ⟩) in (5.58).
We will introduce a symmetric space of Euclidean type via a symmetric pair (G, K) of
noncompact type. Let G0 = K ×s p be the semi-direct product defined by
(k, ξ) · (h, η) = (kh, ξ + Ad(k)η).
Then G0 is a Lie group with identity element (e, 0) and inverse (k, ξ)−1 = (k −1 , −Ad(k −1 )ξ).
The map: (k, ξ) 7→ (k, −ξ) is a Cartan involution on G0 with fixed point set K0 = K ×s {0},
and X0 = G0 /K0 is a symmetric space of Euclidean type, which may be identified with p via
the map: (k, ξ)K0 7→ ξ. By IV.Proposition 4.8 in [34], the spherical functions ϕ on X0 ≡ p
are given by
∫
ψλ (ξ) =
ei⟨Hλ , Ad(k)ξ⟩ dk
(5.59)
K
∗
for ξ ∈ p and λ ∈ aC . Moreover, ψλ = ψµ if and only if µ = λ ◦ w for some w ∈ W . It is
∑
easy to see that for λ ∈ a∗ , ψλ is positive definite on p, that is, p,q ψ(ξp − ξq )ap āq ≥ 0 for
any finite set of ξp ∈ p and ap ∈ C. Therefore, ψλ is bounded on p for λ ∈ a∗ .
Note that with the identification of p with Rn , the convolution product of Ad(K)-invariant
measures on p, regarded as the homogeneous space X0 = G0 /K0 , is the usual convolution
product on Rn .
The spherical transform of an Ad(K)-invariant function f on p is defined by
fˆ(λ) =
∫
p
f (ξ)ψ−λ (ξ)dξ, λ ∈ a∗ ,
(5.60)
where dξ is the Lebesgue measure on p determined by the inner product ⟨·, ·⟩ on g restricted
to p, provided the integral exists.
Let SK (p) and SW (a∗ ) be respectively the Ad(K)-invariant and W -invariant rapidly
decreasing functions on p and on a∗ . Here a function f on Rn is called rapidly decreasing if
for any integer m > 0 and polynomial P (x1 , . . . , xn ),
sup |x|m |P (
x
∂
∂
,...,
)f (x)| < ∞.
∂x1
∂xn
154
By the discussion in [34, IV.9], the spherical transform f 7→ fˆ is a bijection from SK (p) onto
SW (a∗ ), and for f ∈ SK (p),
∫
f (ξ) = c1
a∗
and
fˆ(λ)ψλ (ξ)δ1 (Hλ )dλ,
∫
∫
n
c1 = Vn [(2π)
|f (ξ)| dξ = c1
∫
2
∏
p
a∗
H∈a+ , ∥H∥≤1
δ1 (H)dH]−1 ,
|fˆ(λ)|2 δ1 (Hλ )dλ
(5.61)
(5.62)
where δ1 (H) = α>0 |α(H)|mα and Vn is the volume of the unit ball in p ≡ Rn . The proof
of these results is similar to the proof of the corresponding statements for Fourier transform
on Euclidean spaces, see the proof of Theorem 9.1 in [34, Chapter IV].
Let µt be a continuous convolution semigroup on p ≡ Rn with µ0 = δ0 , which is necessarily
Ad(K)-invariant. Because there is no nonzero vector of p fixed by Ad(K), the simplified
Lévy -Khinchin formula (5.10) holds. The following result, like Theorem 5.27, may be proved
in essentially the same way as Theorem 5.18.
Theorem 5.29 Assume either µt has a nondegenerate diffusion part, or its Lévy measure
η asymptotically dominates ∥ξ∥−β dξ at the origin 0 for some β > n, where ∥ξ∥ = ⟨ξ, ξ⟩1/2 .
Then µt has a density pt = dµt /dξ for t > 0, smooth in (t, ξ) ∈ (0, ∞) × p, given by
∫
pt (ξ) = c1
a∗
et[β(T,ψλ )−
∫
(1−ψλ )dη]
ψλ (ξ)δ1 (Hλ )dλ,
(5.63)
where the integrand is absolutely and uniformly integrable for (t, ξ) ∈ [ε, ∞) × p for any
ε > 0.
Remark 5.30 A simple computation yields ∆p ψλ = −⟨λ, λ⟩ψλ , where ∆p is the Laplace
operator on p = Rn . If G/K is irreducible, then T = a∆p for some constant a ≥ 0 and
β(T, ψλ ) = −a⟨λ, λ⟩ in (5.63).
Example 5.31 Let G = SO(1, n)+ be the connected Lorentz group and let K = SO(n) be
regarded as a subgroup of G as in §4.5. When p is identified with Rn and ⟨·, ·⟩ on p with
the standard inner product on Rn , the Ad(K)-action on p is just the usual SO(n)-action on
Rn . Then G0 = K ×s p becomes the connected Euclidean motion group M0 (Rn ) on Rn and
hence an SO(n)-invariant Lévy process in Rn is a Lévy process in X0 = G0 /K0 .
155
156
Chapter 6
Inhomogeneous Lévy processes in Lie
groups
Inhomogeneous Lévy processes in topological groups, defined by independent increments,
were introduced in §1.4. More useful representation of these processes may be obtained on
a Lie group G. The main purpose of this chapter is to present a martingale representation,
which characterizes an inhomogeneous Lévy process in a Lie group by a triple (b, A, η) of a
deterministic drift bt in G, a matrix function A(t) and a measure function η(t, ·), in close
analogy with the representation of a homogeneous Lévy process. The result for stochastically
continuous processes was obtained in Feinsilver [25], generalizing an earlier work in StroockVaradhan [84] for continuous processes. A martingale representation via an abstract Fourier
analysis is obtained in Heyer-Pap [38], where the processes considered are also stochastically
continuous. We will establish the representation for processes that are not assumed to be
stochastically continuous, as obtained in [61].
We will begin in section §6.1 with the classical Lévy -Khinchin formula and some of its
extensions on Euclidean spaces, that provide motivation for the results on Lie groups. The
main results are stated in §6.3. When the drift has a finite variation, the representation takes
a more direct form, this is discussed in §6.7. Some additional properties are considered in
the other sections, including a simpler representation when the Lévy measure function has a
finite first moment. The proofs of the main results in §6.3 will be given in Chapter 7.
6.1
Additive processes in Euclidean spaces
Recall that a rcll process xt in a Lie group with independent increments x−1
s xt , s < t, and
an infinite life time is called an inhomogeneous Lévy process (see §1.4). Such a process
in a Euclidean space Rd (with additive group structure) is also called an additive process.
157
The purpose of this section is to describe the classical Lévy -Itô formula for stochastically
continuous additive processes and its extension to additive processes that are not necessarily
stochastic continuous. This discussion will provide motivation for the results on Lie groups,
but will not be needed for proving these results.
A rcll process xt is said to have a fixed jump at time t > 0 if P (xt ̸= xt− ) > 0. It is
easy to show ([22, 3.Lemma 7.7] that a rcll process can have at most countably many fixed
jumps. It is clear that xt is stochastically continuous if and only if it has no fixed jump.
An additive process xt in Rd with x0 = 0 and with no fixed jump has the the following
well known Lévy -Itô representation (see for example [46, Theorem 15.4]):
∫
xt = bt + Bt +
0
t
∫
∫
Rd
x1[|x|≤1] Ñ (ds, dx) +
0
t
∫
Rd
x1[|x|>1] N (ds, dx),
(6.1)
where bt is a continuous path in Rd with b0 = 0, called a drift, Bt = (Bt1 , . . . , Btd ) is a
d-dimensional inhomogeneous Brownian motion, that is, a d-dimensional continuous Gaussian process of mean zero and independent increments, N (dt, dx) is an independent Poisson
random measure on R+ × Rd with intensity measure η(dt, dx) = E[N (dt, dx)] satisfying
∫
t
0
∫
(|x|2 ∧ 1)η(ds, dx) < ∞
and
η(dt × {0}) = 0,
∫
∫
and Ñ = N − η is the compensated form of N . Recall by our convention, ab = (a, b] .
Conversely, given b, B and N as above, xt in (6.1) is an additive process in Rd with no fixed
jump.
Let Nt (·) = N ([0, t] × ·) and ηt (·) = η([0, t] × ·). By the definition of stochastic integrals
under Poisson random measures, given in Appendix A.6, the first integral in (6.1), denoted
as It , is the L2 -limit of
∫
Itε =
x1[ε<|x|≤1] Nt (dx) −
∫
x1[ε<|x|≤1] ηt (dx)
as ε ↓ 0. Because It and Itε are L2 -martingales, by Doob’s norm inequality (A.4), along a
sequence of ε ↓ 0, almost surely, Itε → It uniformly for t in any finite interval. Because
∫
Nt ({x ∈ Rd ; |x| > ε}) is a finite Poisson random variable, both x1[ε<|x|≤1] Nt (dx) and the
second integral in (6.1) are sums of finitely many nonzero terms almost surely. Because
∫
x1[ε<|x|≤1] ηt (dx) is continuous in t, it is then easy to see from (6.1) that N is the jump
counting measure of process xt , which is defined by (2.11) on a general Lie group and now
takes the following form,
N ([0, t] × B) = #{u ∈ (0, t]; ∆u x ∈ B and ∆u x ̸= 0},
(∆u x = xu − xu− ),
this being the number of the jumps in B during the time interval [0, t], for t > 0 and
B ∈ B(Rd ).
158
We will present an extension of the Lévy -Itô representation to additive processes with
fixed jumps. Recall from Appendix A.6, an extended Poisson random measure N on R+ ×Rd
with intensity measure η = E[N (·)] is an integer-valued random measure such that almost
surely, N ({t}×Rd ) ≤ 1 for all t ∈ R+ , ηt = η([0, t]×·) is σ-finite on Rd , and Nt = N ([0, t]×·)
has independent increments. Note that the measure-valued function ηt = η([0, t] × ·) is
nondecreasing and right continuous in the sense that ηs ≤ ηt for s < t and ηt ↓ ηs as t ↓ s
(that is, ηt (B) ↓ ηs (B) for any B ∈ B(Rd )). The left limit ηt− at any t > 0 is defined as
the nondecreasing limit of ηs as s ↑ t. The extended Poisson random measure N becomes
a Poisson random measure when its intensity η is continuous in t, that is, ηt− = ηt (or
equivalently E[η({t} × Rd )] = 0) for t > 0.
If an additive process xt in Rd has fixed jumps, then its jump counting measure N is not
a Poisson random measure because its intensity is not continuous in t, but by Theorem 6.1
below, it is an extended Poisson random measure on R+ × Rd .
For any extended Poisson random measure N on R+ × Rd with intensity η, let J be the
set of time points t > 0 such that η({t} × Rd ) = E[N ({t} × Rd )] > 0. By the σ-finiteness of
E[N ([0, t] × ·)], J is countable and for each u ∈ J, N ({u} × ·) = δσu 1[σu ̸=0] for some random
variable σu ∈ Rd . Then
∑
N = Nc +
δ(u,σu ) 1[σu ̸=0] ,
(6.2)
u∈J
where N is a Poisson random measure. Moreover, N c and σu , u ∈ J, are independent.
The truncated mean h of a random variable σ in Rd or of its distribution ν is defined to
be
∫
h = E(σ1[|σ|≤1] ) = x1[|x|≤1] ν(dx).
(6.3)
c
It is clear that |h| ≤ 1. Let hu be the truncated mean of σu and set σu = 0 for u ̸∈ J. Note
that the distribution of σu is νu = η({u} × ·) + [1 − η({u} × Rd )]δ0 for any u > 0.
The following theorem, taken from [62] and stated now in a more precise form, provides
an extended Lévy -Itô formula in which the fixed jumps are expressed in a canonical and
convenient form.
Theorem 6.1 Let xt be an additive process in Rd with x0 = 0. Then there is a triple
(b, B, N ), unique almost surely, of a rcll path bt in Rd with b0 = 0, an inhomogeneous
Brownian motion Bt in Rd , and an independent extended Poisson random measure N on
R+ × Rd satisfying, with the notation above, for all t > 0, ∆t b = ht , and
∫
|x|>1
ηt (dx) < ∞,
∫
|x|≤1
|x|2 ηtc (dx)
< ∞,
∑∫
|x1[|x|≤1] − hu |2 νu (dx) < ∞,
u≤t
such that
∫
xt = bt + Bt +
t
0
∫
x1[|x|≤1]
gc (ds, dx) +
N
∫
0
159
t
∫
x1[|x|>1] N c (ds, dx)
(6.4)
+
∑
(σu 1[|σu |≤1] − hu ) +
u≤t
∑
σu 1[|σu |>1] ,
(6.5)
u≤t
where the two integrals in (6.5) are well defined with meanings as in (6.1), the first of the two
∑
∑
sums u≤t (· · ·) = u∈J,u≤t (· · ·) in (6.5) converges in L2 with mutually orthogonal terms, and
the second has finitely many nonzero terms almost surely. Moreover, N is the jump counting
measure of xt .
Conversely, given any triple (b, B, N ) as stated before (6.5), the process xt given by (6.5)
is an additive process in Rd with x0 = 0.
Note 1: The first integral and the the first sum in (6.5) are L2 -martingales in t. Because the
terms of the sum are independent, its L2 -convergence implies the almost sure convergence,
∑
and hence the two sums in (6.5) may be combined into a single sum u≤t (σu − hu ) =
∑
u∈J, u≤t (σu − hu ), which converges almost surely (regardless of the order on J). Moreover,
by the L2 -convergence of the first sum and Doob’s norm inequality (A.4), there is a sequence
∑
of finite subsets J(n) of J such that J(n) ↑ J, and almost surely, u∈J(n), u≤t (σu − hu )
∑
converges to u∈J,u≤t (σu − hu ) uniformly for t in any finite interval.
Note 2: The 1 in (6.3), (6.4) and (6.5) may be replaced by any constant a > 0. This will
only cause a change in the drift bt .
Note 3: The distribution of the inhomogeneous Brownian motion Bt is determined by its
covariance matrix function Aij (t) = E[Bti Btj ], and the distribution of the Poisson random
measure N is determined by its intensity measure η. Thus, the distribution of an additive
process xt in Rd , with x0 = 0, is completely determined by the triple (b, A, η).
Proof: Let xt be an additive process in Rd with x0 = 0, and let N be its jump counting
measure with associated η, νu and σu . Then for any t > 0 and ε > 0,
∑
νu ({x ∈ Rd ; |x| > ε}) =
∑
P (|σu | > ε) < ∞.
(6.6)
u≤t
u≤t
This follows easily from the independence of the fixed jumps and the Borel-Cantelli lemma
because if the sum in (6.6) is ∞, then almost surely xt would have infinitely many fixed
jumps of size > ε by time t, which is impossible because of the rcll paths. It is also clear
∑
that u≤t σu 1[|σu |>1] has finitely many nonzero terms almost surely.
Let ϕ: Rd → Rd be a smooth function with compact support and ϕ(x) = x for x near
the origin 0, and let h′u = νu (ϕ). Then
∑∫
u≤t
Rd
|ϕ(x − h′u )|2 νu (dx) < ∞.
(6.7)
This follows from [45, II.Theorem 5.2] (see in particular (iii) and (v) in 5.5, and the statement
about the law of Xt , noting ϕ and ht here are denoted as h and ∆Bt in [45]). It will also
160
follow directly from the discussion in §6.3, because by (6.20) and Theorem 6.11 (noting h′u
∫
∑
here corresponds to hu in §6.3), u≤t Rd |ϕ(x−h′u )|2 νu (dx) < ∞, which is equivalent to (6.7)
as h′u can be made arbitrarily close to 0 after excluding finitely many u ∈ J by (6.6). The
∫
∑
same reason also implies that (6.7) is equivalent to u≤t Rd |x1[|x|≤1] − h′u |2 νu (dx) < ∞.
∫
Because |x1[|x|≤1] − hu |2 ≤ 2|x1[|x|≤1] − h′u |2 + 2|hu − h′u |2 and hu − h′u = |x|>ε [x1[|x|≤1] −
∑
∑
ϕ(x)]νu (dx) for some ε > 0, and by (6.6), u≤t |hu − h′u |2 ≤ u≤t |hu − h′u | < ∞, it follows
that
∑∫
|x1[|x|≤1] − hu |2 νu (dx) < ∞.
(6.8)
u≤t
By the independence of fixed jumps, E[(σu 1[|σu |≤1] − hu )(σv 1[|σv |≤1] − hv )] = 0 for u ̸= v,
and then by (6.8),
E{[
∑
u≤t
(σu 1[|σu |≤1] − hu )] } =
2
∑∫
|x1[|x|≤1] − hu |2 νu (dx) < ∞.
u≤t
∑
It follows that u≤t (σu 1[|σu |≤1] − hu ) converges in L2 with mutually orthogonal terms. As
∑
∑
noted in Note 1 after Theorem 6.1, the two sums u≤t (σu 1[|σu |≤1] − hu ) and u≤t σu may be
∑
combined into a single sum u≤t (σu − hu ) which converges almost surely. Moreover, because
∑
2
u≤t (σu 1[|σu |≤1] − hu ) is a L -martingale, by Doob’s norm inequality (A.4), it can be shown
that there is a sequence of finite subsets J(n) of J such that J(n) ↑ J, and almost surely,
∑
∑
u≤t (σu − hu ) uniformly for t in any finite interval.
u∈J(n), u≤t (σu − hu ) →
∑
Let yt = xt − u≤t (σu − hu ). Then yt is an additive process whose fixed jumps are non∑
random, indeed, ∆t y = ht almost surely for all t > 0. Because ytn = xt − u∈J(n), u≤t (σu − hu )
is independent of {σu : u ∈ J(n)}, and ytn → yt almost surely, it follows that the process yt
is independent of {σu ; u ∈ J}.
The central value γ(x) of a real-valued random variable x is defined (by Doob) as the
unique real number γ such that E[tan−1 (x − γ)] = 0, see [44, section 0.2]. It is easy to see
that γ(x+r) = γ(x)+r for any real number r, and if xn → x weakly, then γ(xn ) → γ(x). For
a random point x = (x1 , . . . , xd ) in Rd , define γ(x) = (γ(x1 ), . . . , γ(xd )). Let zt = yt − γ(yt ).
Because ∆t y is non-random, γ(yt ) = γ(yt− + ∆t y) = γ(yt− ) + ∆t y. Then zt is an additive
process with no fixed jump, and the Lévy -Itô representation (6.1) holds with xt and N
replaced by zt and N z , where N z is the jump counting measure of zt .
∑
Let bzt be the drift of zt . Because xt = γ(yt ) + zt + u≤t (σu − hu ), we obtain (6.5) with
N c = N z and bt = γ(yt ) + bzt . Comparing jumps on the both sides of this equation, it is
easy to see that the jump counting measure N of xt is given by (6.2), and the jumps of bt
∫
are precisely ∆u γ(y) = hu for u ∈ J. Because (|x| ∧ 1)η z (dx) < ∞, where η z = η c is the
intensity of N z = N c , and by (6.6) and (6.8), the intensity η of N satisfies (6.4). Because xt
has independent increments, so does N . It follows that N is an extended Poisson random
measure.
161
Because N z is independent of the inhomogeneous Brownian motion Bt in (6.1), and zt is
∑
∑
independent of σu , u ∈ J, it follows that N = N c + u>0 δσu 1[σu ̸=0] = N z + u>0 δσu 1[σu ̸=0]
is independent of Bt .
To prove the uniqueness of the triple (b, B, N ), note that by (6.5) and the condition
∆t b = ht , N must be the jump counting measure of xt . If there is another choice for b and
B, say b′ and B ′ , then bt + Bt = b′t + Bt′ and Bt − Bt′ = b′t − bt . Now the zero mean of Bt and
Bt′ implies b′t − bt = 0, and hence Bt = Bt′ .
It is now easy to see that given any triple (b, B, N ) as stated in the second half of
Theorem 5.21, because of (6.4), the integrals and sums in (6.5) are well defined, xt given by
(6.5) is an additive process in Rd . 2
Theorem 6.2 Let xt be an additive process in Rd with x0 = 0 as in Theorem 6.1, let
zt = xt − bt , and let Aij (t) = E(Bti Btj ). Then for f ∈ C0∞ (Rd ),
f (zt ) −
−
∫ t∫
∑
1 ∑∫ t
Di Dj f (zs )dAij (s) −
xi 1[|x|≤1] Di f (zs )]η c (ds, dx)
[f (zs + x) − f (zs ) −
2 i,j 0
0
i
∑∫
[f (zu− + x − hu ) − f (zu− )]νu (dx) (Di =
u≤t
∂
)
∂xi
(6.9)
is a martingale under the natural filtration {Ftx } of the process xt . In particular, if xt has
no fixed jump, then
f (zt ) −
−
∫
t
∫
0
1 ∑∫ t
Di Dj f (zs )dAij (s)
2 i,j 0
[f (zs + x) − f (zs ) −
∑
xi 1[|x|≤1] Di f (zs )]η(ds, dx)
(6.10)
i
is a martingale under Ftx .
∫
gc (dx)
Proof: In (6.5), there are three Rd -valued L2 -martingales, that is, Bt , Yt = |x|≤1 xN
t
∑
and Zt = u≤t (σu 1[|σu |≤1] − hu ). The quadratic variation of the inhomogeneous Brownian motion Bt satisfies [B, B]ct = [B, B]t = ⟨B, B⟩t = A(t). By (A.19) in Appendix A.6,
∫
[Y i , Y j ]t = |x|≤1 xi xj Nt (dx), which is purely discontinuous in t. By the Lemma below,
[Z i , Z j ]t is also purely discontinuous in t.
Lemma 6.3 For the L2 -martingale Zt above,
⟨Z , Z ⟩t =
i
j
[Z i , Z j ]t =
∑∫
(xi 1[|x|≤1] − hiu )(xj 1[|x|≤1] − hju )νu (dx)
d
R
u≤t
∑
(σui 1[|σu |≤1] − hiu )(σuj 1[|σu |≤1] − hju ),
u≤t
where hiu and σui are, respectively, the i-coordinates of hu and σu .
162
i
Proof: For s < t, let Zs,t
=
i j
x
E(Zs Zs,t | Fs ) = 0, and
∑
i
s<u≤t (σu 1[|σu |≤1]
− hiu ). By the independence of σu , u ∈ J,
j
i
E(Zti Ztj | Fsx ] = E[(Zsi + Zs,t
)(Zsj + Zs,t
) | Fsx ]
∑ ∫
j
i
= Zsi Zsj + E(Zs,t
Zs,t
| Fsx ] = Zsi Zsj +
(xi 1[|x|≤1] − hiu )(xj 1[|x|≤1] − hju )νu (dx).
s<u≤t
∫
∑
This shows that Zti Ztj − u≤t (xi 1[|x|≤1] − hiu )(xj 1[|x|≤1] − hju )νu (dx) is a martingale, and
proves the formula for ⟨Z, Z⟩t .
It is easy to show that
∑
(σui 1[|σu |≤1]
−
−
hiu )(σuj 1[|σu |≤1]
hju )
u≤t
−
∑∫
(xi 1[|x|≤1] − hiu )(xj 1[|x|≤1] − hju )νu (dx)
u≤t
is a martingale, and so is Zti Ztj −
∑
i
u≤t (σu 1[|σu |≤1]
− hiu )(σuj 1[|σu |≤1] − hju ). Because
(∆v Z i )(∆v Z j ) = (σvi 1[|σv |≤1] − hiv )(σvj 1[|σv |≤1] − hjv ) = ∆v
∑
(σui 1[|σu |≤1] − hiu )(σuj 1[|σu |≤1] − hju ),
u≤·
it follows that [Z i , Z j ]t =
∑
i
u≤t (σu 1[|σu |≤1]
− hiu )(σuj 1[|σu |≤1] − hju ). Lemma 6.3 is proved.
We now return to the proof of Theorem 6.2. Because [Y, Y ]t and [Z, Z]t are purely discontinuous, [Y, Y ]c = [Z, Z]c = 0. It follows that for any semimartingale X, [X, Y ]c = [X, Z]c =
∫
∑
0. By (6.5), zt = xt − bt is the sum of Bt , Yt , Zt , |x|>1 xNtc (dx) = u̸∈J, u≤t (∆u x)1[|∆u x|>1]
∑
∑
and u≤t σu 1[|σu |>1] = u∈J, u≤t (∆u x)1[|∆u x|>1] . In the rest of the proof, we will write Mt for
a martingale which may change from expression to expression. Applying Itô’s formula (A.9)
to f (zt ) yields
∑∑
1 ∑∫ t
f (zt ) = f (z0 ) + Mt +
Di Dj f (zs )dAij (s) +
Di f (zu− )∆u xi 1[|∆u x|>1]
2 i,j 0
u≤t i
+
∑
[f (zu ) − f (zu− ) −
u≤t
= f (z0 ) + Mt +
+
∑
∫
1∑
2
i,j
0
∑
i
t
Di Dj f (zs )dAij (s)
[f (zu ) − f (zu− ) −
∑
Di f (zu− )(∆u xi 1[|∆u x|≤1] − hu )].
i
u≤t
∑
Di f (zu− )∆u z i ]
∫ ∫
∑
Let Ut = u̸∈J, u≤t [f (zu )−f (zu− )− i Di f (zu− )∆u xi 1[|∆u x|≤1] ] = 0t [f (zu− +x)−f (zu− )−
∑
c
c
c
c
gc
i Di f (zu− )xi 1[|x|≤1] ]N (du, dx). Replacing N by N = N − η , Ut becomes a martingale,
∫t∫
∑
it follows that Ut = Mt + 0 [f (zu− + x) − f (zu− ) − i Di f (zu− )xi 1[|x|≤1] ]η c (du, dx).
∑
∑
Let Vt = u∈J, u≤t [f (zu ) − f (zu− ) − i Di f (zu− )(∆u xi 1[|∆u x|≤1] − hiu )]. It is easy to show
∑
∑
that u∈J, u≤t i Di f (zu− )(∆u xi 1[|∆u x|≤1] − hiu ) is a martingale in t. Because for s < u ∈ J,
E[f (zu ) − f (zu− ) | Fs ] = E{E[f (zu− + σu − hu ) − f (zu− ) | Fu− ] | Fs }
∫
= E{ [f (zu− + x − hu ) − f (zu− )]νu (dx) | Fs },
163
∑
∑
∫
it follows that u∈J, u≤t [f (zu ) − f (zu− )] − u≤t [f (zu− + x − hu ) − f (zu− )]νu (dx) is a mar∫
∑
tingale, and hence Vt = Mt + u≤t [f (zu− + x − hu ) − f (zu− )]νu (dx).
Summarizing the preceding computation shows that (6.9) is a martingale under Ftx . 2
The martingale property in Theorem 6.2 is stated for the shifted process zt = xt − bt ,
not directly for xt , because an additive process xt may not be a semimartingale. A rcll path
bt in Rd will be said to have a finite variation if each of its components bit has a finite total
∑
variation on any finite time interval. Then bt = bct + u≤t ∆u b, where bct is the continuous
part of bt . When the drift bt of an additive process xt has a finite variation, then xt is a
semimartingale. In this case, a martingale property holds directly for xt as stated below.
Theorem 6.4 Let xt be an additive process in Rd with x0 = 0 as in Theorem 6.1. Assume
its drift bt has a finite variation. Then for f ∈ Cc∞ (Rd ),
f (xt ) −
−
∑
∑∫
i
0
t
Di f (xs )dbcs i −
∫ t∫
1 ∑∫ t
[f (xs + x) − f (xs )
Di Dj f (xs )dAij (s) −
2 i,j 0
0 Rd
xi 1[|x|≤1] Di f (xs )]η c (ds, dx) −
i
∑∫
d
u≤t R
[f (xu− + x) − f (xu− )]νu (dx)
(6.11)
is a martingale under Ftx .
Proof: Apply Itô’s formula (A.9) to f (xt ) and repeat the computation in the proof of
∑ ∫
∑ ∫
∑
∑
Theorem 6.2, noting i 0t Di f (xs− )dbs = i 0t Di f (xs )dbcs i + u≤t i Di f (xu− )hu . 2
6.2
Measure functions
This section is devoted to some preparation for the main results to be represented in the
next section.
A measure function on a measurable space (S, S) is a family of σ-finite measures η(t, ·)
on S, t ∈ R+ , that is nondecreasing and right continuous in the sense that η(s, ·) ≤ η(t, ·)
for s < t, and η(t, B) → η(s, B) as t ↓ s for any B ∈ S such that η(u, B) < ∞ for some
u > s. The left limit η(t−, ·) at t > 0, defined as the nondecreasing limit of measures η(s, ·)
as s ↑ t, exists and is ≤ η(t, ·).
A measure function η(t, ·) may be regarded as a σ-finite measure on R+ × S, determined
by η((s, t] × B) = η(t, B) − η(s, B) for s < t and B ∈ S with η(t, B) < ∞. Conversely, any
measure η on R+ × S such that η(t, ·) = η([0, t] × ·) is a σ-finite measure on S for any t > 0
may be identified with the measure function η(t, ·).
A measure function η(t, ·) is called continuous at t > 0 if η(t, ·) = η(t−, ·), or equivalently
η({t} × ·) = 0, and continuous if it is continuous at all t > 0. In general, the set J = {t > 0:
164
η({t} × G) > 0}, of discontinuity times, is at most countable, and
η = ηc + ηd,
(6.12)
∫
where η c (t, ·) = [0, t]∩J c η(ds, ·) is a continuous measure function, called the continuous part
∑
of η(t, ·), and η d (t, ·) = s≤t η({s} × ·) is called the discontinuous part of η(t, ·). Here, we
∑
∑
have written s≤t for the sum s≤t,s∈J when the terms indexed by s ̸= J are zero. It is
clear that a measure function η is continuous if η = η c .
Recall that an inhomogeneous Lévy process xt in a Lie group G may be defined as a
rcll process with independent increments x−1
s xt , s ≤ t, and an infinite life time. The jump
intensity measure of xt is the measure function η(t, ·) on G defined by
−1
η(t, B) = E{#{s ∈ (0, t]; x−1
s− xs ∈ B and xs− xs ̸= e}},
B ∈ B(G),
(6.13)
the expected number of jumps in B by time t. The required σ-finiteness of η(t, ·) will follow
from Proposition 6.7 later, and then the required right continuity, η(t, ·) ↓ η(s, ·) as t ↓ s,
follows from (6.13). It is clear that the process xt is continuous if and only if η = 0.
For the jump intensity measure η, η(0, ·) = 0, η(t, {e}) = 0 and η({t} × G) ≤ 1 for t > 0,
where e is the identity element of G. Moreover, the distribution of x−1
t− xt is
νt = η({t} × ·) + [1 − η({t} × G)]δe
(where δe is the unit mass at e).
(6.14)
It is clear that xt is stochastically continuous if and only if its jump intensity measure η(t, ·)
is continuous.
As defined in (2.11), the jump counting measure of the process xt is the random measure
on R+ × G given by
−1
N ([0, t] × B) = #{s ∈ (0, t]; x−1
s− xs ̸= e and xs− xs ∈ B}
(6.15)
It is clear that η = E[N (·)]. Because xt has independent increments, so does N . It is then
easy to show that N is an extended Poisson random measure on R+ × G with intensity
measure η as defined in Appendix A.6.
Let J = {u1 , u2 , u3 , . . .} be the set of fixed jump times of the process xt (the set may
be finite or empty, and may not be ordered in magnitude), and let Jm = {u1 , u2 , . . . , um },
setting Jm = Jk for m > k if J has only k points. Let
∆n : 0 = tn0 < tn1 < tn2 < · · · < tnk ↑ ∞ (as k ↑ ∞)
be a sequence of partitions of R+ with mesh ∥∆n ∥ → 0 as n → ∞, such that Jn ⊂ ∆n ⊂ ∆n+1
for all n ≥ 1.
165
For i = 1, 2, 3, . . ., let xni = x−1
tn i−1 xtni . Then for each fixed n, xni are independent random
variables in G. Let µni be their distributions, and define the measure function
∑
ηn (t, ·) =
µni
(setting ηn (0, ·) = 0).
(6.16)
tni ≤t
Proposition 6.5 For any T > 0 and any neighborhood U of e, ηn (T, U c ) is bounded in n
and ηn (T, U c ) ↓ 0 as U ↑ G uniformly in n.
Proof We first establish an equi-continuity type property for ηn (t, ·).
Lemma 6.6 For any T > 0, neighborhood U of e and ε > 0, there is an integer m > 0
such that if n > m and s, t ∈ ∆n ∩ [0, T ] with (s, t] ∩ Jm = ∅ (the empty set), then
ηn (t, U c ) − ηn (s, U c ) < ε.
We note that the condition (s, t] ∩ Jm = ∅ means that (s, t] is contained in the interior
of a single interval of the partition ∆m .
By the Borel-Cantelli Lemma, the independent increments and rcll paths imply that for
any neighborhood U of e,
∑
c
P (x−1
(6.17)
u− xu ∈ U ) < ∞.
u≤T, u∈J
Suppose the claim of the lemma is not true. Then for some ε > 0, and for any integer
m > 0, there are n > m and sn , tn ∈ ∆n ∩ [0, T ] with (sn , tn ] ∩ Jm = ∅ such that ηn (tn , U c ) −
ηn (sn , U c ) ≥ ε. Letting n → ∞ yields a subsequence of n → ∞ such that sn and tn converge
to a common limit t ≤ T as n → ∞.
Let sn = tni and tn = tnj . In the following computation, we will write an ≈ bn if there is
a constant c > 0 such that (1/c)an ≤ bn ≤ can for all n. Then
ηn (tn , U ) − ηn (sn , U ) =
c
c
j
∑
c
P (x−1
tn p−1 xtnp ∈ U )
p=i+1
≈ −
j
∑
j
c
−1
log[1 − P (x−1
tn p−1 xtnp ∈ U )] = − log P [∩p=i+1 (xtn p−1 xtnp ∈ U )] = − log P (An ),
p=i+1
c
c
where An = ∩jp=i+1 (x−1
tn p−1 xtnp ∈ U ). Because ηn (tn , U ) − ηn (sn , U ) ≥ ε, P (An ) ≤ 1 − ε1
for some constant ε1 > 0. Then P (Acn ) > ε1 . Note that on Acn , the process xt makes a
U c -oscillation during the time interval [sn , tn ].
There are three possible cases and we will reach a contradiction in all these cases. Case
one, there are infinitely many sn ↓ t. This is impossible by the right continuity of paths at
time t. Case two, there are infinitely many tn ↑ t. This is impossible by the existence of
path left limit at time t. Case three, there are infinitely many sn < t ≤ tn . This implies
166
c
P (x−1
t− xt ∈ U ) ≥ ε1 . Then t ∈ J. Because (sn , tn ] ∩ Jm = ∅, t ̸∈ Jm . By (6.17), m may be
∑
c
chosen so that u≤T, u∈J−Jm P (x−1
u− xu ∈ U ) < ε1 , a contradiction. Lemma 6.6 is proved.
To prove Proposition 6.5, fix ε > 0 and let m > 0 be as in Lemma 6.6. It suffices to
prove for n > m. For each n > m, Jm ∩ (0, T ] may be covered by at most m sub-intervals
of the form (tn i−1 , tni ], and the rest of the interval (0, T ] may be covered by at most m + 1
sub-intervals (s, t] as in Lemma 6.6. Then ηn (T, U c ) ≤ (m + 1)ε + m, and hence ηn (t, U c ) is
bounded.
Now let V be a neighborhood of e such that V −1 V = {x−1 y; x, y ∈ V } ⊂ U . Then
ηn (T, U c ) =
∑
tni ≤T
=
c
P (x−1
tn i−1 xtni ∈ U ) ≈ −
− log P {∩tni ≤T [x−1
tn i−1 xtni
∑
tni ≤T
c
log[1 − P (x−1
tn i−1 xtni ∈ U )]
∈ U ]} ≤ − log P {∩tni ≤T [xtn i−1 ∈ V and xtni ∈ V ]}
≤ − log P {∩t≤T [xt ∈ V ]} ↓ 0
as V ↑ G uniformly in n because P {∩t≤T [xt ∈ V ]} ↑ 1. 2
Let η(t, ·) be the jump intensity measure of the process xt defined by (6.13). Recall Cb (G)
is the space of bounded continuous functions on G.
Proposition 6.7 For any t > 0 and f ∈ Cb (G) vanishing in a neighborhood of e,
ηn (t, f ) → η(t, f )
as n → ∞, and the convergence is uniform for t in any finite interval. Moreover, η(t, U c ) <
∞ for any neighborhood U of e.
∑
∑
p
Proof We may assume f ≥ 0. Let F = u≤t f (x−1
i=1 f (xni ), where p is
u− xu ) and Fn =
the largest index i such that tni ≤ t. Then η(t, f ) = E(F ) and ηn (t, f ) = E(Fn ). Note that
∑
the sum u≤t in F contains only finitely many nonzero terms. The following proof is similar
to the proof of Theorem 2.6. We have Fn → F almost sure as n → ∞ (see fδ → Fϕ in the
proof of Theorem 2.6). By the independence of xn1 , xn2 , xn3 , . . .,
p
∑
E(Fn2 ) = E{[
=
p
∑
i=1
f (xni )]2 } =
i=1
p
∑
µni (f 2 ) + [
i=1
µni (f )]2 −
p
∑
i=1
p
∑
µni (f 2 ) +
∑
µni (f )µnj (f )
i̸=j
µni (f )2 ≤ ηn (t, f 2 ) + [ηn (t, f )]2 ,
i=1
which is bounded in n by Proposition 6.5. Therefore, E(Fn2 ) are uniformly bounded in n,
and hence Fn are uniformly integrable. It follows that ηn (t, f ) = E(Fn ) → E(F ) = η(t, f ).
The finiteness of η(t, U c ) now follows from Proposition 6.5.
To prove the convergence ηn (t, f ) → η(t, f ) is uniform for t in any finite interval, it
suffices to show that if tn → t, then ηn (tn , f ) → η(t, f ) as n → ∞. Note that if F and Fn
167
∑
∑
′
above are replaced by F ′ = u<t f (x−1
tni <t f (xni ), then repeating the above
u− xu ) and Fn =
argument shows that ηn (t−, f ) → η(t−, f ) as n → ∞. We may assume f ≥ 0, and either
all tn ≥ t or all tn < t. If tn ≥ t, then for any δ > 0, ηn (t, f ) ≤ ηn (tn , f ) ≤ ηn (t + δ) and
η(t, f ) ≤ η(tn , f ) ≤ η(t + δ, f ) when n is sufficiently large. Then ηn (tn , f ) → η(t, f ) follows
from ηn (t, f ) → η(t, f ), ηn (t + δ, f ) → η(t + δ, f ), and the right continuity of η(t, f ) in t. The
case of tn < t can be treated similarly using ηn (t−, f ) → η(t−, f ) and ηn (t−δ, f ) → η(t−δ, f ).
2
Recall Jm = {u1 , u2 , . . . , um }. Define
−1
η m (t, B) = E[#{s ≤ t; x−1
s− xs ∈ B, xs− xs ̸= e and s ̸∈ Jm }].
(6.18)
Then η(t, ·) ≥ η m (t, ·) ↓ η c (t, ·) as m ↑ ∞. Let ∆m
n be the subset of ∆n consisting of tni such
∑
that u ∈ (tn i−1 , tni ] for some u ∈ Jm , and let ηnm (t, ·) = tni ≤t, tni ̸∈∆m
µni . The following
n
lemma will be needed later.
Lemma 6.8 Fix m > 0. Then as n → ∞, ηnm (t, f ) → η m (t, f ) for any t > 0 and f ∈ Cb (G)
vanishing in a neighborhood of e.
∑
∑
f (xni ). Then η m (t, f ) = E(F̃ )
Proof Let F̃ = s≤t, s̸∈Jm f (x−1
tni ≤t, tni ̸∈∆m
s− xs ) and F̃n =
n
and ηnm (t, f ) = E(F̃n ). We can show F̃n → F̃ almost surely as n → ∞. We may assume
f ≥ 0. Because F̃n ≤ Fn in the notation of the proof of Proposition 6.7, so F̃n are uniformly
integrable and hence ηnm (t, f ) = E(F̃n ) → E(F̃ ) = η m (t, f ). 2
6.3
Martingale representation
In §1.4, an inhomogeneous Lévy process xt in a Lie group G, with no fixed jumps, is associated to a continuous two-parameter convolution semigroup µs,t with µt,t = δe , where µs,t is
the distribution of x−1
s xt and e is the identity element of G, and conversely, given such a convolution semigroup µs,t and an initial distribution, there is an inhomogeneous Lévy process
in G with no fixed jumps, unique in distribution.
In this section, we will present a representation of an inhomogeneous Lévy process in a
Lie group, possibly with fixed jumps, in terms of a triple (b, A, η) of a rcll path bt in G, a
matrix function A(t) and a measure function η(t, ·), which can be explicitly constructed. In
contrast, one usually does not have an explicit expression for a convolution semigroup µs,t .
We will extend the martingale property for additive processes in Euclidean spaces, as
discussed in §6.1, to inhomogeneous Lévy processes in a general Lie group G. To state this
result, let ξ1 , . . . , ξd be a basis of the Lie algebra g of G and let ϕ1 , . . . , ϕd be associated
exponential coordinate functions on G. Recall those are functions in Bb (G) such that ϕi
168
vanish outside a compact subset of G and are smooth
near the identity element e of G,
∑
ϕ
(x)ξ
i
i
ϕi (e) = 0 and satisfy the exponential relation x = e i
for x near the identity element
∑
e of G. We will let ϕ(x) = (ϕ1 (x), . . . , ϕd (x)) with Euclidean norm ∥ϕ(x)∥ = [ i ϕi (x)2 ]1/2 .
The local mean of a G-valued random variable x or of its distribution µ is defined to be
b = exp[
d
∑
µ(ϕj )ξj ].
(6.19)
j=1
The distribution µ is called moderate if its local mean b has coordinates µ(ϕ1 ), . . . , µ(ϕd ),
that is,
ϕj (b) = µ(ϕj ), 1 ≤ j ≤ d.
This is the case when µ is sufficiently concentrated near e.
As defined in [25], a Lévy measure function on G is a continuous measure function η(t, ·)
with η(0, ·) = 0 such that η(t, ·) is a Lévy measure on G for all t > 0, that is, η(t, {e}) = 0,
η(t, U c ) < ∞ for any neighborhood U of e and η(t, ∥ϕ∥2 ) < ∞ for any t ≥ 0.
The notion of Lévy measure functions is now extended. A measure function η(t, ·) on G,
not necessarily continuous, will be called an extended Lévy measure function if (a) and (b)
below hold.
(a) η(0, ·) = 0, η(t, {e}) = 0 and η(t, U c ) < ∞ for any t > 0 and neighborhood U of e;
(b) For t > 0, η c (t, ∥ϕ∥2 ) < ∞, η({t} × G) ≤ 1, and with νt given by (6.14),
∑
νs (∥ϕ − ϕ(hs )∥2 ) < ∞,
(6.20)
s≤t
where η c is the continuous part of η and ht = exp[
∑
j
νt (ϕj )ξj ] is the local mean of νt .
∑
Note that νt = δe and ht = e at a continuous point t of η(t, ·), and hence the sum s≤t
in (6.20) has at most countably many nonzero terms. It is clear that a continuous extended
Lévy measure function is a Lévy measure function.
Because for two different bases {ξi } and {ξi′ } of g with associated exponential coordinate
∑
∑
functions ϕi and ϕ′i , i ϕi ξi = i ϕ′i ξi′ in a neighborhood of e, it is easy to show that condition
(b) does not depend on the choice for basis {ξi } and coordinate functions ϕi .
Note that νt in (b), as defined by (6.14), is obtained from η({t}, ·) by adding a suitable
charge at e to make it into a probability measure. Later we will see that any extended
Lévy measure function η is the jump intensity measure of some inhomogeneous Lévy process
xt in G, so νt is the distribution of x−1
t− xt , which is a fixed jump if P (xt− ̸= xt ) > 0, and the
set of discontinuity times of η agrees with the set of fixed jump times of xt . Therefore, the
family of probability measures νt , t ∈ R+ , will be called the jump measures of η.
A continuous path bt in G with b0 = e will be called a drift in G. A function A(t), taking
values in the space of d×d symmetric real matrices, will be called a covariance matrix function
169
if A(0) = 0, A(t) − A(s) ≥ 0 (nonnegative definite) for s < t, and t 7→ A(t) is continuous. A
triple (b, A, η) of a drift bt , a covariance matrix function A(t) and a Lévy measure function
η(t, ·) will be called a Lévy triple on G.
In the following result ([25]), a stochastically continuous inhomogeneous Lévy process in
a Lie group is represented by a Lévy triple through a martingale property.
Theorem 6.9 Let xt be a stochastically continuous inhomogeneous Lévy process in G with
x0 = e. Then there is a unique Lévy triple (b, A, η) such that xt = zt bt and for f ∈ Cc∞ (G),
d ∫ t
1 ∑
f (zt ) −
[Ad(bs )ξj ][Ad(bs )ξk ]f (zs ) dAjk (s)
2 j,k=1 0
−
∫
0
t
∫
G
{f (zs bs xb−1
s ) − f (zs ) −
d
∑
ϕi (x)[Ad(bs )ξi ]f (zs )}η(ds, dx)
(6.21)
i=1
is a martingale under the natural filtration Ftx of process xt . Moreover, η(t, ·) is the jump
intensity measure of process xt given by (6.13).
Conversely, given a Lévy triple (b, A, η), there is a rcll process xt = zt bt in G with x0 = e
such that (6.21) is a martingale under Ftx for f ∈ Cc∞ (G). Moreover, such a process xt is
unique in distribution and is a stochastically continuous inhomogeneous Lévy process in G.
The proof of Theorem 6.9 will be postponed to the next chapter, except the uniqueness
of the Lévy triple will be established later in this chapter (Lemma 6.32). By Proposition 6.10
later, the expression in (6.21) is a bounded random variable for each finite t > 0.
By Theorem 6.9, a stochastically continuous inhomogeneous Lévy process in G is represented by a triple (b, A, η) just like its counterpart in Rd . The complicated form of the
martingale in (6.21) with the presence of the drift bt , as compared with its counterpart (6.10)
on Rd , is caused by the possible non-commutativity of G.
For simplicity, the representation in Theorem 6.9 is stated for a process starting at e.
A general inhomogeneous Lévy process xt = x0 xet , where xet = x−1
0 xt , is represented by the
e
triple (b, A, η) for xt and the distribution of x0 .
In the course of the proof of Theorem 6.9, it will be shown, see Theorem 7.17, that if a
sequence of Lévy triples converge to a Lévy triple in some sense, then the associated processes
also converge weakly under the Skorohod metric.
A rcll path bt in G with b0 = e will be called an extended drift in G. A triple (b, A, η)
of an extended drift bt , a covariance matrix function A(t) and an extended Lévy measure
function η(t, ·) will be called an extended Lévy triple on G. It will be called proper, or a
properly extended Lévy triple, if b−1
t− bt = ht for any t > 0, where ht is the local mean of νt as
defined in (6.19). Otherwise, it will be called improper. In the rest of this work, all extended
170
Lévy triples will automatically be assumed proper unless when explicitly stated otherwise.
It is clear that a Lévy triple is an extended Lévy triple.
Let (b, A, η) be an extended Lévy triple, possibly improper. For f ∈ Cc∞ (G) and a rcll
path zt in G, let
d ∫ t
1 ∑
[Ad(bs )ξj ][Ad(bs )ξk ]f (zs )dAjk (s)
Tt f (z; b, A, η) =
2 j,k=1 0
∫
t
∫
+
0
+
G
{f (zs bs xb−1
s ) − f (zs ) −
∑∫
s≤t G
d
∑
ϕj (x)[Ad(bs )ξj ]f (zs )}η c (ds, dx)
j=1
−1
[f (zs− bs− xh−1
s bs− ) − f (zs− )]νs (dx).
(6.22)
For simplicity, we may write Tt f for Tt f (z; b, A, η), and Ts,t f = Tt f − Ts f for s < t. Note
∫
∑
that the last term in (6.22) may be written as s≤t G [f (zs− bs− xb−1
s ) − f (zs− )]νs (dx) when
(b, A, η) is proper.
Proposition 6.10 For any t ∈ R+ , |Tt f | is bounded for all rcll paths z in G.
∑
Proof: By (6.22), Tt f has three terms, an A-integral, an η c -integral and a sum u≤t . It is
easy to see that ∑
the A-integral is bounded in z. For the η c -integral, by the Taylor expansion
of g(t) = f (zbet i ϕi (x)ξi b−1 ) at t = 0,
∑
1
b ) = g(1) = g(0) + g ′ (0) + g ′′ (v) (for some 0 ≤ v ≤ 1)
2
∑
∑
1∑
ϕj (x)ϕk (x)[Ad(b)ξj ][Ad(b)ξk ]f (zev i ϕi (x)Ad(b)ξk ).
= f (z) +
ϕi (x)[Ad(b)ξi ]f (z) +
2 j,k
i
f (zbe
i
ϕi (x)ξi −1
It follows that when x belongs to a small neighborhood U of e, the integrand of the η c -integral
is equal to
1∑
ϕj (x)ϕk (x)[Ad(bs )ξj ][Ad(bs )ξk ]f (zs σ)
2 j,k
for some σ ∈ G. It follows that the η c -integral is bounded by c1 [η c (t, ∥ϕ∥2 ) + η c (t, U c )] for
some constants c1 > 0 not depending
on z.
∑
∑
t i [ϕi (x)−ϕi (h)]ξi + i ϕi (h)ξi −1 −1
By Taylor expansion of f (zbe
h b ) at t = 0, when x and h
belong to some small neighborhood U of e,
f (zbxh−1 b−1 ) − f (z) = f (zbe
=
d
∑
∑
[ϕi (x) − ϕi (h)]fi (z, b, h, 0) +
i=1
i
[ϕi (x)−ϕi (h)]ξi +
∑
i
ϕi (h)ξi −1 −1
h b ) − f (z)
d
∑
1
[ϕj (x) − ϕj (h)][ϕk (x) − ϕk (h)]fjk (z, b, h, t)
2 j,k=1
∑
∑
∂
f (zbe i ψi ξi + i ϕi (h)ξi h−1 b−1 ) |ψ=t[ϕ(x)−ϕ(h)] and
∂ψi
∑
∑
∂2
f (ze i ψi ξi + i ϕi (x)ξi h−1 b−1 ) |ψ=t[ϕ(x)−ϕ(h)] . Because bt and ht are bound∂ψj ∂ψk
for some t ∈ (0, 1), where fi (z, b, h, t) =
fjk (z, b, h, t) =
ed over any finite t-interval, these partial derivatives are bounded for (z, b, h) = (zs− , bs− , hs ).
171
∫
Because except for finitely many s ≤ t, νs is moderate so that [ϕi (x) − ϕi (hs )]νs (dx) = 0,
then
∫
| [f (zbxh−1 b−1 ) − f (z)]νs (dx)| ≤ c2 [νs (U c ) + νs (∥ϕ − ϕ(h)∥2 )]
G
for some constant c2 > 0 not depending on z. It follows that the sum
∑
by c2 [ s≤t νs (∥ϕ − ϕ(hs )∥2 ) + η(t, U c )] < ∞. 2
∑
s≤t
in Tt f is bounded
A rcll process xt in G is said to be represented by an extended Lévy triple (b, A, η),
possibly improper, if with xt = zt bt ,
Mt f (z; b, A, η) = f (zt ) − Tt f (z; b, A, η)
(6.23)
is a martingale under the natural filtration Ftx of xt for any f ∈ Cc∞ (G). We note that by
Proposition 6.10, Mt f (z; b, A, η) is a bounded random variable for any fixed t ∈ R+ .
If (b, A, η) is a Lévy triple, so that η c = η, νu = δe and hu = e, then f (zt ) − Tt f (z; b, A, η)
becomes (6.21). Thus by Theorem 6.9, there is a rcll process xt with x0 = e, unique in distribution, represented by (b, A, η). Moreover, xt is a stochastically continuous Lévy process.
Theorem 6.11 Let xt be an inhomogeneous Lévy process in G with x0 = e. Then there
is a unique extended Lévy triple (b, A, η) on G such that xt is represented by (b, A, η) as
defined above. Moreover, η(t, ·) is the jump intensity measure of process xt given by (6.13).
Consequently, xt is stochastically continuous if and only if (b, A, η) is a Lévy triple.
Conversely, given an extended Lévy triple (b, A, η) on G, possibly improper, there is an
inhomogeneous Lévy process xt in G with x0 = e, unique in distribution, that is represented
by (b, A, η).
Remark 6.12 In the first half of Theorem 6.11, η(t, ·) as the jump intensity measure clearly
does not depend on the choice for the basis {ξj } of g and associated coordinate functions ϕj .
It will be shown in Proposition 6.33 that A(t) does not depend on {ϕj }, and so the operator
∑d
j,k=1 Ajk (t)ξj ξk does not depend on {ξj } and ϕj .
The proof of Theorem 6.11 will be given in the next chapter, except that the uniqueness
of the extended Lévy triple will be established later in this chapter (Lemma 6.32). For an
inhomogeneous Lévy process xt represented by an extended Lévy triple (b, A, η), zt = xt b−1
t
is also an inhomogeneous Lévy process. See Remark 6.30 for how zt is represented.
The representation on G = Rd in terms of the Fourier transform is obtained in JacodShiryaev [45, II.5]. The assumptions in [45, Theorem II 5.2] correspond to a properly extended Lévy triple defined here. In particular, (i) - (iii) in [45] correspond to (a) and (b),
and (v) to b−1
t− bt = ht , but (iv) in [45] is redundant as it is implied by the other conditions.
Note that in the second part of Theorem 6.11, the extended Lévy triple, when not proper,
is not unique, and η may not be the jump intensity measure. The following trivial example
172
may help illustrate some special features of the representation in Theorem 6.11, see later
sections for more properties.
Example 6.13 Let xt be a non-random rcll path in G with x0 = e. Then xt is an inhomogeneous Lévy process and is represented by the extended Lévy triple (b, 0, 0) with bt = xt . In
this case, zt = e and Mt f in (6.23) is the constant f (e). If xt is not continuous, then (b, 0, 0)
is not proper because the condition b−1
t− bt = ht is not satisfied.
To get a properly extended Lévy triple, let yt = x−1
t− xt and let νt = δyt with local mean
ht . Note that ht = yt except for finitely many t in any finite interval. Let J be the countable
∑
set of time points t such that yt ̸= e. Then η(t, ·) = u∈J, u≤t νu is an extended Lévy measure
function with η c = 0. Let the time points t ∈ J, yt ̸= ht , be listed as t1 < t2 < · · · < tn · · ·.
For t < t1 , let bt = xt , and inductively, for tn ≤ t < tn+1 , let bt = btn − htn x−1
tn xt . Then
−1
bt− bt = ht for any t > 0. Let A(t) = 0 and define zt by xt = zt bt . Then (b, A, η) is a properly
extended Lévy triple representing xt , and because zt is constant between tn and tn+1 , and
−1 −1
−1
−1
−1 −1
−1
−1
−1
btn − (x−1
tn − xtn )htn btn − = btn − (xtn − xtn )(btn − btn ) btn − = btn − (xtn − xtn )btn = ztn −1 ztn ,
it follows that the martingale Mt f in (6.23) is the constant f (e).
6.4
Finite variation
A rcll path bt in a manifold X is said to have a finite variation if for any f ∈ Cc∞ (X), f (bt )
has a finite variation on any finite t-interval. Let G be a Lie group, let {ξ1 , . . . , ξd } be a basis
of its Lie algebra g, and let ϕi be the associated exponential coordinate functions as before.
Proposition 6.14 (a) If bt is a rcll path in G with a finite variation, then there is a unique
set of real-valued continuous functions bj (t) of finite variation, 1 ≤ j ≤ d, with bj (0) = 0,
such that for any f ∈ Cc∞ (G),
∫
f (bt ) = f (b0 ) +
d
t∑
0 j=1
ξj f (bs )dbj (s) +
∑
[f (bs ) − f (bs− )],
(6.24)
s≤t
where dbj (s) is regarded as a signed measure on R+ .
(b) Given a set of real-valued continuous functions bj (t) of finite variation, 1 ≤ j ≤ d, and
x ∈ G, there is a unique continuous path bt in G, with b0 = x and finite variation, such that
∑
(6.24) holds (without s≤t [· · ·]).
Proof Fix a finite T > 0. Let u1 , u2 , u3 , . . . be all the discontinuous time points of bt for
t ≤ T , which are not necessarily ordered in magnitude, and may form a finite or even empty
set, but without loss of generality, we may assume they form a countably infinite set. Fix an
173
open neighborhood U of e and assume its closure is contained in another neighborhood V such
that the map x 7→ ϕ(x) = (ϕ1 (x), . . . , ϕd (x)) is difformorphic from V onto a neighborhood
of the origin in Rd . For g ∈ G, gU is a coordinate neighborhood of g with coordinates
ψi (x) = ϕi (g −1 x).
First assume the path bt , t ≤ T , is covered by b0 U . Then ψj (bt ) is a real-valued rcll
function with finite variation, which may be written as a sum of its continuous part ψj (bt )c
and a pure jump part. For each integer n > 0, let ∆n : 0 = t0 < t1 < t2 < · · · < tk ↑ ∞ as
k → ∞ be a partition of R+ with mesh ∥∆n ∥ = supi>0 (ti − ti−1 ) → 0 as n → ∞, and assume
ui ∈ ∆n for 1 ≤ i ≤ n. Fix t > 0 and let m = max{i; ti ≤ t}. We may assume bt = bt− for
simplicity. Then
f (bt ) = f (b0 ) +
m
∑
[f (bti ) − f (bti−1 )] + [f (bt ) − f (btm )]
i=1
= f (b0 ) +
= f (b0 ) +
+rn +
∑m
[
i=1
m
∑
[f (bti − ) − f (bti−1 )] + [f (bt ) − f (btm )] +
i=1
m ∑
d
∑
m
∑
[f (bti ) − f (bti − )]
i=1
d
∑
∂
∂
f (bti−1 )[ψj (bti − ) − ψj (bti−1 )] +
f (btm )[ψj (bt ) − ψj (btm )]
j=1 ∂ψj
i=1 j=1 ∂ψj
m
∑
[f (bti ) − f (bti − )] (by the Taylor expansion, where rn is controlled by
i=1
∥ψ(bti − ) − ψ(bti−1 )∥2 + ∥ψ(bt ) − ψ(btm )∥2 ] → 0 as n → ∞)
= f (b0 ) +
m ∑
d
∑
∂
i=1 j=1 ∂ψj
+rn′ + rn +
f (bti−1 )[ψj (bti − )c − ψj (bti−1 )c ] +
m
∑
∂
j=1 ∂ψj
f (btm )[ψj (bt )c − ψj (btm )c ]
m
∑
[f (bti ) − f (bti − )] (where rn′ is controlled by the sum of
i=1
∥ψ(bs ) − ψ(bs− )∥ for s < t with s ̸= u1 , . . . , un , and so → 0 as n → ∞)
→ f (b0 ) +
∫
∑
∂
f (bs )dψj (bs )c + [f (bs ) − f (bs− )] (as n → ∞)
j=1 ∂ψj
s≤t
0
∫
= f (b0 ) +
d
t∑
t
d
∑
0 k=1
ξk f (bs )dbk (s) +
∑
[f (bs ) − f (bs− )],
s≤t
∫ ∑
∑
where bk (t) = 0t j βjk (bs )dψj (bs )c with βjk (x) at x ∈ G given by ∂/∂ψj = k βjk (x)ξk .
This proves (6.24). Because ψj (bt )c are continuous in t with finite variation, so are bj (t).
To show that such bj (t) are unique, let f = ψj in (6.24), and let {αij } = {βij }−1 . Then
∫ ∑
∑
∑
ξj = k αjk (∂/∂ψk ), and ψj (bt ) = ψj (b0 ) + 0t k αkj (bs )dbk (s) + s≤t [ψj (bs ) − ψj (bs− )]. This
∑
∑
implies dψj (bt )c = k αkj (bt )dbk (t), and hence dbk (t) = j βjk (bt )dψj (bt )c . This proves (a)
under the assumption that the path bt , t ≤ T , is covered by a single neighborhood b0 U .
In general, the path is covered by finitely many coordinate neighborhoods of the form
gU , g ∈ G. For simplicity, assume it is covered by two coordinate neighborhoods U1 and
174
U2 , and there is t1 ∈ (0, T ) such that bt ∈ U1 for t ≤ t1 and bt ∈ U2 for t1 ≤ t ≤ T . Let
b′t = bt1 +t for t ∈ [0, T − t1 ]. Then there are real valued continuous functions bi (t) and b′i (t),
1 ≤ i ≤ d, of finite variation, such that (6.24) holds for t ≤ t1 , and it also holds, when bt
and bi (t) are replaced by b′t and b′i (t), for t ≤ T − t1 . Letting bi (t) be re-defined for t > t1 as
bi (t) = bi (t1 ) + b′i (t − t1 ) shows that (6.24) holds for t ≤ T .
For part (b), given bj (t) as above, and some x ∈ G, a unique continuous path bt with
∑
b0 = x, of finite variation and satisfying (6.24) (without s≤t [· · ·]), may be obtained by
solving the integral equation
ψj (bt ) = ψj (b0 ) +
∑∫
k
t
0
αkj (bs )dbk (s)
for ψj (bt ) by the usual successive approximation method, noting αkj (bs ) is a smooth function
of ψi (bs ), 1 ≤ i ≤ d. More precisely, the integral equation is solved piecewise in a collection
of neighborhoods of the form gU , g ∈ U , starting in b0 U . In each neighborhood, the solution
can be obtained on a time interval of length at least δ > 0 for some fixed constant δ > 0, so
bt may be obtained for t ∈ [0, T ] for any T > 0 after solving the equation in finitely many
neighborhoods. This proves (b). 2
Let bt be a continuous path in G of finite variation. By (a) of Proposition 6.14, it is
completely determined by its initial value b0 and a set of real valued functions bj (t) of finite
variation via the equation
∫
f (bt ) = f (b0 ) +
d
t∑
0 j=1
ξi f (bs )dbj (s)
for f ∈ Cc∞ (G). The bj (t) will be called the components of bt under the vector fields ξ1 , . . . , ξd .
When bt is a rcll path in G of finite variation, the bj (t) in (a) determine a unique continuous
path bct of finite variation, with bj (t) as components and bc0 = b0 , which will be called the
continuous part of bt . In this case, we may also call bj (t) the components of bt .
The components of a rcll path bt in G, in particular, a drift or an extended drift bt in
G, of finite variation, will always be defined under the basis {ξ1 , . . . , ξd } of the Lie algebra
g chosen before. Note that an extended drift bt of finite variation, as part of an extended
Lévy triple (b, A, η), is determined by its components as its jumps are determined by the
discontinuous part of η.
Some basic properties are summarized below.
Proposition 6.15 Let bt be a rcll path in G of finite variation with components bi (t).
∑
(a) If bt = etξ with ξ = di=1 βi ξi and βi ∈ R, then bj (t) = βj t.
′
(b) For s > 0, b′t = b−1
s bs+t has components bj (t) = bj (s + t) − bj (s).
175
(c) The components βj (t) of of b−1
are given by
t
dβj (t) = −
∑
[Ad(bt )]jk dbk (t),
k
∑
where [Ad(b)]jk is the matrix representing Ad(b), that is, Ad(b)ξj = k [Ad(b)]kj ξk .
(d) Let K be a subgroup of G. If bt is K-conjugate invariant for any t ≥ 0, that is,
∑
kbt k −1 = bt for k ∈ K, then i bi (t)ξi is Ad(K)-invariant for any t ≥ 0.
Proof: (a) follows from f (etξ ) = f (e) +
For s > 0,
∫
f (bs+t ) = f (bs ) +
∫
= f (bs ) +
t
0
∑
s+t
∫t
0
∑
s
ξf (esξ )ds = f (e) +
∑
ξi f (bu )dbi (u) +
i
∫t∑
0
i
βi ξi f (esξ )ds.
[f (bu ) − f (bu− )]
s<u≤t
∑
ξi f (bs+u )dbi (s + u) +
i
[f (bs+u ) − f (bs+u− )].
u≤t
Replacing f by f ◦ lz with z = b−1
s proves (b).
To prove (c), let J: G → G be the inverse map. Then
f (b−1
t )
= (f ◦ J)(bt ) =
= f (b−1
0 )+
∫
0
t
f (b−1
0 )
∫
t
+
0
∑
ξi (f ◦ J)(bs )dbi (s) +
i
∑
[(f ◦ J)(bs ) − (f ◦ J)(bs− )]
s≤t
∑
∑
i
s≤t
[−Ad(bt )ξi ]f (b−1
s )dbi (s) +
−1
[f (b−1
s ) − f (bs− )].
To prove (d), replace f in (6.24) by f ◦ ck (recall ck is the conjugation map). Because
∫ ∑
kbt k −1 = bt , (6.24) is unchanged except the integral term becomes 0t dj=1 Ad(k)ξj f (bs )dbj (s).
∑
∑
By the uniqueness of bj (t), j Ad(k)bj (s)ξj = j bj (s)ξj . 2
For a rcll path bt in a Lie group G, a possible jump at time u > 0 may be removed to
obtain a new path b′t defined by
b′t = bt
for t < u and b′t = bu− b−1
u bt
for t ≥ u.
(6.25)
Similarly, a jump σ ∈ G may be added to the path bt at time u > 0, replacing a possible
existing jump at u, to obtain a new path b′t defined by
b′t = bt
for t < u and b′t = bu− σb−1
u bt
for t ≥ u.
(6.26)
Several jumps at different times may be removed from or added to the path bt to obtain a
new path, which does not depend on the order in which the jumps are removed or added.
A sequence of rcll paths bkt in G is said to converge to a rcll path bt uniformly for
bounded t if for some Riemannian metric ρ on G, sups≤t ρ(bks , bs ) → 0 as k → ∞ for any
176
finite t > 0. Because any two Riemannia metrics are equivalent on any compact subset of G,
this definition does not depend on the choice of ρ. To show the equivalence of Riemannian
metrics, note that as they are expressed by smooth and positive definite tensors, they are
equivalent on an open neighborhood of any point, and as a compact set F is covered by
∑
finitely many such open sets, so they are equivalent on F . Recall ∥ϕ(x)∥ = [ di=1 ϕi (x)2 ]1/2 .
Proposition 6.16 (a) Let bt be a rcll path in G of finite variation, and let uj > 0,
j = 1, 2, . . ., be all its jump times, where the set {u1 , u2 , . . .} is not necessarily ordered
in magnitude. Let bnt be the path obtained from bt after the jumps at times u1 , u2 , . . . , un
n
0
are removed, setting bnt = bm
t for n > m if bt has only m jumps, with bt = bt . Then bt
converges to bct , the continuous part of bt , as n → ∞ uniformly for bounded t. Moreover,
∑
−1
u≤t ∥ϕ(bu− bu )∥ < ∞ for any finite t > 0
(b) Let bt be a continuous path in G of finite variation, and for j = 1, 2, . . ., let uj > 0 be
∑
distinct time points, and hj ∈ G − {e}. Assume for any finite t > 0, uj ≤t ∥ϕ(hj )∥ < ∞.
Let b̄kt be the path obtained from bt after the jumps h1 , . . . , hk at times u1 , . . . , uk are added.
Then b̄kt converges to a rcll path b̄t of finite variation as k → ∞, uniformly for bounded t.
Moreover, b̄0 = b0 and b̄ct = bt for all t ≥ 0, and b̄t jumps only at t = uj with b̄−1
uj − b̄uj = hj .
Proof: For (a), let bt be a rcll path in G of finite variation. Because t 7→ ϕj (bt ) is
∑
a real valued function of finite variation, u≤t ∥ϕ(bu ) − ϕ(bu− )∥ < ∞. Let ρ be a left
invariant Riemannian metric on G. We may assume, without loss of generality, that all b−1
u− bu
are contained in a relatively compact coordinate neighborhood U of e on which ρ0 (x, y) =
∥ϕ(x)−ϕ(y)∥ is a Riemannian metric. By the equivalence of metrics, for some finite constant
a > 0,
∑
u≤t
∥ϕ(b−1
u− bu )∥ ≤ a
∑
ρ(o, b−1
u− bu ) = a
u≤t
∑
ρ(bu− , bu ) ≤ a2
u≤t
∑
∥ϕ(bu− ) − ϕ(bu )∥ < ∞.
u≤t
Let ρ′ be a right invariant Riemannian metric on G. For k > 1, bkt = bk−1
for t < uk , and for
t
t ≥ uk ,
k−1 −1 k−1 k−1
k−1 −1 k−1 k−1
, bt )
, bt ) ≤ aρ′ (bk−1
) = ρ(bk−1
ρ(bkt , bk−1
uk − (buk ) bt
uk − (buk ) bt
t
3
k−1 −1 k−1
2
k−1 −1 k−1
2
k−1 k−1
k−1
= aρ′ (bk−1
uk − , buk ) ≤ a ρ(buk − , buk ) = a ρ(o, (buk − ) buk ) ≤ a ∥ϕ((buk − ) buk )∥.
∑
−1
−1 k−1
) ≤ a3 ∥ϕ((bk−1
Then ρ(bkt , bk−1
t
u≤t ∥ϕ(bu− bu )∥ < ∞, it is
uk − ) buk )∥ for any t ≥ 0. Because
now easy to see that as k → ∞, bkt converges to a rcll path b′t uniformly for bounded t.
Let f ∈ Cc∞ (G). For t < u1 , b1t = bt . For t ≥ u1 , by (6.24), with f1 (b) = f (bu1 − b−1
u1 b),
∫
f (b1t ) = f1 (bt ) = f1 (bu1 ) +
t
d
∑
u1 j=1
ξj f1 (bs )dbj (s) +
177
∑
u1 <u≤t
[f1 (bu ) − f1 (bu− )]
∫
= f (bu1 − ) +
∫
= f (b10 ) +
t
d
∑
u1 j=1
d
t∑
0 j=1
∑
ξj f (b1s )dbj (s) +
[f (b1u ) − f (b1u− )]
u1 <u≤t
ξj f (b1s )dbj (s) +
∑
[f (b1u ) − f (b1u− )].
u≤t,u̸=u1
By a simple induction on k, it is easy to show that for f ∈ Cc∞ (G),
∫
f (bkt ) = f (b0 ) +
d
t∑
0 j=1
∑
ξj f (bks )dbj (s) +
[f (bku ) − f (bku− )],
u≤t,u̸∈J(k)
where J(k) = {u1 , . . . , uk }. Note that (bku− )−1 bku = b−1
u− bu for u ̸∈ J(k), and it can be made
arbitrarily close to e when k is large. By Taylor’s expansion, when (bku− )−1 bku is close to e,
f (bku ) − f (bku− ) = f (bku− exp(
∑
ϕj ((bku− )−1 bku )) − f (bku− )
j
∫ ∑
is controlled by ∥ϕ((bku−1 )−1 bku )∥. Letting k → ∞ yields f (b′t ) = f (b0 ) + 0t dj=1 ξj f (b′s )dbj (s).
This shows b′t = bct , and completes the proof of (a).
To prove (b), fix an arbitrary finite T > 0, and let U be a neighborhood of e which is a
diffeomorphic image of the exponential map, so on which ϕ1 , . . . , ϕd form local∑coordinates.
∑
When νj is moderate, νj (ϕi ) = ϕi (hj ). Because uj ≤T ∥ϕ(hj )∥ < ∞, hj = e i νj (ϕi )ξi ∈ U
for all uj ≤ T except finitely many. For simplicity, we may assume hj ∈ U for all uj ≤ T .
By a computation similar to the one in the proof of (a), it can be shown that ρ(b̄kt , b̄k−1
)≤
t
a3 ∥ϕ(hk )∥ for any t ≤ T , and
∫
f (b̄kt ) = f (b̄k0 ) +
d
t∑
0 j=1
∑
ξj f (b̄s )k )dbj (s) +
[f (b̄u ) − f (b̄u− )].
u≤t,u∈J(k)
It then follows that b̄kt converges to b̄t uniformly for t ≤ T , and (6.24) holds for bt = b̄t , where
bj (t) are components of bt . By (a), after jumps are removed, b̄t will converge to b̄ct , and (6.24)
∑
without s≤t [· · ·] holds for b̄ct . The uniqueness of path in Proposition 6.14 (b) now implies
b̄ct = bt . 2
6.5
A martingale property
Let (b, A, η) be a Lévy triple on G with bt of finite variation, and let ν be a family of
probability measures νu on G, u ∈ R+ , such that νu = δe except for countably many u > 0,
and for any finite t > 0 and any neighborhood U of e,
∑
u≤t
νu (U c ) < ∞,
d
∑∑
|νu (ϕj )| < ∞,
u≤t j=1
∑
u≤t
178
νu (∥ϕ∥2 ) < ∞.
(6.27)
Note that the first inequality in (6.27) implies that νu (U c ) = 0 except for countably many
u > 0, so the condition that νu = δe except for countably many u > 0 may be regarded as
part of this inequality.
A rcll process zt in G is said to have the martingale property under the quadruple
(b, A, η, ν) as described above, or the (b, A, η, ν)-martingale property, if for f ∈ Cc∞ (G),
f (zt ) − St f (z; b, A, η, ν),
(6.28)
is a martingale under the natural filtration Ftz of process zt , where
∫
St f (z; b, A, η, ν) =
−f (zs ) −
∑
i
t
0
∑
∫
ξi f (zs )dbi (s) +
i
ϕi (x)ξi f (zs )]η(ds, dx) −
t
0
∑∫
s≤t G
∫ t∫
1∑
[f (zs x)
ξj ξk f (zs )dAjk (s) +
2 j,k
0 G
[f (zs− x) − f (zs− )]νs (dx),
(6.29)
and bi (t) are the components of bt . We may write St f for St f (z; b, A, η, ν) for simplicity. By
the proposition below, St f in (6.29) is a bounded random variable for any rcll process zt .
Proposition 6.17 For any t ∈ R+ , |St f | is bounded for all rcll paths z in G.
Proof: The boundedness of the first two terms in (6.29) is trivial. The boundedness of
the η-integral can be verified as in the proof of Proposition 6.10. The last term in (6.29)
∫
∫
∑
∑
can be written as s≤t U [f (zs− x) − f (zs− )]νs (dx) + s≤t U c [f (zs− x) − f (zs− )]νs (dx) for a
∫
∑
small neighborhood U of e. The boundedness of s≤t U c follows from the first inequality in
∫
∑
(6.27), and that of s≤t U follows from a Taylor expansion using the second and the third
inequalities in (6.27). 2
In the sequel, a (b, A, η, ν)-martingale property always refers to a Lévy triple (b, A, η)
with bt of finite variation and a family ν = {νu : u ∈ R+ } of probability measures satisfying
(6.27). Such a quadruple (b, A, η, ν) will be called an admissible quadruple.
For any measure λ on G, let λ∗ denote the measure obtained from λ after its charge at
e is removed, that is, λ∗ (B) = λ(B ∩ (G − {e})) for B ∈ B(G). For ν = {νu ; u ∈ R+ }
in a admissible quadruple (b, A, η, ν), let ν ∗ = {νu∗ : u ∈ R+ }. We may regarded ν ∗ as the
∑
measure function ν ∗ (t, ·) = u≤t νu∗ . In fact, ν ∗ is an extended Lévy measure function with
ν ∗d = ν ∗ by the following proposition, which says that an admissible quadruple is just an
extended Lévy triple with a drift of finite variation.
Proposition 6.18 (a) Let (b, A, η, ν) be an admissible quadruple. Then ν ∗ defined above
∑
is an extended Lévy measure function on G with ν ∗d = ν ∗ and u≤t ∥ϕ(hu )∥ < ∞ for any
finite t > 0, where hu is the local mean of νu . Moreover, if b′t is the extended drift obtained
179
from bt by adding all local means hu of νu as jumps at times u (as in Proposition 6.16 (b)),
then (b′ , A, η + ν ∗ ) is an extended Lévy triple with b′t of finite variation.
(b) If (b, A, η) is an extended Lévy triple with bt of finite variation, then (bc , A, η c , ν) is an
admissible quadruple, where bc is the continuous part of b and ν = {νt ; t ∈ R+ } is the family
of jump measures of η as defined by (6.14).
Proof: In (a), to show ν ∗ is an extended Lévy measure function, we need to verify (6.20) for
ν ∗ . Because ϕi (hu ) = νu (ϕi ) for all u ≤ t except finitely many, by the second inequality in
∑
(6.27), we see that u≤t ∥ϕ(hu )∥ < ∞. Now (6.20) for ν ∗ follows from the third inequality in
(6.27). The rest of the proof of (a) is trivial. In (b), we need to verify the three inequalities
in (6.27) for ν. The first inequality follows from η(t, U c ) < ∞. By Proposition 6.16 (a),
∑
−1
u≤t ∥ϕ(hu )∥ < ∞, where hu = bu− bu is the local mean of νu . This implies the second
inequality in (6.27). Now the third inequality follows from (6.20). 2
For g ∈ G, let [Ad(g)] be the matrix representing the linear map Ad(g): g → g under
∑
the basis {ξi }, given by Ad(g)ξj = i [Ad(g)]ij ξi . Let [Ad(g)]′ be its transpose.
Lemma 6.19 Let (b, A, η) be a possibly improper extended Lévy triple. Then a rcll process
xt in G, with x0 = e, is represented by (b, A, η) if and only if, with xt = zt bt , zt has the
(b̄, Ā, η̄, ν̄)-martingale property, where (b̄, Ā(t), η̄, ν̄) are determined by
dĀ(t) = [Ad(bt )]dA(t)[Ad(bt )]′ ,
∫
b̄i (t) =
G
{ϕi (bt xb−1
t )−
η̄(dt, ·) = cbt η c (dt, ·)
∑
ϕp (x)[Ad(bt )]ip }η c (t, dx)
(components of b̄t ),
p
(cb = lb ◦ rb−1 is the conjugation map),
ν̄t (·) = (cbt− ◦ rh−1
)νt ,
t
where η c is the continuous part of η and νt are jump measures of η, defined by (6.14), with
local means ht , noting ν̄t = δe if η(t, ·) is continuous at time t, and ν̄t = (lbt− ◦ rb−1
)νt if
t
(b, A, η) is proper.
Proof It is easy to see that in the notation of this lemma,
Tt f (z; b, A, η) = St f (z; b̄, Ā, η̄, ν̄),
(6.30)
but we need to verify that b̄i (t) have finite variation, η̄(t, ·) is a Lévy measure function, and
(6.27) holds for
∑ ν̄t .
∑
−1
ϕ (x)Ad(bu )ξj
j j
Because e j ϕj (bu xbu )ξj = bu xb−1
=
e
for u ≤ t and x in a small neighu
borhood U of e, the integrand in the integral defining b̄i (t) above vanishes in U . Noting
η c (t, U c ) < ∞, it is now easy to show that b̄i (t) have finite variation.
180
Because η c (t, U c ) < ∞ for any neighborhood U of e,
∫
c
t
∫
η̄(t, U ) =
0
U
c
(bs xb−1
s )η (ds, dx)
c
∑
∫
=
0
t
∫
c
b−1
s U bs
η(ds, dx) < ∞.
∑
∑
Because x = e p ϕp (x)ξp for x near o, bt xb−1
= e p ϕp (x)[Ad(bt )ξp ] = e p,q ϕp (x)[Ad(bt )]qp ξq . It
t
∑
follows that when U is small, ϕi (bt xb−1
t ) =
j [Ad(bt )]ij ϕj (x) for x ∈ U . It is then easy to
2
c
derive η̄(t, ∥ϕ∥ ) < ∞ from η (t, ∥ϕ∥) < ∞. This shows that η̄ is a Lévy measure function.
∑
∑
The first inequality in (6.27) for ν̄, u≤t ν̄u (U c ) < ∞, follows from u≤t νu (U c ) < ∞ for
any neighborhood U of e, noting that bu and hu , for u ∈ [0, t], are contained in a∑compact
subset of G. Write b and h and for bu− and hu , and∑assume νu is moderate, so h = e p ϕp (h)ξp .
By the Taylor expansion of ϕj (bxh−1 b−1 ) = ϕj (be p ϕp (x)ξp h−1 b−1 ) at x = h,
ϕj (bxh−1 b−1 ) =
∑
[ϕp (x) − ϕp (h)]ϕj,p + rj ,
p
∑
˙ − ϕ(h)∥2 for some
where ϕj,p = (∂/∂ψp )ϕj (be r ψr ξr h−1 b−1 ) |ψr =ϕr (h) and |rj | ≤ c∥ϕ(x)
∫
constant c > 0 not depending on u ≤ t. Because νu is moderate, [ϕp (x) − ϕp (h)]νu (dx) = 0,
and hence, after excluding finitely many terms,
∑ ∫
|
−1
ϕj (bu− xh−1
u bu− )νu (dx)| ≤ c
∑
νu (∥ϕ − ϕu (hu )∥2 ) < ∞.
u≤t
u≤t
This proves the second inequality in (6.27) for ν̄t . The third inequality is proved in a similar
fashion noting |ϕj (bxh−1 b−1 )|2 ≤ c∥ϕ(x) − ϕ(h)∥2 for some constant c > 0. 2
In the course of the proof of Theorem 6.11, we will show (see Lemmas 7.24 and 7.26)
that given any admissible quadruple (b, A, η, ν), there is an inhomogeneous Lévy process zt
in G with z0 = e, unique in distribution, such that the (b, A, η, ν)-martingale property holds.
By Lemma 6.19, whose proof does not depend on Theorems 6.9 and 6.11, this implies the
second half of Theorem 6.11.
Later we will see (Lemma 6.31) that for a rcll process zt in G, the (b, A, η, ν)-martingale
property can hold for at most one quadruple (b, A, η, ν).
Lemma 6.20 If zt is a rcll process in G having a (b, A, η, ν)-martingale property, then for
−1
z
and has distribution νt .
any t > 0, zt−
zt is independent of Ft−
z
] = 0, that is,
Proof Let Mt be the martingale (6.28). Then E[Mt − Mt− | Ft−
∀f ∈
Cc∞ (G),
z
] =
Then E[f (zt ) | Ft−
z
Ft− ] = νt (f ). 2
E{f (zt ) − f (zt− ) −
∫
G
∫
G
z
[f (zt− x) − f (zt− )]νt (dx) | Ft−
} = 0.
−1
−1
f (zt− x)νt (dx). Replacing f (x) by f (zt−
x) shows E[f (zt−
zt ) |
181
Lemma 6.21 If xt is a rcll process in G represented by an extended Lévy triple (b, A, η),
x
then for any t > 0, x−1
t− xt is independent of Ft− and has distribution νt , the jump measure
of η at time t given in (6.14).
Proof: Let xt = zt bt . By Lemma 6.19, zt has the (b̄, Ā, η̄, ν̄)-martingale property, and by
−1
x
Lemma 6.20, zt−
zt is independent of Ft−
and has distribution ν̄. Then for f ∈ Cc∞ (G),
x
−1 −1
x
E[f (x−1
t− xt ) | Ft− ] = E[f (bt− zt− zt bt ) | Ft− ] =
6.6
∫
G
f (b−1
t− xbt )ν̄t (dx) = νt (f ). 2
A transformation rule
Let zt be a rcll process in G and assume it has the (b, A, η, ν)-martingale property for an
admissible quadruple (b, A, η, ν), that is, f (zt ) − St f (z; b, A, η, ν) is a martingale under Ftz ,
the natural filtration of zt . Let ut be an extended drift of finite variation with components
ui (t). We would like to derive a martingale property for the process xt = zt ut . We will
obtain some formulas that may be regarded as a version of Itô’s formula, but they do not
seem to follow easily from the standard Itô’s formula, and so will be established directly.
For this purpose, let J be the set of time points s > 0 such that either νs ̸= δe or
u−1
s− us ̸= e. Let ∆n : 0 = tn0 < tn1 < · · · < tni ↑ ∞ (as i ↑ ∞) be a sequence of partitions
of R+ with mesh ∥∆n ∥ → 0, and assume Jn ⊂ ∆n ⊂ ∆n+1 for all n. For f ∈ Cc∞ (G), let
fni (z) = f (zutni ). Because zt has the (b, A, η, ν)-martingale property,
fni (zt ) = fni (e) + Mtni + St fni ,
where Mtni is a martingale with M0ni = 0 and St f = St f (z; b, A, η, ν). We will write Ss,t f for
St f − Ss f for s < t. It follows that for t ∈ [tni , tn i+1 ),
f (zt utni ) = f (e) +
i
∑
[f (ztnj utnj ) − f (ztn j−1 utn j−1 )] + [f (zt utni ) − f (ztni utni )]
j=1
i
∑
= f (e) +
[f (ztnj utn j−1 ) − f (ztn j−1 utn j−1 )] + [f (zt utni ) − f (ztni utni )]
j=1
+
i
∑
[f (ztnj utnj ) − f (ztnj utn j−1 )]
j=1
= f (e) +
i
∑
[Mtnnjj−1 − Mtnnj−1
+
j−1
j=1
+
i
∑
i
∑
Stn j−1 , tnj fn j−1 ] + [Mtni − Mtni
+ Stni , t fni ]
ni
j=1
[f (ztnj utnj ) − f (ztnj utn j−1 )]
j=1
(n)
= f (e) + Mt
+
∑
Stn j−1 ,tnj fn j−1 + Stni ,t fni
tnj ≤t
182
i
∑
+
[f (ztnj utnj ) − f (ztnj utn j−1 )],
(6.31)
j=1
∑
(n)
where Mt = ij=1 [Mtnnjj−1 −Mtnnj−1
]+[Mtni −Mtni
], t ∈ [tni , tn i+1 ), is a martingale. Because
j−1
ni
Jn ⊂ ∆n , Stni ,t fni → 0 as n → ∞ for t ∈ [tni , tn i+1 ).
∑
Lemma 6.22 In (6.31),
t ∈ R+ , and converges to
∫
t
0
Stn j−1 ,tnj fn j−1 is bounded in absolute value for each fixed
∫
∑
(f ◦ rus )(zs )dbp (s) +
∫
p
t∫
+
0
+
tnj ≤t
G
1∑
ξp ξq (f ◦ rus )(zs )dApq (s)
2 p,q
t
0
{f (zs xus ) − f (zs us ) −
∑∫
s≤t G
∑
ϕp (x)ξp (f ◦ rus )(zs )}η(ds, dx)
p
[f (zs− xus− ) − f (zs− us− )]νs (dx)
(6.32)
as n → ∞.
Proof: By (6.29),
∑ ∫
{
tni ≤t
tni
∑
tn i−1
p
∑
tni ≤t
∫
Fp dbp (s)+
Stn i−1 ,tni fn i−1 may be written as
tni
∑
tn i−1 p,q
∫
Fpq dApq (s)+
tni
tn i−1
∫
G
∫
∑
c
Fη η (ds, dτ )+
tn,i−1 <s≤tni
G
Fs νs (dτ )}
for suitable expressions Fp , Fpq , Fη and Fs .
We have Fp = ξp fn i−1 (zs ) = ξp (f ◦ ru )(zs ) for tn i−1 < s ≤ tni , where u = utn i−1 . Because
Jn ⊂ ∆n , it is easy to see that for tn i−1 < s < tni , Fp → ξp (f ◦ rus− )(zs ) as n → ∞. By the
continuity of bp (t) in t,
∑ ∫
tni
tni ≤t tn i−1
∑
Fp dbp (s) →
p
∫
t
∑
0
ξp (f ◦ rus )(zs )dbp (s) as n → ∞.
p
∑
∫
∑
It is also easy to see that |Fp | is bounded for any fixed t, so is | tni ≤t ttnnii−1 p Fp dbp (s)|.
∫
∑
∑
Similarly, we can show that tni ≤t ttnnii−1 p,q Fpq dApq (s) is bounded in absolute value for
∫ ∑
any fixed t, and converges to 0t p,q ξp ξq (f ◦ rus )(zs )dApq (s) as n → ∞.
For tn i−1 < s < tni ,
Fη = fn i−1 (zs x) − fn i−1 (zs ) −
∑
ϕp (x)ξp fn i−1 (zs )
p
−1
= f (xs u−1
s xutn i−1 ) − f (xs us utn i−1 ) −
→ f (zs xus ) − f (zs us ) −
∑
∑
ϕp (x)ξp (f ◦ ru )(zs ) (u = utn i−1 )
p
ϕp (x)ξp (f ◦ rus )(zs ) as n → ∞.
p
∑
By the Taylor expansion of f (zs xutn i−1 ) = f (zs e p ϕp (x)ξp utn i−1 ) at x = e, it can be shown
that |Fη | ≤ c∥ϕ∥2 on U and |Fη | ≤ c on U c for a small neighborhood U of e and a constant
183
∫
∑
∫
c > 0 depending only on (f, t, η, u). It follows that tni ≤t ttnnii−1 G Fη η(ds, dx) is bounded in
absolute value for any fixed t, and by the continuity of η(t, ·) in t, it converges to
∫
t
∫
0
G
{f (zs xus ) − f (zs us ) −
∑
ϕp (x)ξp (f ◦ rus )(zs )}η(ds, dx)
p
as n → ∞.
For tn i−1 < s ≤ tni ,
Fs = fn i−1 (zs− x) − fn i−1 (zs− ) = f (zs− xutn i−1 ) − f (zs− utn i−1 ).
Because Jn ⊂ ∆n , the jumps of ut eventually occur at tni , it follows that
∑
∫
∑
G
tni ≤t tn i−1 <s≤tni
Fs νs (dx) →
∑∫
s≤t G
[f (zs− xus− ) − f (zs− us− )]νs (dx).
∑
By the Taylor expansion of f (zs− xutn i−1 ) at x = e, it can be shown that Fs = p cp ϕp (x) + r
with |r| ≤ c∥ϕ∥2 on a small neighborhood U of e for some constants cp and c, then by
∫
∑
∑
the three inequalities in (6.27), it is easy to show that | tni ≤t tn i−1 <s≤tni G Fs νs (dx)| is
bounded for any fixed t. 2
∑
Lemma 6.23 In (6.31), ij=1 [f (ztnj utnj ) − f (ztnj utn j−1 )] is bounded in absolute value for
each fixed t ∈ R+ , and converges to
∫
t
∑
0
∫
ξp f (zs us )dup (s) +
p
t
=
[f (zs us ) − f (zs us− )]
s≤t
∑
0
∑
ξp f (xs )dup (s) +
p
∑∫
s≤t G
[f (zs− xus ) − f (zs− xus− )]νs (dx) + Mt′ ,
(6.33)
where Mt′ is a martingale under Ftz .
Proof: The boundedness is a consequence of finite variation of ut . By (6.24),
i
∑
[f (ztnj utnj ) − f (ztnj utn j−1 )]
j=1
=
i ∫
∑
{
j=1
tnj
∑
tn j−1
∑
ξp f (ztnj us )dup (s) +
p
[f (ztnj us ) − f (ztnj us− )]}.
tn j−1 <s≤tnj
Because bp (t) is continuous,
i ∫
∑
tnj
j=1 tn j−1
∑
ξp f (ztnj us )dup (s) →
p
∫
t
0
∑
ξp f (zs us )dup (s)
p
almost surely as n → ∞. Because Jn ⊂ ∆n , so jumps of bu eventually occur at tnj , and
hence
i
∑
∑
[f (ztnj us ) − f (ztnj us− )] →
j=1 tn j−1 <s≤tnj
∑
[f (zs us ) − f (zs us− )]
s≤t
184
almost surely as n → ∞. It then follows that
i
∑
[f (ztnj utnj ) − f (ztnj utn j−1 )] →
j=1
∫
t
=
0
where Mt′ =
∑
ξp f (xs )dup (s) +
p
∑
∑∫
s≤t G
∫
t
∑
0
ξp f (zs us )dup (s) +
p
[f (zs us ) − f (zs us− )]
s≤t
−1
′
[f (xs− u−1
s− xus ) − f (xs− us− xus− )]νs (dx) + Mt ,
s≤t {f (zs us ) − f (zs us− ) − E[f (zs us ) − f (zs us− )
E[f (zs us ) − f (zs us− ) |
∑
z
Fs−
]
∫
=
G
z
| Fs−
]} is a martingale, noting
[f (zs− xus ) − f (zs− xus− )]νs (dx)
−1
z
because by Lemma 6.20, zs−
zs has distribution νs and is independent of Fs−
. 2
For any admissible quadruple (b, A, η, ν), an extended drift ut of finite variation with components ui (t), a rcll process zt in G, and f ∈ Cc∞ (G), let Stu f (z; b, A, η, ν) be the expression
St f (z; b, A, η, ν) given in (6.29) but with f replaced by f ◦ rus− . Because bi (s), Ajk (s) and
η(s, ·) are continuous in s, we see that Stu f = Stu f (z; b, A, η, ν) is just (6.32) in Lemma 6.22.
Because Jn ⊂ ∆n , it is clear that for t ∈ [tni , tn i+1 ), f (zt utni ) → f (zt ut ). By (6.31), (6.32)
(n)
and (6.33), Mt + Mt′ is a martingale bounded on any finite time interval, so it converges
to a martingale Mt . It follows that if zt has the (b, A, η, ν)-martingale property, then
f (zt ut ) − Stu f (z; b, A, η, ν) −
−
∑∫
s≤t G
∫
t
0
∑
ξp f (zs us )dui (s)
p
[f (zs− xus ) − f (zs− xus− )]νs (dx)
(6.34)
is a martingale. This is summarized in the following lemma.
Lemma 6.24 Let zt be a rcll process in G and assume it has the (b, A, η, ν)-martingale
property for an admissible quadruple (b, A, η, ν). Let ut be an extended drift of finite variation
with components ui (t). Then for any f ∈ Cc∞ (G), (6.34) is a martingale under Ftz .
For any extended Lévy triple (b, A, η), possibly improper, an extended drift ut of finite
variation with components ui (t), a rcll process zt , and f ∈ Cc∞ (G), let Ttu f = Ttu f (z; b, A, η)
be the expression Tt f (z; b, A, η) given in (6.22) but with f replaced by f ◦ rus− . Because
Tt f (z; b, A, η) = St f (z; b̄, Ā, η̄, ν̄), where b̄, Ā, η̄, ν̄ are given in Lemma 6.19, and it is easy
to see that Ttu f (z; b, A, η) = Stu f (z; b̄, Ā, η̄, ν̄), it follows that if f (zt ) − Tt f (z; b, A, η) is a
martingale for any f ∈ Cc∞ (G), then so is
f (zt ut ) − Ttu f (z; b, A, η) −
∫
t
0
∑
ξi f (zs us )dui (s) −
i
185
∑∫
s≤t G
[f (zs− xus ) − f (zs− xus− )]ν̄s (dx).
We note that Ttu f (z; b, A, η) = Tt f (zu; u−1 b, A, η), and hence
−1
f (zt ut ) − Tt f (zu; u b, A, η) −
−
∑∫
s≤t G
∫
t
∑
0
ξi f (zs us )dui (s)
i
−1
−1
−1 −1
[f (zs− bs− xh−1
s bs− us ) − f (zs− bs− xhs bs− us− )]νs (dx)
(6.35)
is a martingale, where hs is the local mean of νs . This result is recorded as part (a) of the
following lemma. Its part (b) is an immediate consequence of (a).
Lemma 6.25 (a) Let xt = zt bt be a rcll process in G represented by an extended Lévy triple
(b, A, η), possibly improper, and let ut be an extended drift of finite variation with components
ui (t). Then for any f ∈ Cc∞ (G), (6.35) is a martingale under Ftx .
(b) In (a), if ut is continuous, that is, if ut is a drift, then for any f ∈ Cc∞ (G),
f (zt ) − Tt f (zu; u−1 b, A, η) −
∫
t
∑
0
ξi f (zs us )dui (s)
(6.36)
i
is a martingale under Ftx .
The following proposition provides a transformation rule for an admissible quadruple
when an extended drift is added to the associated process.
Proposition 6.26 Let (b, A, η, ν) be an admissible quadruple and let ut be an extended drift
of finite variation with components ui (t). Let (bu , Au , η u , ν u ) be defined as follows: but is a
drift of finite variation with components bui (t) given by
dbui (t)
=
∑
[Ad(u−1
t )]ip dbp (t)
p
∫
+ dui (t) +
G
{ϕi (u−1
t xut ) −
∑
[Ad(u−1
t )]ip ϕp (x)}η(dt, dx),
p
−1 ′
Au (t) is the covariance matrix function given by dAu (t) = [Ad(u−1
t )]dA(t)[Ad(ut )] (prime
′
is the matrix transpose), η u (dt, ·) = cu−1
η(dt, ·) (cu = lu ◦ ru−1 is the conjugation map),
t
u
u
u u
u
and νt = (lu−1
◦ rut )νt . Then (b , A , η , ν ) is an admissible quadruple. Moreover, if a rcll
t−
process zt with has the (b, A, η, ν)-martingale property, then xt = zt ut has the (bu , Au , η u , ν u )martingale property.
Proof: As in the proof of Lemma 6.19 for b̄ and η̄, we can show that but has a finite
variation and η u is a Lévy measure function. To prove (6.27) for ν u , first note that because
ut is contained in a compact set for t in any finite time interval, the first inequality in (6.27)
∑
∑
for ν u , s≤t νs (U c ) < ∞ for any neighborhood U of e, follows easily from s≤t νs (U c ) < ∞.
By the Taylor expansion of ϕj (u−1
s− xus ) at x = e,
−1
ϕj (u−1
s− xus ) = ϕj (us− us ) +
∑
p
186
ϕp (x)ϕj,p + rj ,
∑
where ϕj,p = (∂/∂ψp )ϕj (u−1
s− e
not depending on s ≤ t. Then
∫
νsu (ϕj ) =
r
ψr ξp
us ) |ψr =0 and |rj | ≤ c∥ϕ(x)∥2 for some constant c > 0
−1
ϕj (u−1
s− xus )νs (ds) = ϕj (us− us ) + νs (ϕj ) + νs (rj ).
∑
Because ut has a finite variation, s≤t |ϕj (b−1
s− bs )| < ∞ (see Proposition 6.16 (a)). It is now
easy to derive the second and the third inequalities in (6.27) for ν u from those inequalities
for ν. We have proved that (bu , Au , η u , ν u ) is an admissible quadruple.
By Lemma 6.24, (6.34) is a martingale. After a little simplification, (6.34) can be written
as
∫
0
t
∑
∫
[Ad(u−1
s )ξp ]f (xs )dbp (s)
p
t∫
+
0
+
G
∫
{f (xs u−1
s xus ) − f (xs ) −
∑∫
s≤t G
t
+
0
1∑
−1
[Ad(u−1
s ξp ][Ad(us )ξq ]f (xs )dApq (s)
2 p,q
∑
ϕp (x)[Ad(u−1
s )ξp ]f (xs )}η(ds, dx)
p
[f (xs− u−1
s− xus ) − f (xs− )]νs (dx) +
∫
0
t
∑
ξp f (xs )dui (s),
p
which is just St f (z; bu , Au , η u , ν u ). 2
We record the following relation. If (b, A, η) is an extended Lévy triple with bt of finite
variation, then using the notations in Lemma 6.19 and Proposition 6.26,
−1
−1
−1
−1
(b̄, Ā, η̄, ν̄) = ((bc )b , Ab , (η c )b , ν b ).
(6.37)
−1
Note that to show b̄ = (bc )b , Proposition 6.15 (c) is used for the components of b−1 .
6.7
Representation in the case of finite variation
For f ∈ Cc∞ (G), a rcll path xt in G and an extended Lévy triple (b, A, η) with bt of finite
variation, with components bi (t), let
∫
S̃t f (x; b, A, η) =
−f (xs ) −
∑
d
t∑
0 j=1
∫
ξj f (xs )dbj (s) +
ϕj (τ )ξj f (xs )}η c (ds, dτ ) +
j
0
t
∫ t∫
d
1 ∑
ξj ξk f (xs ) dAjk (s) +
{f (xs τ )
2 j,k=1
0 G
∑∫
s≤t G
[f (xs− τ ) − f (xs− )]νs (dτ ).
(6.38)
By Proposition 6.18 (b) and (6.29),
S̃t f (x; b, A, η) = St f (x; bc , A, η c , ν).
(6.39)
Then by Proposition 6.17, S̃t f = S̃t f (x; b, A, η) in (6.38) is bounded for fixed t for all rcll
paths xt .
187
The following theorem provides a direct martingale representation in the case when the
extended drift has a finite variation. We will provide a proof that does not depend on
Theorems 6.9 and 6.11, so it may may used in the proofs of these main theorems.
Theorem 6.27 Let (b, A, η) be an extended Lévy triple on G with bt of finite variation. Then
a rcll process xt in G with x0 = e is represented by (b, A, η) if and only if
f (xt ) − S̃t f (x; b, A, η)
(6.40)
is a martingale under Ftx for any f ∈ Cc∞ (G).
Proof: Let xt = zt bt . Assume xt is represented by (b, A, η). By Lemma 6.25, (6.35) with
u = b is a martingale for any f ∈ Cc∞ (G), that is,
f (zt bt ) − Tt f (zb; ê, A, η) −
∫
0
t
∑
ξi f (zs bs )dbi (s) −
i
∑∫
s≤t G
[f (zs− bs− x) − f (zs− bs− xh−1
s )]νs (dx)
is a martingale, where ê denotes the trivial drift that is identically equal to e. The above
expression is just (6.40).
Now assume (6.40) is a martingale. Then xt has the (bc , A, η c , ν)-martingale property.
By Lemma 6.24 with (z, u, η) being (x, b−1 , η c ), noting the components ui (t) of ut = b−1
are
t
∑
given by dui (t) = − j [Ad(bt )]ij dbi (t) (see Proposition 6.15 (c)), we see that
−1
b
c
c
f (xt b−1
t ) − St f (x; b , A, η , ν) +
−
∑∫
s≤t G
∫
0
t
∑
[Ad(bs )ξi ]f (xs b−1
s )dbi (s)
i
−1
[f (xs− xb−1
s ) − f (xs− xbs− )]νs (dx).
is a martingale. After some simplification, the above is just f (zt ) − Tt f (z; b, A, η). This
proves that xt = zt bt is represented by (b, A, η). 2
Remark 6.28 Let (b, A, η) be an extended Lévy triple with bt of finite variation. By (6.39)
and Theorem 6.27, a rcll process in G is represented by (b, A, η) if and only if it has the
(bc , A, η c , ν)-martingale property. Conversely, for an admissible quadruple (b, A, η, ν), a rcll
process in G has the (b, A, η, ν)-martingale property if and only if it is represented by the
extended Lévy triple (b′ , A, η + ν ∗ ), where b′ and ν ∗ are defined in Proposition 6.18.
In particular, if (b, A, η) is a Lévy triple with bt of finite variation, then (b, A, η, δˆe ) is an
admissible quadruple, where δˆe denotes the special family ν of probability measures νt on
G, t ∈ R+ , with all νt = δe . In this case, a rcll process in G is represented by (b, A, η) if and
only if it has the (b, A, η, δˆe )-martingale property.
188
Remark 6.29 Let an inhomogeneous Lévy process xt in G be represented by an extended
Lévy triple (b, A, η). By Theorem 6.27, if bt is of finite variation, then xt is a semimartingale
in G. Conversely, if xt is a semimartingale, then bt = xt zt−1 as a non-random semimartingale
must be of finite variation (see Proposition 4.28 in [45, chapter I]). Therefore, bt is of finite
variation if and only if xt is a semimartingale.
Remark 6.30 Let xt be an inhomogeneous Lévy process in G represented by an extended
Lévy triple (b, A, η), and let xt = zt bt . Then by Lemma 6.19 and Proposition 6.18 (a), zt as
an inhmogeneous Lévy process is represented by the extended Lévy triple (b̄′ , Ā, η̄ + ν̄ ∗ ).
6.8
Uniqueness of triple and quadruple
In this section, we will prove that a rcll process zt can have the (b, A, η, ν)-martingale property for at most one quadruple (b, A, η), then we will establish the uniqueness of the extended
Lévy triple in Theorem 6.11, which also implies the uniqueness of the Lévy triple in Theorem 6.9. Our proofs do not depend on Theorems 6.9 and 6.11.
Let (b, A, η) be a Lévy triple with bt of finite variation. We will describe a non-random
time change used in [25], under which Aij (t), η(dt, dx) and bi (t) become absolutely continuous
in t. Choose f0 ∈ C ∞ (G) such that
f0 =
d
∑
ϕ2i near e,
f0 = 1 on U c ,
0 < f0 ≤ 1 on G − {e},
(6.41)
i=1
∫
where U is a relatively compact neighborhood of e. Then F (t) = G f0 (x)η(t, dx) is nondecreasing and continuous in t. The measure f0 (x)η(dt, dx) on R+ × G is finite on [0, t] × G
for any t > 0, and hence factors into the marginal measure dF (t) of t and the conditional
distribution m0 (t, dx) of x given t. Then η(dt, dx) = [1/f0 (x)]m0 (t, dx)dF (t). Let b̄i (t) be
the total variation of bi (·) on [0, t]. Define
ψ(t) = Tr[A(t)] +
∑
b̄i (t) + F (t) + t,
(6.42)
i
a(s) =
dA −1
(ψ (s)),
ds
βi (s) =
dbi −1
(ψ (s)),
ds
m(s, dx) =
1
dF
m0 (ψ −1 (s), dx) (ψ −1 (s)),
f0 (x)
ds
where the derivatives are the Radon-Nikodym derivatives and ψ −1 is the inverse of ψ. Substituting ψ −1 (s) for t in (6.42) and then differentiating with respect to s, we obtain
1 = Tr[a(s)] +
∑
i
|βi (s)| +
d
d
F (ψ −1 (s)) + ψ −1 (s).
ds
ds
189
∫
Because G f0 (x)m(s, dx) = m0 (ψ −1 (s), G)(d/ds)F (ψ −1 (s)) = (d/ds)F (ψ −1 (s)), it follows
that after the time change by ψ(t) in (6.42), A(t), bi (t) and η(t, dx) will have density a(t),
βi (t) and m(t, dx) with respect to the Lebesgue measure dt on R+ such that
Tr[a(t)] +
∑
|βi (t)| +
i
∫
G
f0 (x)m(t, dx) ≤ 1.
(6.43)
In particular, m(t, ∥ϕ∥2 1V ) + δm(t, V c ) ≤ 1 for some neighborhood V of e, where δ =
inf x∈V c f0 (x) > 0. It follows that m(t, ·) is a Lévy measure on G for each t ≥ 0.
Let L(t) be the operator on Cc∞ (G) given by
L(t) =
∫
∑
∑
1∑
aij (t)ξi ξj +
βi (t)ξi + m(t, dy)[ry − 1 −
ϕi (y)ξi ],
2 i,j
G
i
i
(6.44)
where the right translation rg is regarded as an operator on functions on G: rg f (x) =
(f ◦ rg )(x) = f (xg).
Recall S̃t f (x; b, A, η) is defined by (6.38). Because (b, A, η) is a Lévy triple, the last term
∫
∑
in (6.38), s≤t G [f (xs− τ ) − f (xs− )]νs (dτ ), does not appear. After the non-random time
∫
change by ψ, we have S̃t f = 0t L(s)f (xs )ds.
Lemma 6.31 Let zt be a rcll process in G. Then there is at most one quadruple (b, A, η, ν)
such that the (b, A, η, ν)-martingale property holds for zt .
Proof Suppose (b′ , A′ , η ′ , ν ′ ) is another quadruple and zt has the (b′ , A′ , η ′ , ν ′ )-martingale
property. By Lemma 6.20, ν ′ = ν. The time change described earlier may be performed
for the two triples (b, A, η) and (b′ , A′ , η ′ ) together, just adding ψ in (6.42) for (b, A, η) and
(b′ , A′ , η ′ ), so that after the time change, b, A, η, b′ , A′ , η ′ have densities
βi (t), aij (t), m(t, dx), βi′ (t), a′ij (t), m′ (t, dx)
respectively, from which the operators L(t) and L′ (t) may be defined from (β, a, m) and
(β ′ , a′ , m′ ) as in (6.44). We want to show that m(t, ·) = m′ (t, ·), aij (t) = a′ij (t) and βi (t) =
βi′ (t).
We will write the martingale in (6.28) as
Mt f = f (zt ) − S̃t f (z; b, A, η) −
∑∫
u≤t G
[f (zu− x) − f (zu− )]νu (dx).
∫
After the time change, S̃t f (z; b, A, η) = 0t L(s)f (zs )ds. Let Mt′ f be Mt f when (b, A, η) is
∫
∫
replaced by (b′ , A′ , η ′ ). Then Mt′ f is a martingale, and 0t L(s)f (zs )ds − 0t L′ (s)f (zs )ds is a
continuous martingale of finite variation, and hence almost surely, for all t ∈ [0, T ],
∫
0
∫
t
L(s)f (zs )ds =
190
0
t
L′ (s)f (zs )ds.
(6.45)
The space Cc∞ (G) is separable in the sense that it has a countable subset Γ such that for any
f ∈ Cc∞ (G), there are fn ∈ Γ with fn → f , ξi fn → ξi f and ξi ξj fn → ξi ξj uniformly on G as
n → ∞, see Theorem A.6 in Appendix A.7. Then for any f ∈ Cc∞ (G), there are fn ∈ Γ such
that L(t)fn (x) → L(t)f (x) and L′ (t)fn (x) → L′ (t)f (x) uniformly for (t, x) ∈ [0, T ] × G. It
follows that almost surely, (6.45) holds for all f ∈ Cc∞ (G) and t ∈ [0, T ]. There is a fixed rcll
path xt in G such that (6.45) holds for all f ∈ Cc∞ (G) and t ∈ [0, T ]. Then there is a subset
E of [0, T ], with complement being a set of zero Lebesgue measure, such that for all t ∈ E,
L(t)f (xt ) = L′ (t)f (xt ). Because the separability of Cc∞ (G), the set E may be chosen for
all f ∈ Cc∞ (G). For any h ∈ Cc∞ (G) vanishing in a neighborhood of e, let f (x) = h(xt−1 x).
Then L(t)f (xt ) = L′ (t)f (xt ) implies m(t, h) = m′ (t, h) and hence m(t, ·) = m′ (t, ·). Next let
−1
f (x) = ϕi (x−1
t x)ϕj (xt x). Then ξp f (xt ) = 0 and ξp ξq f (xt ) = δip δjq + δiq δjp . It follows that
∫
aij (t) +
=
a′ij (t)
∫
G
+
G
[f (xt y) − f (xt ) −
∑
ϕp (y)ξp f (xt )]m(t, dy)
p
[f (xt y) − f (xt ) −
∑
ϕp (y)ξp f (xt )]m(t, dy).
p
′
This shows aij (t) = a′ij (t). Now letting f (x) = ϕi (x−1
t x) will show βi (t) = βi (t). 2
Lemma 6.32 There is at most one extended Lévy triple (b, A, η) which represents a given
inhomogeneous Lévy process xt in G as in Theorem 6.11.
Proof Suppose two extended Lévy triples (b1 , A1 , η 1 ) and (b2 , A2 , η 2 ) represent the same
inhomogeneous Lévy process xt in G. Let ν i be the family of jump measures of η i , i = 1, 2.
By Lemma 6.21, ν 1 = ν 2 , and hence b1t and b2t have the same jumps. Let xt = zt1 b1t = zt2 b2t .
Then zt1 = zt2 ut , where ut = b2t (b1t )−1 is a rcll path in G. Note that ut = (zt2 )−1 zt1 is a
semimartingale, that is, f (ut ) is a real semimartingale for any f ∈ Cc∞ (G). Because ut is nonrandom, by Remark 6.29, ut has a finite variation. We now show ut is continuous and hence
1
2 −1 2 1 −1
1 −1
1
1 −1 1 −1
2 −1 2
is a drift. Let x = u−1
t− ut = bt− (bt− ) bt (bt ) , then (bt ) xbt = [(bt− ) bt ] [(bt− ) bt ] = e
because b1t and b2t have the same jumps. This implies x = e and hence ut is continuous.
We have f (zti ) − Tt f (z i ; bi , Ai , η i ) is a martingale for any f ∈ Cc∞ (G), i = 1, 2. Then
f (zt1 ) − St f (z 1 ; b̄1 , Ā1 , η̄ 1 , ν̄ 1 )
(6.46)
is a martingale, where (b̄1 , Ā1 , η̄ 1 , ν̄ 1 ) is (b̄, Ā, η̄, ν̄) in Lemma 6.19 with (b, A, η) = (b1 , A1 , η 1 ).
Because zt1 = zt2 ut , by Lemma 6.25 (b) with zt = zt2 and (b, A, η) = (b2 , A2 , η 2 ),
f (zt2 ut )
−1 2
− Tt f (z u; u b , A , η ) −
2
2
191
∫
2
0
t
∑
i
ξi f (zs2 us )dui (s)
is a martingale. Because ut = b2t (b1t )−1 and zt1 = zt2 ut ,
f (zt1 )
− Tt f (z ; b , A , η ) −
1
1
2
∫
t
2
∑
0
ξi f (zs1 )dui (s)
i
= f (zt1 ) − St f (z 1 ; b̄1′ , Ā2 , η̄ 2 , ν̄ 2 ) −
∫
t
0
∑
ξi f (zs1 )dui (s)
(6.47)
i
is a martingale, where (b̄1′ , Ā2 , η̄ 2 , ν̄ 2 ) is (b̄, Ā, η̄, ν̄) in Lemma 6.19 with (b, A, η) = (b1 , A2 , η 2 ).
By (6.46), zt1 has the (b̄1 , Ā1 , η̄ 1 , ν̄ 1 )-martingale property. By the uniqueness of the
quadruple in Lemma 6.31, and (6.47), we have Ā1 = Ā2 and η̄ 1 = η̄ 2 . Because dĀ1 (t) =
[Ad(b1t )]dA1 (t)[Ad(b1t )]′ and dĀ2 (t) = [Ad(b1t )]dA2 (t)[Ad(b1t )]′ , we obtain A1 = A2 . Because
η̄ 1 (dt, ·) = cb1t (η 1 )c (dt, ·) and η̄ 2 (dt, ·) = cb1t (η 2 )c (dt, ·), it follows that (η 1 )c = (η 2 )c . Because
ν 1 = ν 2 , we have η 1 = η 2 . Using (η 1 )c = (η 2 )c , we see that b̄1 = b̄1′ . Let b̄1i (t) be the
components of b̄1 . By (6.47), we see that b̄1i (t) = b̄1i (t) + ui (t), and hence ui (t) = 0. This
shows that b1 = b2 . 2
Recall a Lévy triple or an extended Lévy triple is defined under a basis {ξ1 , . . . , ξd } of g
and associated exponential coordinate functions ϕi . Let an inhomogeneous Lévy process xt
be represented by an extended Lévy triple (b, A, η). As noted in Remark 6.12, η as the jump
intensity measure of xt clearly does not depend on ξi and ϕi . The next proposition shows
that A does not depend on ϕi . It also shows that when a rcll process is represented by an
improperly extended Lévy triple (b, A, η), the A and η c are uniquely determined, although
the triple is not as noted before.
Proposition 6.33 Let xt be an inhomogeneous Lévy process in G represented by an extended Lévy triple (b, A, η) (under exponential coordinate functions ϕi ), and represented by an
extended Lévy triple (b′ , A′ , η ′ ) under another set of exponential coordinate functions ϕ′i (associated to the same basis of g), both may be improper. Then A = A′ and η c = η ′c .
Proof: Let xt = zt bt = zt′ b′t . Then zt′ = zt ut , where ut = bt b′−1
is an extended drift of
t
finite variation (as noted in the proof of Lemma 6.32). Then for any f ∈ Cc∞ (G), f (zt ) −
Tt f (z; b, A, η) is a martingale, and so is f (zt′ ) − Tt1 f (z ′ ; b′ , A′ , η ′ ), where Tt1 f is defined as Tt f
in (6.22) but replacing ϕi in the η c -integral by ϕ′i . Then
f (zt′ )
′
′
′
′
− Tt f (z ; b , A , η ) −
∫
t
∑
0
ξi f (zs′ )dvi (s)
i
∫
∑
is a martingale, where dvi (s) = j [Ad(b′s )]ij G [ϕj (x)−ϕ′j (x)](η ′ )c (ds, dx) are the components
of a drift v of finite variation. It follows that
f (zt′ )
′
′
′
′
′
− St f (z ; b̄ , Ā , η̄ , ν̄ ) −
192
∫
0
t
∑
i
ξi f (zs′ )dvi (s)
(6.48)
is a martingale, where (b̄′ , Ā′ , η̄ ′ , ν̄ ′ ) is (b̄, Ā, η̄, ν̄) in Lemma 6.19 with (b, A, η) = (b′ , A′ , η ′ ).
Because zt′ = zt ut and b′t = u−1
t bt , by Lemma 6.25 (a),
f (zt′ ) − Tt f (z ′ ; b′ , A, η) −
=
f (zt′ )
′
∫
t
∑
0
i
− St f (z ; b̄ , Ā , η̄ , ν̄ ) −
1
1
1
ξi f (zs′ )dui (s) −
∫
t
1
∑
0
∑∫
s≤t G
ξi f (zs′ )dui (s)
i
−
(· · ·)νs (dx)
∑∫
s≤t G
(· · ·)νs (dx)
(6.49)
is a martingale, where (b̄1 , Ā1 , η̄ 1 , ν̄ 1 ) is (b̄, Ā, η̄, ν̄) in Lemma 6.19 with (b, A, η) being (b′ , A, η).
Comparing (6.48) and (6.49), by the uniqueness of the quadruple in Lemma 6.31, we have
Ā′ = Ā1 and η̄ ′ = η̄ 1 , from which it is easy to obtain A′ = A and (η ′ )c = η c . 2
6.9
Representation in the reduced form
The purpose of this section is to present a simpler and a more convenient representation of
an inhomogeneous Lévy process by an extended Lévy triple (b, A, η) when η c has a finite first
moment, that is, when
∫
∀t ∈ R+ ,
G
∥ϕ(x)∥η c (t, dx) < ∞.
(6.50)
Note that this condition holds when η c (t, G) is finite for any t ∈ R+ .
Let xt be an inhomogeneous Lévy process in G with x0 = e, represented by an extended Lévy triple (b, A, η). Recall this means that with xt = zt bt , f (zt ) − Tt f (z; b, A, η) is a
martingale for f ∈ Cc∞ (G). The expression Tt f (z; b, A, η) contains the integral
∫
t
0
∫
G
{f (zu bu yb−1
u ) − f (zu ) −
∑
ϕi (y)[Ad(bu )ξi ]f (zu )}η c (du, dy).
i
∑
Note that without the term i ϕi (y)[Ad(bu )ξi ]f (zu ) in the integrand, the above integral may
∫ ∫
not exist. However, when η c has a finite first moment, the simpler integral 0t G [f (zu bu yb−1
u )−
c
f (zu )]η (du, dy) exists. We will show that in this case, with a suitable change in the extended
drift bt , the martingale representation in Theorem 6.11 holds with this simpler η c -integral,
that is, we will show that after a suitable change in bt , for any f ∈ Cc∞ (G),
f (zt ) − Tt′ f (z; b, A, η)
(6.51)
is a martingale under the natural filtration Ftx of xt , where for any rcll path zt in G,
Tt′ f (z; b, A, η)
−
∫
0
t∫
G
= f (zt ) −
∫
t
0
1∑
[Ad(bs )ξi ][Ad(bs )ξj ]f (zs )dAij (s)
2 i,j
c
[f (zs bs yb−1
s ) − f (zs )]η (ds, dy) −
∑∫
s≤t G
193
−1
[f (zs− bs− yh−1
s bs− ) − f (zs− )]νs (dy),
(6.52)
and νs are the jump measures of η, as given by (6.14), with local means hs .
Recall that by Propostion 6.10, for any t ∈ R+ , |Tt f (z; b, A, η)| is bounded for all rcll
paths zt in G. The proof of this proposition also works for Tt′ f (z; b, A, η), except that the
first Taylor expansion should be modified to use the finite first moment of η c , to show that
for any t ∈ R+ , |Tt′ f (z; b, A, η)| is bounded for all rcll paths zt .
A rcll process xt in G with x0 = e will be said to be represented by an extended Lévy triple
(b, A, η) in the reduced form, where η c has a finite first moment, if with xt = zt bt , (6.51) is
a martingale under Ftx for any f ∈ Cc∞ (G). Such a representation is more convenient as it
does not directly involve the coordinate functions ϕi .
Let (b, A, η, ν) be an admissible quadruple as defined in §6.5, and assume η c = η has
a finite first moment. For f ∈ Cc∞ (G) and a rcll path zt in G, let St′ f = St′ f (z; b, A, η, ν)
∑
be defined as for St f in (6.29) but removing − i ϕi (x)ξi f (zs ) from the integrand of the ηintegral. Proposition 6.17 is easily seen to be valid for St′ f in place St f , and so for any t ∈ R+ ,
|St′ f | is bounded for all rcll paths zt in G. We will say a rcll process zt in G has the (b, A, η, ν)martingale property in the reduced form if for any f ∈ Cc∞ (G), f (zt ) − St′ f (z; b, A, η, ν) is a
martingale under Ftz .
The results in the last four sections all have versions for the reduced form, and these new
versions can be proved in the same fashion, in fact, the proofs will be simpler. We will state
below some of these results in the reduced form as they will be needed later.
Let (b, A, η) be an extended Lévy triple with η c having a finite first moment. The version
of Lemma 6.19 in the reduced form can be stated as follows: if a rcll process xt in G is
represented by (b, A, η) in the reduced form, then zt = xt b−1
has the (ê, Ā, η̄, ν̄)-martingale
t
property in the reduced form, where ê, as defined before, is the trivial drift that is identically
equal to e, and Ā, η̄, ν̄ are as defined in Lemma 6.19.
Lemmas 6.24 and 6.25 hold also in the reduced form with St f , Stu f and Tt f replaced by
∑
St′ f ,St′u f and Tt′ f respectively, where St′u f is obtained from Stu f after − i ϕi (x)ξi f (zs ) is
removed from the integrand of the η-integral.
The transformation rule, Proposition 6.26, is also valid for the reduced form, that is, if
(b, A, η, ν) is an admissible quadruple with η having a finite first moment, and if u is an
extended drift of finite variation, then (bu′ , Au , η u , η u ) is also an admissible quadruple with
η u having a finite first moment, where Au , η u , ν u are as defined in Proposition 6.26, and
∫ ∑
bu′ is given in components as bui ′ (t) = 0t p [Ad(u−1
s )]ip dbp (s) + ui (t). Moreover, if a rcll
process zt has the (b, A, η, ν)-martingale property in the reduced form, then zt ut has the
(bu′ , Au , η u , ν u )-martingale property in the reduced form.
Let (b, A, η) be an extended Lévy triple with bt of finite variation and η c having a finite
first moment. For a rcll path xt in G and f ∈ Cc∞ (G), let S̃t′ f = S̃t′ f (x; b, A, η) be defined
∑
as for S̃t f in (6.38) but removing − j ϕj (τ )ξj f (xs ) from the integrand of the η c -integral.
194
Then S̃t′ f (x; b, A, η) = St′ f (x; bc , A, η c , ν), where ν = {νt ; t ∈ R+ } is the family of the jump
measures of η, as given in (6.14). The proof of Theorem 6.27 works also when Tt f and S̃t f
are replaced by Tt′ f and S̃t′ f , so we have the following version of Theorem 6.27 in the reduced
form.
Theorem 6.34 Let (b, A, η) be an extended Lévy triple with bt of finite variation and η c
having a finite first moment. Then a rcll process xt with x0 = e is represented by (b, A, η)
in the reduced form if and only if for any f ∈ Cc∞ (G), f (xt ) − S̃t′ f (x; b, A, η) is a martingale
under Ftx .
The versions of Lemmas 6.31 and 6.32 for the reduced form are as follows: A rcll process
xt in G can have at most one admissible quadruple (b, A, η, ν), with η having a finite first
moment, such that the (b, A, η, ν)-martingale property in the reduced form holds for xt , and
there is at most one extended Lévy triple (b, A, η), with η c having a finite first moment, such
that xt is represented by (b, A, η) in the reduced form.
Theorem 6.35 Let xt be a rcll process in G with x0 = e, represented by an extended
Lévy triple (b, A, η), and assume η c has a finite first moment. Then there is an extended
drift brt such that (br , A, η) is an extended Lévy triple, and xt is represented by (br , A, η) in
the reduced form. Moreover, the extended Lévy triple (br , A, η), with η c having a finite first
moment, representing a given rcll process in the reduced form, is unique.
Conversely, given an extended Lévy triple (br , A, η) with η c having a finite first moment,
there is an inhomogeneous Lévy process xt in G with x0 = e, unique in distribution, represented by (br , A, η) in the reduced form. Moreover, xt is represented by the extended Lévy triple
(b̃, A, η) for some b̃. Furthermore, if (br , A, η) is a Lévy triple, then the above uniqueness in
distribution holds among all rcll processes in G starting at e.
Proof: We will first consider an extension of Lemma 6.25 (b), which says that for a rcll
process zt in G and an extended Lévy triple (b, A, η), if f (zt ) − Tt f (z; b, A, η) is a martingale
for any f ∈ Cc∞ (G), then for any drift ut of finite variation, (6.36) is a martingale. We can
∫ ∑
extend this result by including in Tt f (z; b, A, η) an term of the form 0t i ξi f (zs )dvi (s) for
a drift v of finite variation with components vi (t) without any essential change in the proof.
Moreover, the reduced form of this extension also holds with the same proof. To be precise,
we have the following extension of Lemma 6.25 (b).
Lemma 6.36 (a) Let zt be a rcll process in G, let (b, A, η) be an extended Lévy triple, and
let vt be a drift of finite variation with components vi (t). If
f (zt ) − Tt f (z; b, A, η) −
195
∫
t
0
∑
i
ξi f (zs )dvi (s)
is a martingale under Ftz for any f ∈ Cc∞ (G), then so is
−1
f (zt ut ) − Tt f (zu; u b, A, η) −
∫
t
0
∑
[Ad(u−1
s )ξi ]f (zs us )dvi (s) −
i
∫
t
∑
0
ξi f (zs us )dui (s) (6.53)
i
for any drift ut of finite variation with components ui (t).
(b) The statement in (a) holds also for the reduced form, that is, it holds when Tt f is replaced
by Tt′ f .
We now begin the proof of Theorem 6.35. Let xt = zt bt be a rcll process in G represented
by (b, A, η) with η c having a finite first moment. Then
f (zt ) − Tt f (z; b, A, η) = f (zt ) −
is a martingale, where
vi (t) = −
∫
t
Tt′ f (z; b, A, η)
−
∫
t
0
∑
ξi f (zs )dvi (s)
i
∑
[Ad(bs )]ij η c (ds, ϕj )
0
(6.54)
j
are the components of a drift vt of finite variation. By Lemma 6.36 (b), (6.53) with Tt f
replaced by Tt′ f is a martingale. In this expression, setting ut = vt−1 and using Proposition 6.15 (c) for the components of ut , we obtain,
f (zt vt−1 ) − Tt′ f (zv −1 ; vb, A, η)
is a martingale. Let brt = vt bt . Then xt = (zt vt−1 )brt . This shows that xt is represented by
(br , A, η) in the reduced form.
The uniqueness of the extended Lévy triple (br , A, η) representing a given rcll process in
the reduced form is already mentioned before Theorem 6.35.
Now let (br , A, η) be an extended Lévy triple with η c having a finite first moment, and let
xt = zt brt be a rcll process represented by (br , A, η) in the reduced form. Then for f ∈ Cc∞ (G),
f (zt ) − Tt′ f (z; br , A, η) is a martingale, and hence so is
f (zt ) − Tt f (z; b , A, η) −
∫
r
where
∫
wi (t) =
0
t
∑
0
t
∑
ξi f (zs )dwi (s),
i
[Ad(brs )]ij η c (ds, ϕj )
(6.55)
j
are components of a drift wt of finite variation. By Lemma 6.36 (a) with ut = wt−1 ,
and using Proposition 6.15 (c) for the components of wt−1 , we obtain that f (zt wt−1 ) −
Tt f (zw−1 ; wbr , A, η) is a martingale. Let b̃t = wt brt . This shows that xt = (zt wt−1 )b̃t is
represented by the extended Lévy triple (b̃, A, η).
196
By Theorem 6.11, there is an inhomogeneous Lévy process xt represented by (b̃, A, η).
The computation in the last paragraph can be reversed to show that xt is represented by
(br , A, η) in the reduced form. Indeed, with xt = zt b̃t = zt wt brt , f (zt ) − Tt f (z; wbr , A, η) is a
martingale. By Lemma 6.25 (b), so is
f (zt wt ) − Tt f (zw; br , A, η) −
∫
t
0
∑
ξi f (zs ws )dwi (s) = f (zt wt ) − Tt′ f (zw; br , A, η).
i
This shows that xt = (zt wt )brt is an in homogeneous Lévy process represented by (br , A, η) in
the reduced form. By the uniqueness of distribution in Theorem 6.11, the inhomogeneous
Lévy process xt with x0 = e represented by (br , A, η) in the reduced form is unique in distribution, and if (br , A, η) is a Lévy triple, then so is (b̃, A, η), and hence, by Theorem 6.9, the
uniqueness in distribution holds among all rcll processes starting at e. 2
Remark 6.37 It is clear from the proof that in Theorem 6.35, br = vt bt and b̃t = wt brt ,
where vt and wt are the drifts of finite variation with components given by (6.54) and (6.55)
respectively.
6.10
Some simple properties and proof of Theorem 2.1
We present some simple properties such as the time shift and the joining of two processes.
They seem to be quite obvious, but because of the complicated form of the representation,
they may still require some proofs. We will also derive Hunt’s generator formula (Theorem 2.1) in this section.
Proposition 6.38 Let xt be an inhomogeneous Lévy process in G with x0 = e represented by
an extended Lévy triple (b, A, η). Then for a fixed v > 0, the time shifted process x′t = x−1
v xv+t
is represented by the extended Lévy triple (b′ , A′ , η ′ ) given by
b′t = b−1
v bv+t ,
A′ (t) = A(v + t) − A(v),
η ′ (t, ·) = η(v + t, ·) − η(v, ·).
Proof: Because xt is represented by (b, A, η), xt = zt bt , and f (zt ) − Tt f (z; b, A, η) is a
′ ′
′
−1
martingale for any f ∈ Cc∞ (G) under Ftx . Let zt′ = b−1
v zv zv+t bv . Then xt = zt bt . It suffices
′
to show that f (zt′ ) − Tt f (z ′ ; b′ , A′ , η ′ ) is a martingale under Ftx for f ∈ Cc∞ (G). Because
f (zt )−Tt f (z; b, A, η) is a Ftx -martingale, f (zv+t )−f (zv )−Tv,v+t f (z; b, A, η) is a Gt -martingale,
x
. By a simple shift in the time variable, this expression is equal to
where Gt = Fv+t
f (zv+t ) − f (zv ) − Tt f (zv+· ; bv+· , A′ , η ′ ).
197
(6.56)
For a, b, z ∈ G, let f ′ (z) = f (azb). Then ξf ′ (z) = [Ad(b−1 )ξ]f (azb) for ξ ∈ g. Now let
−1
f ′ (z) = f (b−1
v zv zbv ). Then
−1 −1
′
′
[Ad(bv+s )ξj ]f ′ (zv+s ) = [Ad(b−1
v )Ad(bv+s )ξj ]f (bv zv zv+s bv ) = [Ad(bs )ξj ]f (zs ),
[Ad(bv+s )ξj ][Ad(bv+s )ξk ]f ′ (zv+s ) = [Ad(b′s )ξj ][Ad(b′s )ξk ]f (zs′ ),
−1 −1
−1
−1
′ ′
′−1
f ′ (zv+s bv+s xb−1
v+s ) = f (bv zv zv+s bv bv bv+s xbv+s bv ) = f (zs bs xbs ),
′
′
′−1
f ′ (zv+u− bv+u− xb−1
v+u ) = f (zu− bu− xbu ) for u > 0,
Replacing f (z) in (6.56) by f ′ (z) yields
f ′ (zv+t ) − f ′ (zv ) − Tt f ′ (zv+· ; bv+· , A′ , η ′ ) = f (zt′ ) − f (e) − Tt f (z ′ ; b′ , A′ , η ′ ),
which as mentioned earlier is a Gt -martingale. Since this expression as a process is adapted
′
′
′
to Ftz = Ftx ⊂ Gt , it follows that f (zt′ ) − Tt f (z ′ ; b′ , A′ , η ′ ) is a Ftx -martingale. 2
It is easy to see that two independent inhomogeneous Lévy processes can be joined one
after another to form a new inhomogeneous Lévy process. This is formalized as follows.
Proposition 6.39 Let x1t and x2t be two independent rcll processes in G with x10 = x20 = e,
represented by extended Lévy triples (b1 , A1 , η 1 ) and (b2 , A2 , η 2 ) respectively. Fix s > 0 and
let
xt = x1t for t ≤ s and xt = x1s x2t−s for t ≥ s.
Then xt is a rcll process in G represented by the extended Lévy triple (b, A, η), where bt = b1t ,
A(t) = A1 (t) and η(t, ·) = η 1 (t, ·) for t ≤ s, and
bt = b1s b2t−s ,
A(t) = A1 (s) + A2 (t − s),
η(t, ·) = η 1 (s, ·) + η 2 (t − s, ·)
for t ≥ s. Moreover, if x1t and x2t are inhomogeneous Lévy processes, then so is xt .
Proof Let xt = zt bt . We want to show that Mt f (z; b, A, η) = f (zt ) − Tt f (z; b, A, η) is a
martingale under Ftz for any f ∈ Cc∞ (X). This clearly holds for t ≤ s.
Let x2t = zt2 b2t . Then Mt f (z 2 ; b2 , A2 , η 2 ) is a martingale under the natural filtration of zt2 .
For t ≥ s,
2
2
(b1s )−1 )(b1s b2t−s ).
b2t−s = (zs1 b1s zt−s
xt = x1s x2t−s = zs1 b1s zt−s
2
(b1s )−1 for t ≥ s. Let f ′ (z) = f (zs1 b1s z(b1s )−1 ).
Because b1s b2t−s = bt , it follows that zt = zs1 b1s zt−s
Then f ′ (e) = f (zs1 ) = f (zs ), and for t ≥ s
f (zt ) = f (zs ) + [f (zt ) − f (zs )]
= Ms f (z; b, A, η) + Ts f (z; b, A, η) + Mt−s f ′ (z 2 ; b2 , A2 , η 2 ) + Tt−s f ′ (z 2 ; b2 , A2 , η 2 ) − f ′ (e).
198
Let Mt = Mt f (z 1 ; b1 , A1 , η 1 ) for t ≤ s and
Mt = Ms f (z 1 ; b1 , A1 , η 1 ) + Mt−s f ′ (z 2 ; b2 , A2 , η 2 ) − f ′ (e)
for t ≥ s. By the independence of the two processes zt1 and zt2 , it is easy to show that Mt is
a martingale under Ftx . Because for t ≥ s,
Tt f (z; b, A, η) = Ts f (z; b, A, η)) + Tt−s f ′ (z 2 ; b2 , A2 , η 2 ),
it follows that f (zt ) − Tt f (z; b, A, η) = Mt . This shows that xt is represented by the extended
Lévy triple (b, A, η). The rest of the proposition is trivial to prove. 2
We note that the proofs of the above two propositions do not depend on Theorems 6.9
and 6.11. Two processes are said to have the same distribution on a time interval if they
have the same distribution when regarded as processes restricted to this time interval.
Proposition 6.40 Let x1t and x2t be two inhomogeneous Lévy processes in G with x10 = x20 =
e, represented by extended Lévy triples (b1 , A1 , η 1 ) and (b2 , A2 , η 2 ) respectively. Fix a > 0.
Then the two processes x1t and x2t have the same distribution on [0, a] if and only if the two
triples (b1 , A1 , η 1 ) and (b2 , A2 , η 2 ) are equal for t ∈ [0, a]. This conclusion holds also when
[0, a] is replaced by [0, a).
Proof: It will be clear from the proof of Theorem 6.11 that the the distribution of an
inhomogeneous Lévy process on [0, a] determines the associated triple on [0, a], and vise
versa. However, we will present a proof based on Proposition 6.39 and the uniqueness in
Theorem 6.11. Suppose the two triples are equal on [0, a]. By Proposition 6.39, x2t may
be modified for t > a to become an inhomogeneous Lévy process represented by (b1 , A1 , η 1 ).
This shows that x11 and x2t have the same distribution on [0, a]. Conversely, if x11 and x2t
have the same distribution on [0, a], then x2t may be modified for t > a to become an
inhomogeneous Lévy process that is equal to x1t in distribution. Then the associated triple
must be equal to (b1 , A1 , η 1 ), and its restriction to [0, a] must agree with (b2 , A2 , η 2 ). This
shows that (b1 , A1 , η 1 ) agrees with (b2 , A2 , η 2 ) on [0, a]. The conclusion for [0, a) follows
immediately because [0, a) = ∪n>1 [0, a − 1/n]. 2
Let xt be an inhomogeneous Lévy process and let b̃t be an extended drift. It is easy to
see that xt b̃t has independent increments, and so is an inhomogeneous Lévy process. The
next proposition shows how the representing triple is affected by adding the extended drift
b̃t . Note that b̃t xt may not have independent increments.
199
Proposition 6.41 Let xt be an inhomogeneous Lévy process in G with x0 = e, represented by
an extended Lévy triple (b, A, η), and let b̃t be an extended drift. Then x̂t = xt b̃t is represented
by the extended Lévy triple (b̂, Â, η̂) for some b̂t , where
−1 ′
dÂ(t) = [Ad(b̃−1
t )]dA(t)[Ad(b̃t )]
and η̂(dt, ·) = (lb̃−1 ◦ rb̃t )η(dt, ·).
t−
(6.57)
In particular, η̂ c (dt, ·) = cb̃−1
η c (dt, ·).
t
−1 −1
Proof: The relation between η̂ and η given in (6.57) follows from x̂−1
t− x̂t = b̃t− xt− xt b̃t . To
prove the relation between  and A, let xt = zt bt and x̂t = ẑt b̂t . Then for any f ∈ Cc∞ (G),
f (zt ) − Tt f (z; b, A, η) is a martingale, and so is
f (ẑt ) − Tt f (ẑ; b̂, Â, η̂) = f (ẑt ) − St f (ẑ; b̄1 , Ā1 , η̄ 1 , ν̄ 1 ),
(6.58)
where (b̄1 , Ā1 , η̄ 1 , ν̄ 1 ) is (b̄, Ā, η̄, ν̄) in Lemma 6.19 with (b, A, η) being (b̂, Â, η̂).
Let ut = bt b̃t b̂−1
t . Then ẑt = zt ut and ut is an extended drift of finite variation. By
Lemma 6.25 (a),
−1
f (ẑt ) − Tt f (ẑ; u b, A, η) −
∫
t
∑
0
= f (ẑt ) − St f (ẑ; b̄2 , Ā2 , η̄ 2 , ν̄ 2 ) −
∫
i
t
0
ξi f (ẑs )dui (s) −
∑
∑∫
s≤t G
ξi f (ẑs )dui (s) −
i
(· · ·)νs (dx)
∑∫
s≤t G
(· · ·)νs (dx)
(6.59)
is a martingale, where (b̄1 , Ā2 , η̄ 2 , ν̄ 2 ) is (b̄, Ā, η̄, ν̄) in Lemma 6.19 with (b, A, η) being
(u−1 b, A, η) = (b̂b̃−1 , A, η).
Comparing (6.58) and (6.59), by the uniqueness of the quadruple in Lemma 6.31, we obtain
−1 ′
Ā1 = Ā2 . This means that [Ad(b̂t )]dÂ(t)[Ad(b̂t )]′ = [Ad(b̂t b̃−1
t )]dA(t)[Ad(b̂t b̃t )] , which
−1 ′
implies dÂ(t) = [Ad(b̃−1
t )]dA(t)[Ad(b̃t )] . 2
Remark 6.42 Propositions 6.38, 6.39, 6.40 and 6.41 hold also for the representation in the
reduced form when it is available, and this is clear from their proofs.
The following lemma will be needed in the proof of Theorem 2.1.
Lemma 6.43 Let xt be an inhomogeneous Lévy process in G with x0 = e represented by a
Lévy triple (b, A, η) as in Theorem 6.9. Then xt is a Lévy process if and only if
bt = exp(tξ0 ),
Ajk (t) = ajk t,
η(t, ·) = tη1
(6.60)
for some ξ0 ∈ g, a covariance matrix ajk and a Lévy measure η1 on G. Moreover, the domain
D(L) of the generator L of xt as a Feller process contains Cc∞ (G), and for f ∈ Cc∞ (G), Lf
is given in (2.3), where ξ0 , ajk , η are precisely ξ0 , ajk , η1 above.
200
Proof: Let xt be a Lévy process, and as an inhomogeneous Lévy process it is represented by
a Lévy triple (b, A, η), that is, the expression (6.23) is a martingale (with η c = η and νu = δe ).
Fix s > 0. By Proposition 6.38, the time shifted process xst = x−1
s xs+t is represented by the
s
s s
s
−1
s
shifted Lévy triple (b , A , η ), where bt = bs bs+t , A (t) = A(s + t) − A(s) and η s (t, ·) =
η(s+t, ·)−η(s, ·). Because xt is time homogeneous, the two processes xt and xst have the same
distribution. By the uniqueness of the Lévy triple in Theorem 6.9, (bs , As , η s ) = (b, A, η),
and hence bs+t = bs bt , A(s + t) = A(s) + A(t) and η(s + t, ·) = η(s, ·) + η(t, ·). This implies
(6.60).
Now assume (b, A, η) is a Lévy triple in the form of (6.60), and let xt be an inhomogeneous
Lévy process represented by (b, A, η). For s > 0, by Proposition 6.38 and (6.60), the time
shifted process xst is represented by by the same triple (b, A, η), and hence by the unique
distribution in Theorem 6.9, xst and xt have the same distribution as processes. This shows
that xt is a Lévy process.
Taking expectation of Mt f in (6.40), using Proposition 6.15 (a), yields E[f (xt )] = f (e) +
E[ 0 Lf (xs )ds] for f ∈ Cc∞ (G), where Lf is given in (2.3) with (ξ0 , ajk , η) = (ξ0 , ajk , η1 ), but
f is not assumed to be in D(L). Then Lf (e) = limt→0 {E[f (xt )] − f (e)}/t. Replacing f by
f ◦ lg for any g ∈ G shows that Lf (g) = limt→0 {E[f (gxt )] − f (g)}/t. This may be written
as Lf (g) = limt→0 [Pt f (g) − f (g)]/t, where Pt f (g) = E[f (gxt )]. It remains to show that
f ∈ D(L) for f ∈ Cc∞ (G).
∫t
For λ > 0, let Rλ : C0 (G) → C0 (G) be the resolvent operator defined by Rλ f (g) =
∫
E[ 0 e−λt f (gxt )dt] = 0∞ e−λt Pt f (g)dt for g ∈ G and f ∈ C0 (G). It is well known that the
range Rλ C0 (G) of Rλ is D(L) (see for example Proposition 1.4 in [79, chapter VII], where
Rλ is denoted as Uλ ). We just have to show that Cc∞ (G) ⊂ Rλ C0 (G). Let f ∈ Cc∞ (G).
∫
Because Pt f (g) − f (g) = E[ 0t Lf (gxs )ds], (1/t)(Pt f − f ) is bounded on G and converges to
f pointwise as t → 0. By the semigroup property Ps+t = Ps Pt ,
∫∞
1
Rλ (λf − Lf ) = λRλ f − lim Rλ (Pt f − f )
t→0 t
∫ ∞
1 λt ∫ ∞ −λs
= λRλ f − lim [e
e Ps f ds −
e−λs Ps f ds]
t→0 t
t
0
e−λt ∫ t −λs
eλt − 1 ∫ ∞ −λs
e Ps f ds + lim
e Ps f ds = f. 2
= λRλ f − lim
t→0 t
t→0
t
0
0
Proof of Theorem 2.1 Theorem 2.1 is stated for a Feller transition semigroup on a Lie
group G, but equivalently it may also be stated in terms of a Lévy process. We want to show
that if xt is a Lévy process in G, then as a Feller process, the domain D(L) of its generator
L contains Cc∞ (G), and for f ∈ Cc∞ (G), Lf is given by (2.3), and conversely, given such an
operator L, there is a Lévy process xt in G with x0 = e, unique in distribution, with L as
generator.
201
If xt is a Lévy process in G, then it is also an inhomogeneous Lévy process. By Lemma 6.43, it is represented by a Lévy triple given by (6.60), and its generator L, restricted to
Cc∞ (G), is given by (2.3), with D(L) ⊃ Cc∞ (G). Conversely, given such an operator L on
Cc∞ (G) with (ξ0 , aij , η) as in (2.3), writing η1 for η and letting (b, A, η) be the Lévy triple
given by (6.60), by Theorem 6.9, there is an inhomogeneous Lévy process represented by
(b, A, η). By Lemma 6.43, it is a Lévy process with L as its generator restricted on Cc∞ (G).
Suppose there is another Lévy process x′t whose generator restricted to Cc∞ (G) is the same
as L on Cc∞ (G). Let x′t be represented by the Lévy triple (b′ , A′ , η ′ ) as an inhomogeneous
Lévy process. By Lemma 6.43,
b′t = exp(tξ0′ ),
A′jk (t) = a′jk t,
η ′ (t, ·) = tη1′
for some ξ0′ ∈ g, a covariance matrix a′jk and a Lévy measure η1′ on G. Moreover, for f ∈
Cc∞ (G), Lf is given in (2.3), where ξ0 , ajk , η are precisely ξ0′ , a′jk , η1′ . By Proposition 2.4, η1′ =
η1 , a′ij = aij (under the same basis {ξi } of g), and hence ξ0′ = ξ0 (under the same coordinate
functions ϕi ). This shows that x′t is represented by the same Lévy triple (b, A, η), so by
Theorem 6.9, must have the same distribution. Theorem 2.1 has now been proved, except
that this theorem is stated for general coordinate functions ϕi , but by using Theorem 6.9, the
exponential coordinate functions are assumed. To see that Theorem 2.1 holds under general
coordinate functions, one just notes that a change in coordinate functions corresponds to a
suitable change in the drift vector ξ0 . 2
6.11
Adding jumps
Let xt be an inhomogeneous Lévy process in G with x0 = e, represented an extended
Lévy triple (b, A, η). We will consider adding jumps determined by an extended Lévy measure
function η ′ (t, ·). We will assume η ′ is finite in the sense that η ′ (t, G) < ∞ for any t ∈ R+ .
Because η ′ is finite, it has at most finitely many discontinuous time points in any finite
time interval. Let N ′ be an extended Poisson random measure on R+ × G with intensity
measure η ′ as defined in Appendix 6, independent of the process xt . For any t ∈ R+ , N ′ has
at most finitely many points (u, σ) in [0, t] × G almost surely. We add successively all σ as
jumps to the process xt at times u, according to (6.26), to obtain a new process x′t , which
will be called the process obtained from xt after adding jumps from N ′ or from η ′ .
It is obvious that the jump intensity measure of x′t is η + η ′ . It is also natural to expect
that x′t has the same covariance matrix function A(t) as xt , but to prove this formally requires
some effort because of the complicated form of the representation when the drift bt is not of
finite variation or when the representation is not in the reduced form.
202
Proposition 6.44 (a) Let xt be an inhomogeneous Lévy process in G with x0 = e, represented by an extended Lévy triple (b, A, η), and let η ′ be a finite extended Lévy measure function.
Assume η(t, ·) and η ′ (t, ·) have no common discontinuity in t. Then the process x′t obtained
from xt after adding jumps from η ′ , as defined above, is an inhomogeneous Lévy process in
G represented by the extended Lévy triple (b̃, A, η + η ′ ) for some extended drift b̃.
(b) If bt has a finite variation with components bi (t), then b̃t also has a finite variation with
∫
components b̃i (t) = bi (t) + G ϕi (y)η ′c (t, dy).
(c) If η c has a finite first moment, and xt is represented by (br , A, η) in the reduced form (see
Theorem 6.35), then x′t is an inhomogeneous Lévy process in G represented by the extended
Lévy triple (b̃r , A, η + η ′ ) in the reduced form, where b̃rt is obtained from brt after adding local
means h′u of νu′ from η ′ as jumps.
′
Proof: Let Ftx and FtN be respectively the natural filtrations of xt and N . For s < t, x′−1
s xt
′
depends only on x−1
u xv for v > u ≥ s and the restriction of N on [s, t] × G. It follows that
′
x
N
′
x′−1
s xt is independent of Fs and Fs . This shows that xt has independent increments and
hence is an inhomogeneous Lévy process in G.
Let x′t be represented by an extended Lévy triple (b2 , A2 , η 2 ). Because N ′ is independent
of the jump counting measure N of xt , it follows that almost surely, N and N ′ have no
common point. This implies that N + N ′ is the jump counting measure of x′t , and hence
η + η ′ is the jump intensity measure of xt . This shows η 2 = η + η ′ . To prove (a), it remains
to show A2 = A.
Recall Tt f = Tt f (z; b, A, η), and St f = St f (x; b, A, η) for bt with a finite variation, are
defined by (6.22) and (6.38).
Lemma 6.45 Let (b, A, η) be an extended Lévy triple, and let xt = zt bt be a rcll process in G
with x0 = e, represented by (b, A, η). Let 0 < u1 < u2 < · · · < um be continuous time points
of η, and σi , hi ∈ G for 1 ≤ i ≤ m. Let x′t be the process obtained from xt when the jumps
σi at times ui are successively added according to (6.26), and let b′t be the extended drift in
G obtained from bt when jumps hi at times ui are successively added. Then with x′t = zt′ b′t ,
for f ∈ Cc∞ (G),
f (zt′ )
′
′
− Tt f (z ; b , A, η) −
m
∑
1[t≥ui ] [f (zu′ i ) − f (zu′ i − )]
(6.61)
i=1
is a martingale under the natural filtration Ftx of xt . Moreover, if bt has a finite variation,
then
f (x′t ) − S̃t f (x′ ; b′ , A, η) −
m
∑
1[t≥ui ] [f (x′ui − σi ) − f (x′ui − )]
i=1
is a martingale under Ftx .
203
(6.62)
Let x1t be the process obtained from xt when the jump σ = σ1 at time u = u1 is
added, and let b1t be the extended drift obtained from bt when the jump h = h1 at time
−1 −1
u is added. Let x1t = zt1 b1t . We have, for t ≥ u, x1t = xu σx−1
u xt = zu bu σbu zu zt bt =
−1
−1 −1 1
1
−1
−1
−1 −1
zu bu σb−1
u zu zt bu h bu bt . It follows that zt = zu (bu σbu )(zu zt )(bu h bu ) for t ≥ u. In
1
1
−1 1
−1
particular, zu1 = zu (bu σh−1 b−1
u ) = zu− (bu− σh (bu− ) ).
−1
−1 −1
For f ∈ Cc∞ (G), let f ′ (z) = f (zu (bu σb−1
u )zu z(bu h bu )). Then for t ≥ u and ξ ∈ g,
1
1
1
[Ad(bt )ξ]f ′ (zt ) = [Ad(bu hb−1
u bt )ξ]f (zt ) = [Ad(bt )ξ]f (zt ),
for τ ∈ G,
1 1
1 −1
−1 −1
−1 −1
−1
f ′ (zt bt τ b−1
t ) = f (zu (bu σbu )zu zt (bt τ bt )(bu h bu )) = f (zt bt τ (bt ) ),
and
−1
−1 −1
−1 −1
−1 −1
1 1
−1 1 −1
f ′ (zt− bt− τ h−1
t bt− ) = f (zu (bu σbu )zu zt− (bt− τ ht bt− )(bu h bu )) = f (zt− bt− τ ht (bt− ) ).
It then follows that for t ≥ u, writing Tu,t f for Tt f − Tu f ,
Tu,t f ′ (z; b, A, η) = Tu,t f (z 1 ; b1 , A, η).
Now for t ≥ u,
1
1
f (zt1 ) = f (zu−
) + [f (zu1 ) − f (zu−
)] + [f (zt1 ) − f (zu1 )]
1
= f (zu ) + [f (zu1 ) − f (zu−
)] + [f ′ (zt ) − f ′ (zu )]
1
= Tu f (z; b, A, η) + Mu + [f (zu1 ) − f (zu−
)] + Tu,t f ′ (z; b, A, η) + Mt′ − Mu′
(Mt and Mt′ are martingales)
1
= Tt f (z 1 ; b1 , A, η) + [f (zu1 ) − f (zu−
)] + Mt1 ,
where Mt1 = Mt for t < u and Mt1 = Mu + Mt′ − Mu′ for t ≥ u. It is easy to see that Mt1 is a
martingale. This shows that the expression in (6.61) when m = 1,
1
)],
f (zt1 ) − Tt f (z 1 ; b1 , A, η) − 1[t≥u] [f (zu1 ) − f (zu−
is a martingale.
The above computation may be repeated with f (zt )−Tt f (z; b, A, η) replaced by the above
expression, and (z, b, u1 , σ1 , h1 ) replaced by (z 1 , b1 , u2 , σ2 , h2 ) to show that the expression in
(6.61) when m = 2 is a martingale. Continuing in this way shows that (6.61) is a martingale
for any m.
204
When bt has a finite variation, a much simpler computation can be used to prove that
(6.62) is a martingale. We will only prove for m = 1. First note that because S̃t f (b, A, η)
depends only on the components of bt , S̃t f (x; b, A, η) = S̃t f (x; b1 , A, η). For t ≥ u,
f (x1t ) = f (x1u− ) + [f (x1u ) − f (x1u− )] + [f (x1t ) − f (x1u )]
= f (xu ) + [f (x1u− σ) − f (x1u− )] + [f ′ (xt ) − f ′ (xu )] (f ′ (x) = f (xu σx−1
u x))
= S̃u f (x; b, A, η) + Mu + [f (x1u− σ) − f (x1u− )] + S̃u,t f ′ (x; b, A, η) + Mt′ − Mu′ ,
where Mt and Mt′ are martingales. It is easy to see that S̃u,t f ′ (x; b, A, η) = S̃u,t f (x1 ; b, A, η).
It follows that (6.62) with m = 1 is a martingale. A general m can be proved as for (6.61).
Lemma 6.45 is proved.
Let νu be the jump measures of η as defined in (6.14). Note that νu = δe if u is not a
discontinuous time of η. Let νu′ be the jump measures of η ′ .
Lemma 6.46 Let xt be a rcll process in G with x0 = e, represented by an extended Lévy triple
(b, A, η). Let η ′ be a finite extended Lévy measure function. Assume η and η ′ have no common
discontinuous time. Let x′t be the process obtained from xt after adding jumps from η ′ , and
let b′t be the extended drift obtained from bt after adding the local means h′u of νu′ as jumps
to bt at all discontinuous times u of η ′ . Then with x′t = zt′ b′t , for f ∈ Cc∞ (G),
f (zt′ )
′
− Tt f (z ; b , A, η) −
∫
∑
−
′
G
u≤t, u∈J ′
∫
t
∫
0
G
′
′c
[f (zu′ b′u σb′−1
u ) − f (zu )]η (du, dσ)
′
′−1
′
′
[f (zu−
b′u− σh′−1
u bu− ) − f (zu− )]νu (dσ)
(6.63)
is a martingale under the natural filtration Ft′ of x′t , where J ′ is the set of the discontinuous
times of η ′ . Moreover, if bt has a finite variation, then
f (x′t ) − S̃t f (x′ ; b, A, η) −
−
∫
∑
u≤t u∈J ′
G
∫
t
0
∫
G
[f (x′u σ) − f (x′u )]η ′c (du, dσ)
[f (x′u− σ) − f (x′u− )]νu′ (dσ)
(6.64)
is a martingale under Ft′ .
To prove Lemma 6.46, let N ′ be an extended Poisson random measure on R+ × G with
intensity measure η ′ , independent of the process xt . Then for any t ∈ R+ , almost surely, N ′
has finitely many points (u, σ) with u ≤ t. These points may be considered as non-random
for the process xt with u as continuous times of xt . We apply Lemma 6.45 to successively
add jumps σ at times u to xt to obtain x′t , and successively add jumps h′u at times u to bt to
obtain b′t , noting h′u = e if u is not a discontinuous time of η ′ . Then with x′t = zt′ b′t ,
f (zt′ ) − Tt f (z ′ ; b′ , A, η) −
∫
0
t
∫
G
′c
′
′
b′u− σb′−1
[f (zu−
u− ) − f (zu− )]N (du, dσ) −
205
∑
u≤t, u∈J ′
′
)]
[f (zu′ ) − f (zu−
is a martingale under Ftx when N ′ is regarded as non-random, and hence is also a martingale
under Ft′ when N ′ is regarded as random and independent of the process xt . Because
∫
∫
t
0
and
∑
u≤t, t∈J ′
G
′
′
′c
[f (zu−
b′u σb′−1
u ) − f (zu− )]Ñ (du, dσ)
′
[f (zu′ ) − f (zu−
)] −
∑
u≤t u∈J ′
′
′
E[f (zu′ ) − f (zu−
) | Fu−
]
are martingales, where Ñ ′c = N ′c − η ′c is the compensated form of N ′c , and because
′
′−1
′
′
b′u− σu h′−1
] = E[f (zu−
E[f (zu′ ) | Fu−
u bu− ) | Fu− ] =
∫
G
′−1 ′
′
b′u− σh′−1
f (zu−
u bu− )νu (dσ),
it follows that (6.63) is a martingale, noting
∫
t
0
∫
G
′
[f (zu−
b′u− σb′−1
u− )
′
f (zu−
)]η ′c (du, dσ)
−
∫
t
∫
=
0
G
′c
′
[f (zu′ b′u σb′−1
u ) − f (zu )]η (du, dσ)
because η ′c (t, ·) is continuous in t.
If bt has a finite variation, by the second conclusion of Lemma 6.45,
f (x′t )
′
− S̃t f (x ; b, A, η) −
∫
0
t
∫
G
[f (x′u− σ) − f (x′u− )]N ′c (du, dσ) −
∑
u≤t, u∈J ′
[f (x′u ) − f (x′u− )]
is an {Ftx }-martingale when N ′ is regarded as non-random, and hence is an {Ft′ }-martingale
when N ′ is regarded as random and independent of the process xt . As in the preceding
proof, this implies that the expression in (6.64) is a martingale. Lemma 6.46 is proved.
We now return to the proof of Proposition 6.44. Recall St f defined by (6.29). Note
that the expression (6.63) may be written as f (zt′ ) − St f (z ′ ; b1 , Ā′ , η̄ ′ , ν̄ ′ ) for a suitably chosen
drift b1 of finite variation, where (Ā′ , η̄ ′ , ν̄ ′ ) are just (Ā, η̄, ν̄) in Lemma 6.19 but with (b, A, η)
replaced by (b′ , A, η + η ′ ). Thus, zt′ has the (b1 , Ā′ , η̄ ′ , ν̄ ′ )-martingale property.
Because x′t is represented by (b2 , A2 , η 2 ). Then with x′t = zt2 b2t , zt2 has the (b̄2 , Ā2 , η̄ 2 , ν̄ 2 )martingale property, where (b̄2 , Ā2 , η̄ 2 , ν̄ 2 ) are just (b̄, Ā, η̄, ν̄) in Lemma 6.19 but with (b, A, η)
replaced by (b2 , A2 , η 2 ). Because zt2 = zt′ ut with ut = b′t (b2t )−1 , by Proposition 6.26,
′
−1 ′
−1
′
′ ′
−1 ′
dĀ2 (t) = [Ad(u−1
t )]dĀ (t)[Ad(ut )] = [Ad(ut )][Ad(bt )]dA(t)[Ad(bt )] [Ad(ut )]
= [Ad(b2t )]dA(t)[Ad(b2t )]′ .
Because dĀ2 (t) = [Ad(b2t )]dA2 (t)[Ad(b2t )]′ , A2 (t) = A(t). This proves (a).
To prove (b), just note that (6.64) may be written as f (x′t ) − St f (x′ ; b̃, A, η + η ′ ).
To prove (c), note that the proof of Lemma 6.45, and hence its conclusions, are still valid
when xt is represented by an extended Lévy triple (b, A, η) in the reduced form, where η is
finite. This means that the η c -integral in Tt f ,
∫
0
t
∫
G
{f (zs bs yb−1
s ) − f (zs ) −
∑
ϕ(y)[Ad(bs )ξi ]f (zs )}η c (ds, dy),
i
206
∫ ∫
c
is to be replaced by 0t G [f (zs bs yb−1
s ) − f (zs )]η (ds, dy). Then it can be shown, as in the
proof of the second part of Lemma 6.46, that (6.51) is a martingale with η replaced by η + η ′
and bt replaced by b̃r . 2
207
208
Chapter 7
Proofs of main results
The chapter is mainly devoted to the proofs of Theorems 6.9 and 6.11.
7.1
Matrix-valued measure functions
As before, let G be a Lie group with identity element e, let {ξ1 , . . . , ξd } be a basis of the
Lie algebra g of G and let ϕ1 , . . . , ϕd be the associated exponential coordinate functions (see
§2.1).
Let xt be an inhomogeneous Lévy process in G with x0 = e. The notion of measure
functions introduced in §6.2 is now extended to matrix-valued measures. A family of d × d
symmetric matrix valued functions A(t, B) = {Ajk (t, B)}j,k=1,2,...,d , for t ∈ R+ and B ∈ B(G),
is called a matrix-valued measure function on G if Ajk (t, ·) is a finite signed measure on G,
A(0, B) and A(t, B) − A(s, B) are nonnegative definite for s < t, and A(t, B) → A(s, B) as
t ↓ s for any s ≥ 0.
The trace of a matrix-valued measure function A(t, ·), q(t, ·) = Tr[A(t, ·)], is a finite
measure function such that for s < t,
|Ajk (t, ·) − Ajk (s, ·)| ≤ q(t, ·) − q(s, ·) = q((s, t] × ·).
(7.1)
As in §6.2, let J = {u1 , u2 , u3 , . . .} be the set of fixed jump times of the process xt (the
set may be finite or empty, and may not be ordered in magnitude), let
∆n : 0 = tn0 < tn1 < tn2 < · · · < tnk ↑ ∞ (as k ↑ ∞)
be a sequence of partitions of R+ with mesh ∥∆n ∥ → 0 as n → ∞, such that ∆n contains J
as n → ∞ in the sense that any u ∈ J is in ∆n when n is sufficiently large.
For any integer m > 0, let Jm = {u1 , u2 , . . . , um }. When J has only a finite number k of
elements, then Jm = Jk for m > k. Although not necessary, to make things easier, we will
assume Jm ⊂ ∆m for any m, and ∆n ⊂ ∆n+1 for any n.
209
Let
xni = x−1
tn i−1 xtni
(7.2)
with distribution µni , and let
ηn (t, ·) =
∑
µni
(setting ηn (0, ·) = 0)
(7.3)
tni ≤t
as defined in (6.16).
By Proposition 6.7, for any t ∈ R+ and f ∈ Cb (G) vanishing in a neighborhood of e,
ηn (t, f ) → η(t, f ) as n → ∞, where η(t, ·) is the jump intensity measure of the process xt
defined by (6.13). Moreover, η(t, U c ) < ∞ for any neighborhood U of e.
Let An (t, ·) be the matrix-valued measure functions on G defined by
Anjk (t, B) =
∑
where bni = e
j
µni (ϕj )ξj
∑ ∫
tni ≤t B
[ϕj (x) − ϕj (bni )][ϕk (x) − ϕk (bni )]µni (dx),
(7.4)
is the local mean of xni , setting An (0, ·) = 0. Its trace is
n
q (t, B) =
∑ ∫
tni ≤t B
∥ϕ(x) − ϕ(bni )∥2 µni (dx).
(7.5)
Recall ϕ(x) = (ϕ1 (x), . . . , ϕd (x)) and ∥ · ∥ is the Euclidean norm on Rd .
Let νt be defined by (6.14) with local mean ht . Then νt and ht are respectively the
distribution and local mean of x−1
t− xt , which are nontrivial only when t ∈ J. We will see that
η(t, ·) is an extended Lévy measure function. This fact is now proved below under an extra
assumption.
Lemma 7.1 Assume q n (t, G) is bounded in n for each t > 0. Then the jump intensity
measure η(t, ·) is an extended Lévy measure function, and hence its continuous part η c (t, ·)
is a Lévy measure function.
Proof Let cq be a constant such that q n (t, G) ≤ cq for all n. Any u ∈ J is contained in
(tn i−1 , tni ] for some i = in and νu (∥ϕ − ϕ(hu )∥2 ) = limn→∞ µni (∥ϕ − ϕ(bni )∥2 ). It follows
∑
that u≤t,u∈J νu (∥ϕ − ϕ(hu )∥2 ) ≤ cq . It remains to prove the finiteness of η c (t, ∥ϕ∥2 ). As
in §6.2, let ∆m
n be the subset of ∆n consisting of tni such that u ∈ (tn i−1 , tni ] for some
u ∈ {u1 , u2 , . . . , um }. Let
max m ∥ϕ(bni )∥.
(7.6)
rnm =
tni ≤t, tni ̸∈∆n
∥µni (ϕ)∥ for large m, and limn→∞ rnm = supu≤t, u̸∈Jm ∥νu (ϕ)∥ → 0
Then rnm = maxtni ≤t, tni ̸∈∆m
n
as m → ∞. For two neighborhoods U ⊂ V of e,
η c (t, ∥ϕ∥2 1V c ) ≤ η m (t, ∥ϕ∥2 1V c ) ≤ limn→∞
≤ 2 limn→∞
≤ 2cq +
∑
∫
Uc
∫
Uc
∥ϕ(x)∥2 ηnm (t, dx)
(by Lemma 6.8)
∥ϕ(x) − ϕ(bni )∥2 µni (dx) + 2 limn→∞ (rnm )2 ηn (t, U c )
tni ≤t, tni ̸∈Jm
2(limn→∞ (rnm )2 )(limn→∞ ηn (t, U c )).
210
Now letting m → ∞ and then V ↓ {e} shows η c (t, ∥ϕ∥2 ) ≤ 2cq . 2
Now consider the following equi-continuity type condition on q n (t, ·).
(A) For any T > 0 and ε > 0, there is an integers m > 0 such that if n > m and s, t ∈ [0, T ]
with (s, t] ∩ ∆m = ∅, q n (t, G) − q n (s, G) < ε.
We note that the condition (s, t]∩∆m = ∅ means that (s, t] is contained in the interior of
a single interval in the partition ∆m , so it does not contain any point in Jm ⊂ ∆m . Moreover,
q n (t, G) − q n (s, G) = 0 for n ≤ m because ∆n ⊂ ∆m .
Lemma 7.2 Under (A), for any T > 0, q n (T, G) is bounded in n, and q n (T, U c ) ↓ 0 uniformly in n as U ↑ G, where U is a neighborhood of e.
Proof The boundedness of q n (T, G) is derived from (A) in the same way as the boundedness of ηn (T, U c ) in Proposition 6.5 is derived from Lemma 6.6. Because the convergence
ηn (T, U c ) ↓ 0, U ↑ G, is uniform in n, so is q n (T, U c ) ↓ 0. 2
Lemma 7.3 Assume (A). Then there is a matrix valued measure function A(t, ·) on G such
that along a subsequence of n → ∞, An (t, f ) → A(t, f ) for all t ≥ 0 and f ∈ Cb (G).
Moreover, there is a covariance matrix function A(t) = {Ajk (t)} on G such that
∫
Ajk (t, f ) = f (e)Ajk (t) +
+
∑
u≤t, u∈J
∫
G
G
f (x)ϕj (x)ϕk (x)η c (t, dx)
f (x)[ϕj (x) − ϕj (hu )][ϕk (x) − ϕk (hu )]νu (dx).
(7.7)
Note: From the proof of Lemma 7.3, it is clear that any sequence of n → ∞ has a further
subsequence along which An (t, f ) converges to the expression in (7.7) for some covariance
matrix function A(t). We will show in Lemma 7.10 that A(t) is the covariance matrix
function in the triple (b, A, η) that represents the inhomogeneous Lévy process xt . Then by
the uniqueness of the triple in Theorem 6.11, all convergent subsequences of An (t, f ) have the
same limit, and hence, the convergence An (t, f ) → A(t, f ) in Lemma 7.3 holds as n → ∞,
not just along a subsequence.
Proof of Lemma 7.3 Let Λ be a countable dense subset of [0, T ] containing J ∩ [0, T ] and
let H be a countable subset of Cb (G). By the boundedness of q n (T, G) and (7.1), along a
subsequence of n → ∞, An (t, f ) converges for all t ∈ Λ and f ∈ H. Let Kn be an increasing
sequence of compact subsets of G such that Kn ↑ G. For each Kn , there is a countable subset
Hn of Cb (Kn ) that is dense in C(Kn ) under the supremum norm. We will extend functions
in Hn to be functions on G without increasing their supremum norms, and let H = ∪n Hn .
211
By Lemma 7.2, q n (T, U c ) ↓ 0 as U ↑ G uniformly in n, where U is a relatively compact
neighborhood of e, it follows that An (t, f ) converges for any f ∈ Cb (G) along a subsequence
of n → ∞. In the rest of this proof, the convergence as n → ∞ will always mean along this
subsequence. By (A) and (7.1), An (t, f ) converges to some limit, denoted as A(t, f ), for any
t ∈ [0, T ] and f ∈ Cb (G), and the convergence is uniform for t ∈ [0, T ] and for f bounded
by a fixed constant. Because An (t, f ) is right continuous with left limits in t, so is A(t, f ),
and hence A(t, ·) is a matrix-valued measure function. Moreover, the jumps of A(t, f ) are
∫
G
f (x)[ϕj (x) − ϕj (ht )][ϕk (x) − ϕk (ht )]νt (dx)
at t ∈ J.
∑
Let the sum tni ≤t in (7.4), which defines An (t, ·), be broken into two partial sums:
∑
∑
and tni ≤t, tni ∈∆m
, where ∆m
tni ≤t, tni ̸∈∆m
n is as in the proof of Lemma 7.1, and write
n
n
An (t, ·) = An,m (t, ·) + B n,m (t, ·),
where An,m (t, ·) =
n → ∞,
∑
and B n,m (t, ·) =
tni ≤t, tni ̸∈∆m
n
n,m
Bjk
(t, f )
→
∫
∑
s≤t, s∈Jm
G
∑
.
tni ≤t, tni ∈∆m
n
f (x)[ϕj (x) − ϕj (ht )][ϕk (x) − ϕk (ht )]νs (dx),
and hence limn→∞ An,m (t, f ) also exists. Note that the sum
jumps of A(s, f ) at s ∈ [0, t] ∩ Jm and it converges to
′
Bjk
(t, f )
=
∑
s≤t, s∈J
∫
G
Then for f ∈ Cb (G), as
∑
s≤t, s∈Jm
(7.8)
in (7.8) contains the
f (x)[ϕj (x) − ϕj (ht )][ϕk (x) − ϕk (ht )]νs (dx)
as m → ∞, and B ′ (t, ·) is a matrix valued measure function. It follows that as m → ∞,
limn→∞ An,m (t, f ) converges to the matrix valued measure function A′ (t, f ) = A(t, f ) −
B ′ (t, f ).
Let ψ ∈ Cc∞ (G) with ψ = 1 near e and 0 ≤ ψ ≤ 1 on G. Because ηn (t, 1 − ψ) is bounded
in n, and limn→∞ rnm → 0 as m → ∞ for rnm defined by (7.6), by Lemma 6.8,
An,m
jk (t, (1 − ψ)f ) =
∫
′
[1 − ψ(x)]f (x)ϕj (x)ϕk (x)η c (t, dx) + rnm
(7.9)
′
→ 0 as m → ∞. Then
with limn→∞ rnm
A′jk (t, (1 − ψ)f ) =
∫
[1 − ψ(x)]f (x)ϕj (x)ϕk (x)η c (t, dx).
Let ψ = ψp ↓ 1{e} with supp(ψp ) ↓ {e} as p ↑ ∞, and define A(t) = limp→∞ A′ (t, ψp ).
Then limp→∞ A′ (t, f ψp ) = f (e)A(t). Because A(t, f ) = A′ (t, f ) + B ′ (t, f ) and A′ (t, f ) =
A′ (t, ψp f ) + A′ (t, (1 − ψp )f )), letting p → ∞ yields (7.7). Because the jumps of A(t, f ) are
212
∑
accounted for by the sum u≤t, u∈J in (7.7), it follows that A(t) is continuous and hence is
a covariance matrix function. 2
Let Y be a smooth manifold equipped with a compatible metric r, and let y n and y be
rcll functions: R+ → Y . Assume for any t > 0, r(y n (tni ), y(tni )) → 0 as n → ∞ uniformly
for tni ≤ t. Let F (y, b, x) be bounded continuous function on Y × G × G and let Fjk (y, b) be
such a function on Y × G.
Lemma 7.4 Assume the above and (A), and let A(t) be the covariance matrix function in
Lemma 7.3. Then for any t > 0 and neighborhood U of e with η(t, ∂U ) = 0, as n → ∞,
∑ ∫
→
c
tni ≤t U
∫ t∫
Uc
0
F (y n (tn i−1 ), bni , x)µni (dx)
∑
F (y(s), e, x)η c (ds, dx) +
u≤t, u∈J
∫
Uc
F (y(u−), hu , x)νu (dx),
(7.10)
and along the subsequence of n → ∞ in Lemma 7.3,
d ∫
∑ ∑
tni ≤t j,k=1 G
→
∫
d
∑
{
j,k=1
+
Fjk (y n (tn i−1 ), bni )[ϕj (x) − ϕj (bni )][ϕk (x) − ϕk (bni )]µni (dx)
∫
t
0
Fjk (y(s), e)dAjk (s) +
∑
u≤t, u∈J
∫
G
t
∫
0
G
Fjk (y(s), e)ϕj (x)ϕk (x)η c (ds, dx)
Fjk (y(u−), hu )[ϕj (x) − ϕj (hu )][ϕk (x) − ϕk (hu )]νu (dx)}.
(7.11)
Proof Let V be a relatively compact neighborhood of e. By Proposition 6.5, ηn (t, V c ) ↓
0 as V ↑ G uniformly in n. Because of the uniform convergence r(y n (tni ), y(tni )) → 0,
F (y n (tni ), b, x) − F (y(tni ), b, x) → 0 as n → ∞ uniformly for tni ≤ t and for (b, x) in a
compact set. Therefore, it suffices to prove (7.10) with y n and U c replaced by y and U c ∩ V
for an arbitrary relatively compact neighborhood V of e with η(t, ∂V ) = 0. Similarly, because
all hu are contained in a fixed compact set, it suffices to prove (7.11) with y n replaced by y.
We now show that for s < t and f ∈ Cb (G) vanishing in a neighborhood of e, as n → ∞,
∑
µni (f ) → η((s, t) × f ).
(7.12)
s<tn i−1 <tni <t
∑
∑
The sum s<tn i−1 <tni <t µni (f ) differs from ηn (t, f ) − ηn (s, f ) = s<tni ≤t µni (f ) by one or two
terms of the form µni (f ). By Proposition 6.7, ηn (t, f ) − ηn (s, f ) → η(t, f ) − η(s, f ) =
η((s, t] × f ) = η((s, t) × f ) + νt (f ). Using Jn ⊂ ∆n , (7.12) may be derived by noting that
as n → ∞, µni (f ) → νt (f ) if t ∈ J ∩ (tn i−1 , tni ] and µni (f ) → 0 otherwise.
Let ∆′m : 0 = sm0 < sm1 < · · · < smk ↑ ∞ be a sequence of partitions of R+ such that
∥∆′m ∥ → 0 as m → ∞. Let J ′ = {v1 , v2 , . . .} be the union of J and the set of discontinuity
213
′
′
times of y(t), and let Jm
= {v1 , v2 , . . . , vm }, setting Jm
= Jk′ for m > k if J ′ has only k
′
points. We may assume Jm
⊂ ∆′m by suitably choosing ∆′m . In the following computation,
we will write A ≈ B if |A − B| → 0 as m → ∞ uniformly in n.
∑
{
∫
∑
smj ≤t sm j−1 <tn i−1 <tni <smj
∑
≈
{
∑
smj ≤t sm j−1 <tn i−1 <tni <smj
∫
U c ∩V
U c ∩V
F (y(tn i−1 ), bni , x)µni (dx)}
F (y(sm j−1 ), e, x)µni (dx)}
′
(because Jm
⊂ ∆′m , ∀ε > 0, may choose m0 so that ∀m > m0 ,
|F (y(sm j−1 ), e, x) − F (y(tn i−1 ), bni , x)| < ε if sm j−1 < tn i−1 < tni < smj )
∑ ∫
→
c
smj ≤t U ∩V
F (y(sm j−1 ), e, x)η((sm j−1 , smj ), dx)
∑ ∫
≈
(as n → ∞, by (7.12))
∫
smj ≤t (sm j−1 , smj )
U c ∩V
F (y(s), e, x)η c (ds, dx).
On the other hand, for tn i−1 < smj ≤ tni (≤ t), because η(t, ∂U ) = η(t, ∂V ) = 0,
∫
U c ∩V
F (y(tn i−1 ), bni , x)µni (dx) → νsmj (F (y(smj −), hsmj , ·)1U c ∩V )
as n → ∞.
This proves (7.10) when y n and U c are replaced by y and U c ∩ V . As noted earlier, this
proves (7.10) in its original form. The convergence in (7.11) is proved in a similar fashion,
which is even simpler because Fjk does not contain the variable x. 2
Now assume x0 = e. For 0 ≤ t < tn1 , let xn0 = bn0 = zt0 = e, and with i ≥ 1, let

n


 xt
bnt
ztn



= xtni = xn1 xn2 · · · xni , tni ≤ t < tn i+1 , (xni = x−1
tn i−1 xtni )
= bn1 bn2 · · · bni , tni ≤ t < tn i+1 ,
= xnt (bnt )−1 = zn1 zn2 · · · zni , tni ≤ t < tn i+1 ,
(7.13)
n
−1
where zni = bntn i−1 (xni b−1
ni )(btn i−1 ) .
For f ∈ Cc∞ (G), let Mtn f = f (ztn ) = f (e) for 0 ≤ t < tn1 , and for t ≥ tn1 , let
Mtn f = f (ztn ) −
∑ ∫
tni ≤t G
−1
n
n
[f (ztnn i−1 bntn i−1 xb−1
ni (btn i−1 ) ) − f (ztn i−1 )]µni (dx).
(7.14)
Lemma 7.5 Mtn f is a martingale under the natural filtration Ftx of process xt .
Proof The martingale property can be verified directly noting
∫
E[
G
n
−1
x
n
x
f (ztnn i−1 bntn i−1 xb−1
ni (btn i−1 ) )µni (dx) | Ftn i−1 ] = E[f (ztn i−1 zni ) | Ftn i−1 ]
= E[f (ztnni ) | Ftxn i−1 ]. 2
Fix a left invariant Riemannian metric r on G. Such a metric is complete. It is well
known (see for example [53, VIII.§6]) that any subset of a complete Riemannian manifold
214
is relatively compact if and only if it is bounded in the Riemannian metric. Therefore, a
subset of G is relatively compact if and only if it is bounded in r. Consider the following
equi-continuity type condition on bnt .
(B) For any T > 0 and ε > 0, there is an integer m > 0 such that if n > m and s, t ∈ [0, T ]
with (s, t] ∩ ∆m = ∅, then r(bns , bnt ) < ε.
Lemma 7.6 Under (B), for any T > 0, bnt is bounded for 0 ≤ t ≤ T and n ≥ 1. Moreover,
there is a rcll path bt in G with b0 = e such that along a subsequence of n → ∞, which may
be taken to be a further subsequence of the subsequence in Lemma 7.3, bnt → bt uniformly for
−1
0 ≤ t ≤ T . Furthermore, b−1
t− bt = ht (the local mean of xt− xt ) for any t > 0.
Note: As in Note after Lemma 7.3, by the uniqueness of (b, A, η) in Theorem 6.11, the
convergence bnt → bt in Lemma 7.6 holds as n → ∞, not just along a subsequence.
Proof of Lemma 7.6 The boundedness of bnt may be derived from (B) in the same way
as the boundedness of ηn (t, U c ) in Proposition 6.5 is derived from Lemma 6.6.
Because bnt is bounded, there is a subsequence of n → ∞ along which bnt converges to
some bt for t in a countable dense subset of [0, T ] including J. The equi-continuity of bnt in
(B) implies that the convergence holds for all t and is uniform for t ≤ T , and the limit bt is
a rcll path with jump b−1
t− bt = ht at t ∈ J, noting ht = limn→∞ bni for t ∈ (tn i−1 , tni ]. 2
∞
Lemma 7.7 Assume (A) and (B), and let zt = xt b−1
t . Then for f ∈ Cc (G), Mt f given in
(6.23) is a martingale under the natural filtration Ftx of xt .
Proof We need only to show that for any t > 0, Mtn f is bounded in n and Mtn f → Mt f .
We will assume, for the time being, that for any u > 0, νu is moderate as defined after
(6.19). Then µni are moderate for tni ≤ t when n ∑
is large.
Let U be a neighborhood of e such that x = e j ϕj (x)ξj for x ∈ U and η(T, ∂U ) = 0. Let
∑
∑
∑
n
n
tni ≤t be the sum in (7.14) that defines Mt f as f (zt ) −
tni ≤t
tni ≤t . A typical term in
∫
′−1 −1
n
n
′
may be written as G [f (zbxb b ) − f (z)]µ(dx), where z = ztn i−1 , b = btn i−1 , b = bni and
∑
µ = µni . By Taylor expansion of f (zbxb′−1 b−1 ) = f (zb exp( dj=1 ϕj (x)ξj )b′−1 b−1 ) at x = b′ ,
∫
∫G
=
[f (zbxb′−1 b−1 ) − f (z)]µ(dx) =
Uc
[f (zbxb′−1 b−1 ) − f (z)]µ(dx) +
∫
+
U
{
1∑
2
∫
∫
c
∫U
U
[. . .]µ(dx) +
∑
{
[. . .]µ(dx)
U
fj (z, b, b′ )[ϕj (x) − ϕj (b′ )]}µ(dx)
j
fjk (z, b, b′ )[ϕj (x) − ϕj (b′ )][ϕk (x) − ϕk (b′ )]}µ(dx) + r,
j,k
where
fj (z, b, b′ ) =
∑
∂
f (zbe p ϕp (x)ξp b′−1 b−1 ) |x=b′ ,
∂ϕj
215
fjk (z, b, b′ ) =
∑
∂2
f (zbe p ϕp (x)ξp b′−1 b−1 ) |x=b′ ,
∂ϕj ∂ϕk
and the remainder r satisfies |r| ≤ cU µ(∥ϕ − ϕ(b′ )∥2 1U ) with constant cU → 0 as U ↓ {e}.
∫
Because b′ is the local mean of the moderate µ = µni , ϕj (b′ ) = µ(ϕj ) and U [ϕj (x) −
∫
∫
∫
∑
ϕj (b′ )]µ(dx) = G − U c = 0 − U c [ϕj (x) − ϕj (b′ )]µ(dx). This implies that the sum tni ≤t in
(7.14) is equal to
∑ ∫
c
tni ≤t U
{f (zbxb′−1 b−1 ) − f (z) −
fj (z, b, b′ )[ϕj (x) − ϕj (b′ )]}µ(dx)
j
∑ ∫
+
∑
tni ≤t U
{
1∑
2
fjk (z, b, b′ )[ϕj (x) − ϕj (b′ )][ϕk (x) − ϕk (b′ )]}µ(dx) + R,
(7.15)
j,k
∑
where the remainder R satisfies |R| ≤ cU tni ≤t µ(∥ϕ − ϕ(b′ )∥2 1U ) = cU q n (t, U ).
Note that the two sums in (7.15) are bounded in absolute value by cηn (t, U c ) and cq n (t, U )
∑
respectively for some constant c > 0. Therefore, tni ≤t and hence Mtn f are bounded in n.
By (7.13), xntni = xtni . Because ztn = xnt (bnt )−1 , by the uniform convergence bnt → bt ,
r(ztnni , ztni ) = r((bntni )−1 , (btni )−1 ) → 0 as n → ∞ uniformly for tni ≤ t.
The above fact allows us to apply (7.10) in Lemma 7.4 to the first sum in (7.15). Let
∑
Y = G × G, y = (z, b), and F (y, b′ , x) = f (zbxb′−1 b−1 ) − f (z) − j fj (z, b, b′ )[ϕj (x) − ϕj (b′ )].
Define y n (t) and y(t) by setting y n (t) = (ztnni , bntni ) for tni ≤ t < tn i+1 , and y(t) = (zt , bt ).
Note that fj (z, b, e) = [Ad(b)ξj ]f (z). It follows that as n → ∞, the first sum in (7.15)
converges to
∫
∑
u≤t,u∈J
t
∫
Uc
0
∫
U
{f (zs bs xb−1
s ) − f (zs ) −
∑
j
−1
{f (zu− bu− xh−1
u bu− )
c
− f (zu− ) −
ϕj (x)[Ad(bs )ξj ]f (zs )}η c (ds, dx) +
∑
fj (zu− , bu− , hu )[ϕj (x) − ϕj (hu )]}νu (dx). (7.16)
j
Similarly, noting fjk (z, b, e) = [Ad(b)ξj ][Ad(b)ξk ]f (z) and applying (7.11) in Lemma 7.4 to
the second sum in (7.15) show that it converges to
1 ∑∫ t
[Ad(bs )ξj ][Ad(bs )ξk ]f (zs )dAjk (s)
2 j,k 0
1∑
+
2 j,k
+
∫
0
t
∫
U
[Ad(bs )ξj ][Ad(bs )ξk ]f (zs )ϕj (x)ϕk (x)η c (ds, dx)
1∑ ∑
2 j,k u≤t,u∈J
∫
U
fjk (zu− , bu− , hu )[ϕj (x) − ϕj (hu )][ϕk (x) − ϕk (hu )]νu (dx). (7.17)
By a computation similar to the one leading to (7.15), using Taylor’s expansion of
∫
∑
−1 −1
−1
f (zu− bu− xh−1
u≤t,u∈J G [f (zu− bu− xhu bu− ) − f (zu− )]νu (dx) is
u bu− ) at x = hu , one can show
∑
∑ ∑
the sum of u≤t,u∈J (· · ·) in (7.16) and (1/2) j,k u≤t,u∈J (· · ·) in (7.17), plus an error term
that converges to 0 as U ↓ {e}. Letting U ↓ {e} in (7.16) and (7.17) shows that Mtn f → Mt f .
216
For u ∈ J, let au be the νu -integral term in Mt f given by (6.23) and let anu be the
µni -integral term in Mtn f given by (7.14) with u ∈ (tn i−1 , tni ]. Then anu → au as n → ∞.
In the preceding computation, we have assumed that all νu are moderate. When νu is not
moderate, µni with u ∈ (tn i−1 , tni ] may not be moderate even for large n, so the computation
leading to (7.15) done for the µni -integral is not valid, but we may still take limit anu → au
for this term and the result is the same. Since there are only finitely many u ∈ J for which
νu are not moderate, the result holds even when not all νu are moderate. 2
7.2
Proofs of (A) and (B)
Recall ∆n are partitions of R+ with mesh ∥∆n ∥ → 0 as n → ∞, and Jn ⊂ ∆n ⊂ ∆n+1 for
all n. Because q n (t, ·) = q n (tni , ·) and bnt = bntni if tni ≤ t < tn i+1 , to verify (A) and (B), we
may assume s, t in (A) and (B) are contained in [0, T ] ∩ ∆n .
If either (A) or (B) does not hold, then for some ε > 0 and any integer m > 0, there
are n > m and sn , tn ∈ [0, T ] ∩ ∆n with (sn , tn ] ∩ ∆m = ∅ such that either q n (tn , G) −
q n (sn , G) ≥ ε or r(bnsn , bntn ) ≥ ε. Because the jumps of q n (t, G) and bnt for t ∈ (sn , tn ]
become arbitrarily small when m is large, by decreasing tn if necessary, we may also assume
q n (tn , G) − q n (sn , G) ≤ 2ε and r(bnsn , bnt ) ≤ 2ε for sn ≤ t ≤ tn .
Letting m → ∞ yields a subsequence of n → ∞ such that sn and tn in [0, T ] ∩ ∆n
converge to a common limit, (sn , tn ] ∩ Jmn = ∅ with mn ↑ ∞, and either
(i) ε ≤ q n (tn , G) − q n (sn , G) ≤ 2ε and r(bnsn , bnt ) ≤ 2ε for t ∈ [sn , tn ], or
(ii) q n (tn , G) − q n (sn , G) ≤ 2ε, r(bnsn , bntn ) ≥ ε and r(bnsn , bnt ) ≤ 2ε for t ∈ [sn , tn ].
We will derive a contradiction from (i) or (ii). Let εn = supsn <t≤tn [q n (t, G) − q n (t−, G)].
Then εn → 0 as n → ∞. Let γn : [0, ∞) → [sn , ∞) be a strictly increasing continuous
function such that γn ([0, 1]) = [sn , tn ]. Because q n (t, G) − q n (sn , G) is a nondecreasing step
function in t, bounded by 2ε on [sn , tn ] with jump size ≤ εn , γn may be chosen so that
|q n (γn (t), G) − q n (γn (s), G)| ≤ 2ε|t − s| + εn for s, t ∈ [0, 1].
γn
γn
−1
Let xγt n = x−1
sn xγn (t) for t ≤ 1 and xt = xsn xtn for t > 1. Then xt is an inhomogeneous
Lévy process in G starting at e, and on [0, 1], it is just the process xt on [sn , tn ] time changed
by γn . Because sn and tn converge to a common limit, and (sn , tn ] ∩ Jmn = ∅ with mn ↑ ∞,
it follows that xγt n → e as n → ∞ uniformly in t almost surely. Define
γ
γ,n
γ
n,γ
γ
(t, ·), q γ,n (t, ·), ztγ,n , Mtγ,n f,
xγ,n
t , µni , ηn (t, ·), bni , bt , A
for the time changed process xγt n and partition ∆γn = {sni }, where sni = γn−1 (tni ), in the
same way as xnt , µni , ηn (t, ·), bni , bnt , An (t, ·), q n (t, ·), ztn , Mtn f are defined for the process xt
217
and partition ∆n = {tni }. Then for s, t ∈ [0, 1],
|q γ,n (t, G) − q γ,n (s, G)| = |q n (γn (t), G) − q n (γn (s), G)| ≤ 2ε|t − s| + εn .
(7.18)
Because εn → 0, the above means that q γ,n (t, G) are equi-continuous in t for large n.
Note that we now have a process xγt n for each n and from which other objects, such
as ηnγ and Aγ,n , are defined, unlike before when we have a single process xt for all n, but
as xγt n → e uniformly in t, the results established for ηn and An hold for ηnγ and Aγ,n in
simpler forms. For example, because ηnγ (t, ·) = ηn (γn (t), ·) − ηn (sn , ·), by Proposition 6.7,
ηnγ (t, f ) → 0 as n → ∞ for f ∈ Cb (G) vanishing near e. Using (7.18), the proofs of Lemmas
7.3 and 7.4 can be easily modified for Aγ,n , and also simplified, to show that there is a
covariance matrix function Aγ (t) such that Aγ,n (t, f ) → f (e)Aγ (t) for any f ∈ Cb (G), and
under the assumption of Lemma 7.4,
∑ ∫
c
sni ≤t U
F (y n (sn i−1 ), bγni , x)µγni (dx) → 0
(7.19)
and
d
∑ ∑
Fjk (y n (sn i−1 ), bγni )[ϕj (x) − ϕj (bγni )][ϕk (x) − ϕk (bγni )]µγni (dx)
sni ≤t j,k=1
→
d ∫
∑
j,k=1 0
t
Fjk (y(s), e)dAγjk (s).
(7.20)
Note that because Tr[Aγ,n (1, 1)] = q γ,n (1, G) = q n (tn , G) − q n (sn , G), if (i) holds, then
Tr[Aγ (1)] as the limit of [q n (tn , G) − q n (sn , G)] is ≥ ε.
Let D(G) be the space of rcll paths: R+ → G. Then D(G) is a complete separable metric
space under the Skorohod metric (see [22, chapter 3]). Some basic properties of this space
are briefly reviewed in §1.6. A rcll process xt in G may be regarded as a random variable
in D(G). A sequence of rcll processes xnt in G are said to converge weakly to a rcll process
xt in D(G) (under the Skorohod metric) if xn· → x· in distribution as D(G)-valued random
variables. A family of rcll processes in G are called relatively weak compact in D(G) if any
sequence in the family has a subsequence that converges weakly.
We will show that ztγ,n are relatively weak compact in D(G). Let U be a neighborhood of
e. The amount of time it takes for a process xt to make a U c -displacement from a stopping
time σ (under the natural filtration of process xt ) is denoted as τUσ , that is,
c
τUσ = inf{t > 0; x−1
σ xσ+t ∈ U }.
(7.21)
For a sequence of processes xnt in G, let τUσ,n be the U c -displacement time for xnt from σ.
218
Lemma 7.8 A sequence of rcll processes xnt in G are relatively weak compact in D(G) if for
any T > 0 and any neighborhood U of e,
lim sup P (τUσ,n < δ) → 0 as δ → 0,
(7.22)
lim sup P [(xnσ− )−1 xnσ ∈ K c ] → 0 as compact K ↑ G,
(7.23)
n→∞ σ≤T
and
n→∞ σ≤T
where supσ≤T is taken over all stopping times σ ≤ T . Here “compact K ↑ G” means that
K = Kn , each Kn is compact and is contained in the interior of Kn+1 , and ∪n Kn = G.
Proof Let r be a left invariant Riemannian metric on G, which is complete and separable.
As in [22, section 3.6], the measurement of δ-oscillation of a rcll path x in G on [0, T ] is
given by
w′ (x, δ, T ) = inf max
sup r(xt , xs ),
(7.24)
{ti } 1≤i≤n s,t∈[ti−1 , ti )
where the infimum inf {ti } is taken over all partitions 0 = t0 < t1 < · · · < tn−1 < T ≤ tn with
min1≤i≤n (ti − ti−1 ) > δ. By [22, 3.Corollary 7.4], xnt are relatively weak compact in D(G) if
for any T > 0,
lim P (xnt ∈ K c for some t ≤ T ) → 0 as compact K ↑ G
n→∞
(7.25)
and for any ε > 0
lim P [w′ (xn· , δ, T ) ≥ ε] → 0 as δ → 0.
(7.26)
n→∞
For a fixed ε > 0, the successive stopping times 0 = τ0ε < τ1ε < τ2ε < · · · < τiε < · · · when
a rcll process xt makes an ε-displacement are defined inductively by
ε
ε ) > ε}
τiε = inf{t > τi−1
; r(xt , xτi−1
ε
for i = 1, 2, 3, . . ., setting inf ∅ = ∞ and τiε = ∞ if τi−1
= ∞. Let τiε,n be the ε-displacement
times of the process xnt . It is easy to see that
ε
[min{τi+1
− τiε ; τiε < T } > δ]
i≥0
implies
[w′ (x, δ, T ) ≤ 2ε],
(7.27)
ε
− τiε ; τiε < T } ≤ δ]. Thus, if
and hence P [w′ (x· , δ, T ) > 2ε] ≤ P [mini≥0 {τi+1
∀T > 0 and ε > 0,
ε,n
lim P [min{τi+1
− τiε,n ; τiε,n < T } < δ] → 0 as δ → 0,
n→∞
i≥0
(7.28)
then (7.26) holds for any T > 0 and ε > 0.
ε,n
Let F (t) = supn supi≥0 P (τi+1
− τiε,n < t, τiε,n < T ). By [22, 3.Lemma 8.2],
F (δ) ≤
ε,n
sup P [min{τi+1
i≥0
n
−
τiε,n ;
τiε,n
< T } < δ] ≤ LF (δ) +
219
∫
∞
0
e−Lt F (t/L)dt
(7.29)
for any δ > 0 and L = 1, 2, 3, . . .. It is clear that supn in F (t) and (7.29) may be replaced
by limn→∞ . Consequently, (7.28) is equivalent to
∀T > 0 and ε > 0,
ε,n
lim sup P (τi+1
− τiε,n < δ; τiε,n < T ) → 0 as δ → 0.
n→∞ i≥0
(7.30)
It is now clear that (7.30), and hence (7.28) and (7.26), are implied by (7.22) for any
neighborhood U of e and T > 0. It remains to verify (7.25). It suffices to show that for any
η > 0, there are a compact K ⊂ G and integer m > 0 such that P [xnt ∈ K c for some t ≤ T ] ≤
η for all n ≥ m. Because (7.22) implies (7.28), there are δ > 0 and integer m > 0 such that
ε,n
for all n ≥ m, P (An ) < η/2, where An = [mini≥0 {τi+1
− τiε,n ; τiε,n < T } < δ]. Let p = [T /δ],
ε,n
the integer part of T /δ. Then on Acn , mini≥0 {τi+1
− τiε,n ; τiε,n < T } ≥ δ, and hence xnt makes
at most p displacements of size ε, before time T , between τ1ε,n , τ2ε,n , . . . , τpε,n , with possible
jumps at these times. By (7.23), there is a compact H ⊂ G such that P (Bn,i ) ≤ η/(2p) for
all n ≥ m and 1 ≤ i ≤ p, where Bn,i = [(xnτ− )−1 xnτ ∈ H c ; τ ≤ T ] with τ = τiε,n . Let U
be the ε-ball around e and let K be a compact subset of G containing U HU H · · · U HU =
{u1 h1 u2 h2 · · · up hp up+1 ; ui ∈ U and hi ∈ H}. Then for n ≥ m,
P [xnt ∈ K c for some t ≤ T ] ≤ P (An ) +
p
∑
P (Acn ∩ Bn,i ) ≤ η. 2
i=1
We will now apply Lemma 7.8 to the processes ztγ,n . Because ztγ,n is constant for t ≥ 1,
to verify the conditions of Lemma 7.8, it is enough to consider only stopping times σ ≤ 1.
Let f ∈ Cc∞ (G) be such that 0 ≤ f ≤ 1 on G, f (e) = 1 and f = 0 on U c . Write τ for the
U c -displacement time for process ztγ,n from a stopping time σ. Let fσ = f ◦ (zσγ,n )−1 . Then
γ,n
γ,n
); τ < δ] ≤ E[fσ (zσγ,n ) − fσ (zσ+τ
P (τ < δ) = E[fσ (zσγ,n ) − fσ (zσ+τ
∧δ )],
γ,n
noting fσ (zσγ,n ) = 1, fσ (zσ+τ
) = 0 and τ = τ ∧ δ on [τ < δ], where a ∧ b = min(a, b). Because
Mtγ,n f = f (ztγ,n ) −
∑ ∫
sni ≤t G
γ
−1
γ,n
[f (zsγ,n
bγ,n x(bγni )−1 (bγ,n
sn i−1 ) ) − f (zsn i−1 )]µni (dx)
n i−1 sn i−1
(7.31)
is a martingale for any f ∈ Cc∞ (G), and σ and σ + τ ∧ δ are stopping times,
γ,n
γ,n
γ,n
E[Mσγ,n fσ − Mσ+τ
∧δ fσ ] = E{E[Mσ fσ − Mσ+τ ∧δ fσ | Fσ ]} = 0.
γ
γ
Writing z, b, b′ , µ for zsγ,n
, bγ,n
sn i−1 , bni , µni , we obtain
n i−1
∫
∑
P (τ < δ) ≤ −E{
σ<sni ≤σ+τ ∧δ G
≤ E{
∑
σ<sni ≤σ+δ
|
∫
G
[fσ (zbxb′−1 b−1 ) − fσ (z)]µ(dx)}
[fσ (zbxb′−1 b−1 ) − fσ (z)]µ(dx)|}.
220
(7.32)
Performing the same computation with Taylor’s expansion as in the proof of Lemma 7.7 and
noting that µ is moderate for large n,
∫
G
∫
=
[f (zbxb′−1 b−1 ) − f (z)]µ(dx)
Uc
[f (zbxb′−1 b−1 ) − f (z) −
∫
∑
fj (z, b, b′ )(ϕj (x) − ϕj (b′ ))]µ(dx)
j
1∑
+ [
fjk (z, b, b′ )(ϕj (x) − ϕj (b′ ))(ϕk (x) − ϕk (b′ ))]µ(dx) + r,
2
U
j,k
(7.33)
where the remainder r satisfies |r| ≤ cU µ(∥ϕ − ϕ(b′ )∥2 1U ) with constant cU → 0 as U ↓ {e}.
By (7.32) and (7.33), P (τ < δ) is controlled by
E[ηnγ (1 + δ, U c ) + q γ,n (σ + δ, G) − q γ,n (σ, G)].
Because ηnγ (t, ·) → 0 weakly outside a neighborhood of e and q γ,n (t, G) is equi-continuous in
t for large n as expressed by (7.18), it follows that limn→∞ supσ≤1 P (τ < δ) → 0 as δ → 0.
This verifies the condition (7.22) in Lemma 7.8 for xnt = ztγ,n .
Because the above method for verifying the condition (7.22) will be used again, it is
convenient to summarize it in a general form. Let Ftn (f, z·γ,n ) denote the sum on the right
hand side of (7.31). Then f (ztγ,n ) − Ftn (f, z·γ,n ) is a martingale. It is shown that P (τ < δ) ≤
n
n
(f, z·γ,n ) − Fσn (f, z·γ,n )| → 0 as δ → 0
E[|Fσ+δ
(f, z·γ,n ) − Fσn (f, z·γ,n )|]. By (7.33), limn→∞ |Fσ+δ
uniformly in z·γ,n . This implies limn→∞ P (τ < δ) → 0 as δ → 0, and hence verifies (7.22).
This is summarized in the following lemma for later reference.
Lemma 7.9 For any integer n ≥ 1 and f ∈ Cc∞ (G), let (t, x) 7→ Ftn (f, x) be a real valued
function on R+ × D(G) such that it is rcll in t for fixed x ∈ D(G) and is measurable under
σ{x(u); 0 ≤ u ≤ t} for fixed t ∈ R+ . Assume ∀ε > 0, ∃δ > 0 such that
limn→∞ |Ftn (f, x) − Fsn (f, x)| < ε
for all s, t ∈ [0, T ] with |t − s| < δ and x ∈ D(G). Let xnt be a sequence of rcll processes
in G such that f (xnt ) − Ftn (f, xn· ) is an martingale for any f ∈ Cc∞ (G). Then the condition
(7.22) in Lemma 7.8 holds.
We now verify (7.23). Because xγ,n
= ztγ,n bγ,n
t
t ,
γ,n −1 γ,n
γ,n −1
−1 γ,n
γ,n c
P [(zσ−
) zσ ∈ K c ] = P [(xγ,n
σ− ) xσ ∈ ((bσ− ) Kbσ ) ].
−1
γ,n
Because by either (i) or (ii), bγ,n
are bounded in n, when K is large, (bγ,n
contains
σ− ) Kbσ
t
a fixed neighborhood H of e. It follows that
γ,n −1 γ,n
c
c
γ
−1 γ,n
P [(zσ−
) zσ ∈ K c ] ≤ P [(xγ,n
σ− ) xσ ∈ H ] ≤ ηn (1, H ) → 0
221
as n → ∞. This establishes (7.23) even before taking K ↑ G.
Now by Lemma 7.8, ztγ,n are relatively weak compact, and hence along a subsequence of
n → ∞, ztγ,n converges weakly to a rcll process ztγ in D(G). By a well known coupling result,
the convergence in distribution implies the almost sure convergence on a suitable probability
space ([46, Theorem 4.30]). There are D(G)-valued random variables z̃· and z̃·n such that
d
d
z·γ = z̃· (equal in distribution) and for each n, z·γ,n = z̃·n , and z̃·n → z̃· almost surely as
d
γ,n
n → ∞. It follows that x̃n· = z̃·n bγ,n
= z·γ,n bγ,n
= xγ,n
→ e (a constant path
·
·
· . Because x·
n
in D(G)) almost surely, x̃· → e in distribution. It is well known that when the limit is a
constant, the convergence in distribution is equivalent to the convergence in probability ([46,
Lemma 4.7]), Thus, x̃n· → e in probability, and then along a subsequence, x̃n· → e almost
surely. This implies that along the same subsequence, bγ,n
= (z̃·n )−1 x̃n· → bγ· = z̃·−1 , where
·
the convergence → is in D(G).
As mentioned in §1.6, the convergence bγ,n
→ bγt in D(G) means that bγ,n
converge to bγt
t
t
uniformly for bounded t up to a time change that is asymptotically an identity. If bγt has a
jump of size r(bγs− , bγs ) > 0 at time s, then bγ,n
would have a jump of size close to r(bγs− , bγs )
t
γ,n
at time t = λ−1
are uniformly small when
n (s), which is impossible because the jumps of bt
γ
γ,n
γ
n is large. It follows that bt is continuous in t and hence bt → bt uniformly in t as n → ∞.
Then the convergence ztγ,n → ztγ is also uniform in t.
By (7.19), (7.20) and (7.33), the martingale Mtγ,n f given by (7.31) converges to
f (ztγ ) −
∫
0
t
d
1 ∑
[Ad(bγs )ξj ][Ad(bγs )ξk ]f (zsγ )dAγjk (s)
2 j,k=1
for any f ∈ Cc∞ (G). It follows that e = ztγ bγt and for any f ∈ Cc∞ (G), the above is a
martingale. Because ztγ = (bγt )−1 is non-random and [Ad(bγs )ξj ][Ad(bγs )ξk ]f (zsγ ) = ξjr ξkr f (zsγ ),
where ξ r is the right invariant vector field on G defined in §2.1, we obtain,
f (ztγ ) =
∫
t
0
d
1 ∑
ξ r ξ r f (z γ )dAγjk (s)
2 j,k=1 j k s
(7.34)
for any f ∈ Cc∞ (G). We now show ztγ = e and Aγ (t) = 0 for all t > 0. Let t0 be the largest
time point such that ztγ = e and Aγ (t) = 0 for t ∈ [0, t0 ]. By a simple time shift, we may
assume t0 = 0 in the following computation. Note that ξjr f (ztγ ) = ξj f (e)+O(∥ϕ(ztγ )∥) for any
∫ ∑
f ∈ C ∞ (G). Letting f = ϕ2j in (7.34) yields ϕj (ztγ )2 = 0t p,q [δpj δqj + O(∥ϕ(zsγ )∥)]dAγpq (s) =
Aγjj (t) + o(TrAγ (t)). This implies Tr[Aγ (t)] = O(∥ϕ(ztγ )∥2 ). Now letting f = ϕj in (7.34)
yields ϕj (ztγ ) = O(∥ϕ(ztγ )∥2 ), which is impossible unless ztγ = e for t close to 0. It then
follows from ϕj (ztγ )2 = Aγjj (t) + o(TrAγ (t)) that Aγ (t) = 0 for t closed to 0. This contracts
the maximality of t0 = 0, and hence proves that ztγ = e and Aγ (t) = 0 for all t > 0.
As mentioned earlier, if (i) holds, then Tr[Aγ (1)] ≥ ε. Therefore, (i) cannot hold. On the
n −1 n
other hand, if (ii) holds, then bγ1 as the limit of bγ,n
1 = (bsn ) btn cannot be equal to e, which
contradicts xγ1 = z1γ = e. This shows that (A) and (B) must hold.
222
Because (A) and (B) have been verified, by Lemmas 7.1 and 7.7, we have proved the
following result which is the first part of Theorem 6.11 except for the uniqueness of (b, A, η).
Lemma 7.10 An inhomogeneous Lévy process xt in G with x0 = e is represented by an
extended Lévy triple (b, A, η) as defined before Theorem 6.11, where η(t, ·) is its jump intensity
measure, and A(t) and bt are given in Lemmas 7.3 and 7.6.
7.3
Approximation
Let (b, A, η) be a Lévy triple. By the general definition in §6.3, a rcll process xt in G is said
to be represented by (b, A, η) if xt = zt bt , and (6.21) is a martingale, that is, for f ∈ Cc∞ (G),
f (zt )−Tt f (z; b, A, η) is a martingale under the natural filtration Ftx of xt , where Tt f (z; b, A, η)
is defined by (6.22). A purpose of this section is to prove the following result, which is the
existence part in the second half of Theorem 6.9.
Lemma 7.11 Given a Lévy triple (b, A, η), there is a inhomogeneous Lévy process xt in G
with x0 = e that is represented by (b, A, η).
To prove Lemma 7.11, we will approximate (b, A, η) by a sequence of piecewise linearized
Lévy triples (bn , An , η n ), construct processes xnt represented by (bn , An , η n ) that are piecewise
(homogeneous) Lévy processes, and show that xnt are relatively weak compact in D(G) and
so a subsequence converges weakly to an inhomogeneous Lévy process xt represented by
(b, A, η).
Let ∆n : 0 ≤ tn0 < tn1 < tn2 < · · · < tnk ↑ ∞ as k ↑ ∞ be a sequence of partitions of R+
with mesh ∥∆n ∥ = supk δnk → 0 as n → ∞, where δnk = tnk − tn k−1 for k ≥ 1.
For any integers n, k ≥ 1, let
ank
ij = [Aij (tnk ) − Aij (tn k−1 )]/δnk
and η nk (·) = [η(tnk , ·) − η(tn k−1 , ·)]/δnk .
−1
Then η nk is a Lévy measure on G. When n is large, b−1
tn k−1 btnk is close to e, and so btn k−1 btnk =
exp(δnk ξ0nk ) for some ξ0nk ∈ g. Note that for any T > 0, when n is large, this definition is
∑
valid for all tnk ≤ T . Let ξ0nk = i βink ξi for some βink ∈ R.
By Theorem 2.15 and (2.33), there is a Lévy process gt in G with g0 = e such that
f (gt ) −
= f (gt ) −
∫
+
G
∫
∫
t
Lf (gs )ds
0
t
0
{
∑
βink ξi f (gs ) +
i
[f (gs y) − f (gs ) −
∑
i
223
1 ∑ nk
a ξi ξj f (gs )
2 i,j ij
ϕi (y)ξi f (gs )]η nk (dy)}ds
(7.35)
is an {Ftg }-martingale for any f ∈ Cc∞ (G).
Let
nk
nk
bnk
Ank
t = exp(tξ0 ),
ij (t) = aij t,
η nk (t, ·) = η nk (·)t.
−1
nk
nk
Then bnk
δnk = btn k−1 btnk , A (δnk ) = A(tnk ) − A(tn k−1 ) and η (δnk , ·) = η(tnk , ·) − η(tn k−1 , ·).
∑
nk
nk
nk
Because bnk
t = exp( i tβi ξi ) is a drift of finite variation with components bi (t) = βi t,
and because (7.35) is a martingale, by Theorem 6.27, whose proof does not depend on
Theorems 6.9 and 6.11, gt is represented by (bnk , Ank , η nk ).
nk
We will write xnk
t for gt . We may and will assume the processes xt , for distinct n and
k, are independent. Let xnt = xn1
t for t ≤ tn1 , and recursively for k ≥ 1, let
k+1
xnt = xntnk xnt−t
nk
for tnk < t ≤ tn k+1 .
(7.36)
n
n1
n
n1
n
n
Let bnt = bn1
t , Aij (t) = Aij (t) and η (t, ·) = η (t, ·) for t ≤ tn1 , and define bt , Aij (t) and
η n (t, ·) inductively for k ≥ 1 and tnk < t ≤ tn k+1 by
k+1
,
bnt = bntnk bnt−t
nk
Anij (t) = Anij (tnk ) + (t − tnk )anijk+1 ,
η n (t, ·) = η n (tnk , ·) + (t − tnk )η n k+1 (·).
Lemma 7.12 (a) xnt defined in (7.36) is an inhomogeneous Lévy process in G with xn0 = e,
and is represented by the Lévy triples (bn , An , η n ).
(b) (bn , An , η n ) → (b, A, η) as n → ∞ in the sense that for any finite constant T > 0 and
f ∈ Cb∞ (G) vanishing near e, bnt → bt , An (t) → A(t) and η n (t, f ) → η(t, f ) as n → ∞
uniformly for t ≤ T , and there are open sets Um ↓ {e} such that
lim sup η n (T, ∥ϕ∥2 1Um ) = 0.
m→∞
(7.37)
n
Proof Part (a) follows directly from Proposition 6.39.
Note that bntnk = btnk , and for t ∈ (tn k−1 , tnk ], bnt = btn k−1 exp((t − tn k−1 )ξ0nk ) with
−1
exp((tnk − tn k−1 )ξ0nk ) = b−1
tn k−1 btnk . By the continuity of bt , for finite T > 0, btn k−1 bt−tn k−1 → e
and (tnk − tn k−1 )ξ0nk → 0 as n → ∞ uniformly for tnk ≤ T . This implies that bnt → bt as
n → ∞ uniformly for t ≤ T . Because An (tnk ) = A(tnk ) and η n (tnk , ·) = η(tnk , ·), and for
t ∈ (tn k−1 , tnk ], An (t) and η n (t, ·) are linear interpolations of A(t) and η n (t, ·), the uniform
convergence of An (t) → A(t) and η n (t, f ) → η(t, f ) as stated in part (c) follows easily from
the continuity of A(t) and η(t, f ) in t. Because for t ∈ (tn k−1 , tnk ], η n (t, ·) ≤ η(tnk , ·), it
follows that limm→∞ supn η n (T, ∥ϕ∥2 1Um ) = 0 for any open sets Um ↓ {e}. 2
A sequence of Lévy triples (bn , An , η n ) is said to converge to a Lévy triple (b, A, η), denoted
as (bn , An , η n ) → (b, A, η), if the properties stated in Lemma 7.12 (b) hold. See Remark 7.18
224
for a weaker form of the convergence of Lévy triples. Note that because A(t) and η(t, ·) are
increasing in t, the uniform convergence of An (t) → A(t) and η n (t, f ) → η(t, f ) for t ≤ T
holds if the convergence holds for all fixed t > 0. Moreover, the convergence η n (t, f ) → η(t, f )
in Lemma 7.12 (b) implies that for compact Km ↑ G,
c
lim sup η n (T, Km
) = 0.
m→∞
(7.38)
n
Lemma 7.13 Let (bn , An , η n ) → (b, A, η), and fix a finite T > 0.
∫
∫
(a) For any real valued rcll function α(t) on R+ , 0t α(s)dAnij (s) → 0t α(s)dAij (s) as n → ∞
uniformly for t ≤ T .
∫
(b) Let f ∈ Cb (G2 ). Then 0t f (bs , x(s))dAij (s) is a bounded continuous function in (t, x) ∈
R+ × D(G), and for any compact subset Γ of D(G),
∫
0
t
f (bns , x(s))dAnij (s) →
∫
t
0
f (bs , x(s))dAij (s)
as n → ∞ uniformly for (t, x) ∈ [0, T ] × Γ.
(c) Let α(t) be a real-valued rcll function on R+ and ψ ∈ Cb (R+ × G) be such that for all s ∈
∫ ∫
R+ and g near e, |ψ(s, g)| ≤ c∥ϕ(g)∥2 for some constant c > 0. Then 0t G ψ(α(s), y)η n (ds, dy)
∫ ∫
→ 0t G ψ(α(s), y)η(ds, dy) as n → ∞ uniformly for t ≤ T .
(d) Let h ∈ Cb (G3 ) satisfy |h(g1 , g2 , g3 )| ≤ c∥ϕ(g3 )∥2 for any g1 , g2 ∈ G and g3 near e, and
∫ ∫
for some constant c > 0. Then 0t G h(bs , x(s), y)η(ds, dy) is a bounded continuous function
in (t, x) ∈ R+ × D(G), and for any compact subset Γ of D(G),
∫
t
0
∫
G
h(bns , x(s), y)η n (ds, dy) →
∫
t
0
∫
G
h(bs , x(s), y)η(ds, dy)
as n → ∞ uniformly for (t, x) ∈ [0, T ] × Γ.
(e) Let f ∈ Cb (G2 ) and h ∈ Cb (G3 ) be as in (b) and (d). Then for any ε > 0, there is δ > 0
such that
∫ t
∫ t
n
n
sup[ |f (bs , x(s))|dAij (s) +
|h(bns , x(s), y)|η n (ds, dy)] < ε
n
s
s
for any s, t ∈ [0, T ] with |s − t| < δ and x ∈ D(G).
∫
∫
Proof: The uniform convergence 0t α(s)dAnij (s) → 0t α(s)dAij (s) clearly holds if α(t) is a
step function. It also holds for a general rcll α(t) because it can be approximated uniformly
by step functions. This proves (a)
As mentioned in §1.6, the convergence xn (s) → x(s) in D(G) is uniform in s ≤ T up to a
time change that is asymptotically an identity, because bs and Aij (s) are continuous in s, it
∫
∫
is easy to see that if tn → t, then 0tn f (bs , xn (s))dAij (s) → 0t f (bs , x(s))dAij (s) as n → ∞,
∫
and hence 0t f (bs , x(s))dAij (s) is continuous in (t, x) ∈ [0, T ] × D(G). This proves the first
part of (b).
225
To prove the second part of (b), first assume f ∈ Cb (G) does not depend on bs . By
Theorem 6.3 and Remark 6.4 in [22, chapter 3], a subset Γ of D(G) is relatively compact if
and only if for any T > 0, there is a compact K ⊂ G such that for any x ∈ Γ, x(s) ∈ K
for s ≤ T , and limδ→0 supx∈Γ w′ (x, δ, T ) = 0, where w′ (x, δ, T ) is the oscillation of the path
x over a δ-partition of [0, T ] as defined in (7.24). Recall a δ-partition of [0, T ] is a collection
of intervals [ti , ti+1 ) that cover [0, T ] and have lengths > δ. Therefore, for any ε > 0, there
is δ-partition ∆ of [0, T ] such that the oscillation of any x ∈ Γ over ∆ is within ε-distance
under a Riemannian metric on G. For each x ∈ Γ, we may choose a step function in G that
is associated to the partition ∆ and approximate x(s) within ε for all s ∈ [0, T ]. It then
∫
∫
follows that 0t f (x(s))dAnij (s) → 0t f (x(s))dA(s) uniformly for (t, x) ∈ [0, T ] × Γ.
∫
For f ∈ Cb (G2 ), because bns → bs uniformly for s ≤ T , to prove 0t f (bns , x(s))dAnij (s) →
∫t
n
0 f (bs , x(s))dAij (s) uniformly for (t, x) ∈ [0, T ] × Γ, it suffices to prove this when bs is
replaced by bs . Then the proof is essentially the same as in the previous paragraph, with
a partition ∆′ chosen to be finer than ∆, and two step functions associated to ∆′ , one
approximating x(s) and the other approximating bs . Part (b) is now proved.
Let U be a neighborhood of e such that η(T, ∂U ) = 0. Then for f ∈ Cb (G), s 7→
∫
∫
η(s, f 1U c ) = U c f (y)η(s, dy) is continuous for s ≤ T . The statements in (c) and (d) with G
∫
replaced by U c can be proved in the same way as (a) and (b). By (7.37), it is then easy to
derive these statements in their original forms.
Part (e) follows easily from the boundedness of f (bns , x(s)) and h(bns , x(s), y), the uniform
convergence of Anij (t) → Aij (t) and η n (t, f ) → η(t, f ), and the continuity of t 7→ Aij (t) and
t 7→ η(t, ·). 2
Let (bn , An , η n ) be a sequence of Lévy triples that converges to a Lévy tripe (b, A, η), and
let xnt be inhomogeneous Lévy processes represented by (bn , An , η n ). We will use Lemma 7.8
to show xnt are relatively weak compact in D(G). Let xnt = ztn bnt . Because bnt → bt as n → ∞
uniformly for t ≤ T , it suffices to show that ztn are relatively compact in D(G).
Because (b, A, η) is a Lévy triple, by (6.22),
∫
Tt f (z; b, A, η) =
t
0
∫
1∑
[Ad(bs )ξi ][Ad(bs )ξj ]f (zs )dAij (s)
2 i,j
t
∫
+
0
G
{f (zs bs σb−1
s ) − f (zs ) −
∑
ϕi (σ)[Ad(bs )ξi ]f (zs )}η(ds, dσ).
i
By (b) and (d) in Lemma 7.13, Tt f (z; b, A, η) is a bounded continuous function in (t, z) ∈
R+ × D(G), and Tt f (z; bn , An , η n ) → Tt f (z; b, A, η) as n → ∞ uniformly for t ≤ T and z
contained in a compact subset of D(G). By Lemma 7.13 (e), Ftn (f, z) = Tt f (z; bn , An , η n )
satisfies the condition in Lemma 7.9, and then by this lemma, the condition (7.22) in Lemma 7.8 holds for ztn .
226
To verify the condition (7.23) in Lemma 7.8, note that η n (t, ·) is the jump intensity
measure of process xnt , so ηzn (t, ·) = c(bnt )η n (t, ·) is the jump intensity measure of ztn , where
c(b) = cb is the conjugation by b. Because bnt is contained in a compact subset of G that does
c
not depend on n and t ≤ T , by (7.38), limm→∞ supn ηzn (T, Km
) = 0 for compact Km ↑ G.
Then for any stopping time σ ≤ T ,
n −1 n
P [(zσ−
) zσ ∈ K c ] ≤ ηzn (T, K c ) → 0
as compact K ↑ G uniformly in n. By Lemma 7.8, the processes ztn are relatively weak
compact in D(G), and hence along a subsequence of n → ∞, ztn converge weakly to a rcll
process zt in D(G). Along the same subsequence, xnt converge weakly to the rcll process
xt = zt bt in D(G). By the lemma below, xt is an inhomogeneous Lévy process in G.
Lemma 7.14 Let xnt be a sequence of rcll processes in G and assume they converge weakly
to a rcll process xt in D(G). Then for any t1 < t2 < · · · < tk , which are not fixed jump times
of the process xt ,
(xnt1 , xnt2 , . . . , xntk ) → (xt1 , xt2 , . . . , xtk ) weakly as n → ∞.
(7.39)
Moreover, if each xnt is an inhomogeneous Lévy process in G, then so is xt .
Proof The weak convergence (7.39), over time points which are not fixed jump times of
xt , follows from [22, 3.Theorem 7.8]. If xnt are inhomogeneous Lévy processes, then xt has
independent increments over time points which are not fixed jump times of xt . By the rcll
paths, this holds over all time points. 2
Lemma 7.15 Let S be a complete and separable metric space, let Fn be uniformly bounded
Borel functions on S converging to a bounded continuous function F uniformly on compact
subsets of S, and let xn be random variables in S converging weakly to a random variable x.
Then E[Fn (xn )] → E[F (x)] as n → ∞. Moreover, the same holds when F is not assumed
to be continuous, only bounded Borel, but satisfies P (x ∈ H) = 1, where H is the set of the
continuous points of F .
Proof Let P and Pn be the probability distributions on S associated x and xn respectively.
Then Pn → P weakly and P (H) = 1, and the result to prove is Pn (Fn ) → P (F ).
Let the constant c > 0 bound both |Fn | and |F |. By Prohorov’s Theorem for probability
measures on complete and separable metric spaces (see [22, 3.Theorem 2.2]), Pn → P weakly
implies Pn is tight, that is, for any ε > 0, there is a compact K ⊂ S such that supn Pn (K c ) <
ε. Then
|Pn (Fn ) − P (F )| ≤
∫
K
|Fn − F |dPn + 2cPn (K c ) + |Pn (F ) − P (F )|
≤ sup |Fn − F | + 2cε + |Pn (F ) − P (F )|,
K
227
(7.40)
which converges to 2cε as n → ∞ when F is continuous.
When F is not continuous, assume P (H) = 1. There is a compact Y1 ⊂ S such that
P (Y1c ) < ε/2. By [75, II.Theorem 1.2], any measure µ defined on the Borel σ-algebra S of a
metric space S is regular in the sense that for any B ∈ S.
µ(B) = sup{µ(V ); closed V ⊂ B} = inf{µ(U ); open U ⊃ B}.
Then there is a closed Y2 ⊂ H with P (Y2c ) < ε/2. Let Y = Y1 ∩ Y2 . Then Y is compact,
Y ⊂ H, and P (Y c ) < ε. There is a continuous function F ′ on S, bounded also by c, such
that F = F ′ on Y . Because Y is compact and contains only continuous points of F , there
is open Z ⊃ Y such that |F − F ′ | < ε on Z. Because Pn → P weakly and Z c is closed,
Pn (Z c ) < ε for all large n. Then
|Pn (F )−P (F )| ≤ |Pn (F ′ )−P (F ′ )|+|Pn (F −F ′ )−P (F −F ′ )| ≤ |Pn (F ′ )−P (F ′ )|+2(ε+2cε).
Because [Pn (F ′ ) − P (F ′ )] → 0 as n → ∞, this shows Pn (F ) → P (F ), and then by (7.40),
Pn (Fn ) → P (F ). 2
Lemma 7.16 Let (bn , An , η n ) → (b, A, η) be the convergence of Lévy triples, and let xnt be
inhomogeneous Lévy processes represented by (bn , An , η n ). Then along a subsequence, xnt
converges weakly in D(G) to an inhomogeneous Lévy process xt represented by (b, A, η).
Proof It has already established that xnt converge weakly to an inhomogeneous Lévy process
xt . To prove xt is represented by (b, A, η), we have to show that, with xt = zt bt , f (zt ) −
Tt f (z; b, A, η) is a martingale for f ∈ Cc∞ (G). Fix s < t. Let Ts,t f = Tt f − Ts f . It suffices
to show
E{[f (zt ) − f (zs ) − Ts,t f (z; b, A, η)]h(z)} = 0
(7.41)
for any bounded function h on D(G) that is measurable under the σ-algebra generated by
the coordinate functions on D(G) up to time s. We may assume
h(x) = f1 (x(s1 ))f2 (x(s2 )) · · · fk (x(sk ))
for s1 < s2 < · · · < sk and f1 , f2 , . . . , fk ∈ Cb (G). Because fi (x(si )) = limε→0 (1/ε)hε (x),
∫
where hε (x) = ssii +ε fi (x(u))du is continuous in x ∈ D(G), we may assume h in (7.41) is
bounded and continuous on D(G).
Because (bn , An , η n ) → (b, A, η), it is easy to see that Tt f (x; bn , An , η n ) are uniformly
bounded in n, and by (b) and (d) in Lemma 7.13, Tt f (x; bn , An , η n ) → Tt f (x; b, A, η) as
n → ∞ uniformly for x in any compact subset of D(G). Then
Gnt (f, x) = [f (x(t)) − f (x(s)) − Ts,t f (x; bn , An , η n )]h(x)
228
are uniformly bounded in n, and
Gnt (f, x) → Gt (f, x) = [f (x(t)) − f (x(s)) − Ts,t f (x; b, A, η)]h(x)
as n → ∞ uniformly on any compact subset of D(G). Because f (ztn ) − Tt f (z n ; bn , An , η n )
is a martingale, E[Gnt (f, z n )] = 0. As mentioned before, Ts,t f (x; b, A, η) is bounded and
continuous in x ∈ D(G). Then Gt (f, x) is bounded, and it is continuous at every x ∈ D(G)
with x(t−) = x(t) and x(s−) = x(s). Because zt is stochastically continuous, P (H) = 1,
where H is the set of continuous points x of Gt (f, x). By Lemma 7.15 with S = D(G),
xn = z n , x = z and F (x) = Gt (f, x), E[Gnt (f, z n )] → E[Gt (f, z)], and hence E[Gt (f, z)] = 0.
This is (7.41). 2
Proof of Lemma 7.11: Applying Lemma 7.16 to the process xnt defined in (7.36), and
using Lemma 7.12, we obtain a proof of Lemma 7.11. 2
By the uniqueness in Theorem 6.9, a rcll process represented by a Lévy triple is unique
in distribution. Therefore, in Lemma 7.16, the whole sequence xnt converge weakly to xt
in D(G). This derives the following result from (not yet proved) Theorem 6.9, which may
be regarded as a partial extension to Lie groups of a more complete result on Rd , see [45,
VII.Theorem 3.4] and Remark 7.18.
Theorem 7.17 Let (bn , An , η n ) be a sequence of Lévy triples on G that converges to a
Lévy tripe (b, A, η), and let xnt and xt be inhomogeneous Lévy processes in G represented by
(bn , An , η n ) and (b, A, η) respectively. Then the processes xnt converge to xt weakly in D(G).
Remark 7.18 By Theorem 7.17, the convergence of Lévy triples, as defined by the properties in Lemma 7.12 (b), implies the weak convergence of the represented inhomogeneous
Lévy processes. We will show that the weak convergence of processes is also implied by a
weaker notion of the convergence of Lévy triples that is found in [45] for processes in Rd . For
a Lévy triple (b, A, η) on G, let Ãij (t) = Aij (t) + η(t, ϕi ϕj ). Consider the following condition
on (b, A, η): For any finite T > 0 and f ∈ Cb (G) vanishing near e,
bnt → bt ,
Ãn (t) → Ã(t),
η n (t, f ) → η(t, f ) as n → ∞ uniformly for t ≤ T .
(7.42)
Here, the convergence bnt → bt is defined under a Riemann metric on G, but it does not depend
on the choice of this metric. Note also that because Ã(t) and η(t, ·) are nondecreasing in t,
the uniform convergence of Ãn (t) → Ã(t) and η n (t, f ) → η(t, f ) in (7.42) is implied by the
convergence at all fixed t > 0.
From the properties in Lemma 7.12 (b), it is easy to derive η n (t, ϕi ϕj ) → η(t, ϕi ϕj ), and
hence the condition (7.42) is implied by (bn , An , η n ) → (b, A, η) as defined here. We will
229
show that if (7.42) holds, and if xnt and xt are inhomogeneous Lévy processes in G with
xn0 = x0 = e, represented by (bn , An , η n ) and (b, A, η) respectively, then xnt converge to xt
weakly in D(G) as n → ∞. We note that by [45, VII.Theorem 3.4], for G = Rd , (7.42) is a
necessary and sufficient condition for the weak convergence xnt → xt .
First it is easy to see that under (7.42), (a) and (b) in Lemma 7.13 hold with A(t) replaced
by Ã(t). We have, with xt = zt bt ,
∫
Tt f (z; b, A, η) =
t
0
∫ t∫
d
1 ∑
F (zs , bs , y)η(ds, dy),
[Ad(bs )ξi ][Ad(bs )ξj ]f (zs )dÃij (s) +
2 i,j=1
0 G
where
−1
F (x, b, y) = f (xbyb ) − f (z) −
d
∑
ϕi (y)[Ad(b)ξi ]f (x)
i=1
−
d
1 ∑
ϕi (y)ϕj (y)[Ad(b)ξi ][Ad(b)ξj ]f (x).
2 i,j=1
For y contained in a small neighborhood U of e that is the exponential image of a small
neighborhood W of 0 in the Lie algebra g of G, with y = ew , using a Taylor expansion of
ψ(t) = f (xbetw b−1 ) at t = 0, we obtain
F (x, b, y) =
d
1 ∑
ϕi (y)ϕj (y){[Ad(b)ξi ][Ad(b)ξj ]f (xesw ) − [Ad(b)ξi ][Ad(b)ξj ]f (x)}
2 i,j=1
for some s ∈ [0, 1]. Because under (7.42), η n (t, ∥ϕ∥2 ) is bounded in n, it follows that
∫t∫
n
0 U F (x(s), bs , y)η (ds, dy) → 0 as U ↓ {e} uniformly in n, t ≤ T , and x(s) and bs contained
in a compact subset of G for 0 ≤ s ≤ T . Then as in the proof of (c) and (d) in Lemma 7.13,
∫ ∫
∫ ∫
it can be shown that 0t G F (x(s), bn , y)η n (ds, dy) → 0t G F (x(s), bs , y)η(ds, dy) as n → ∞
uniformly for t ≤ T and x in any compact subset of D(G). The rest of the proof for the
weak convergence ztn → zt , and hence for xnt → xt , is the same as before.
7.4
Uniqueness in distribution under finite variation
Let (b, A, η) be a Lévy triple on G. It has been established in Lemma 7.11 that there is
an inhomogeneous Lévy process xt in G with x0 = e represented by (b, A, η). The following
lemma provides the uniqueness of the process in distribution in the second half of Theorem 6.9
under the additional assumption that the drift bt is of finite variation. Theorem 6.9 will be
proved completely at the end of this section.
Lemma 7.19 Let (b, A, η) be a Lévy triple on G with bt of finite variation, and let xt be a
rcll process in G with x0 = e represented by (b, A, η). Then the distribution of the process xt
is uniquely determined by (b, A, η).
230
By Theorem 6.27, whose proof does not depend on Theorems 6.9 and 6.11, xt is represented by (b, A, η) if and only if f (xt ) − S̃t f (x; b, A, η) is a martingale under Ftx for
any f ∈ Cc∞ (G). After the non-ramdon time change described in §6.8, S̃t f (x; b, A, η) =
∫t
∞
0 L(s)f (xs )ds, where L(t) be the operator on Cc (G) given by
L(t) =
∫
∑
∑
1∑
aij (t)ξi ξj +
βi (t)ξi + m(t, dy)[ry − 1 −
ϕi (y)ξi ].
2 i,j
G
i
i
(7.43)
Here the right translation rg is regarded as an operator on functions on G: rg f (x) = (f ◦
rg )(x) = f (xg). Similarly the left translation lg will be regarded as an operator later. The
operator L(t) is left invariant in the sense that L(t)(f ◦ lg )(x) = L(t)f (gx) for g, x ∈ G.
After the above mentioned non-random time change by ψ, a rcll process xt in G, with
x0 = e, is represented by (b, A, η) if and only if for any f ∈ Cc∞ (G),
f (xt ) −
∫
t
0
L(s)f (xs )ds
(7.44)
is a martingale under Ftx . A rcll process xt in G will be said to have the generator L(t)
if (7.44) is a martingale for any f ∈ Cc∞ (G). Thus, a rcll process xt is represented by the
Lévy triple (b, A, η) if and only if, after the time change, it has the generator L(t).
Recall an inhomogeneous Lévy process xt in G is characterized by the independence of
−1
−1
xs xt and Fsx for s < t. If x−1
s xt is replaced by xt xs , we obtain a different type of processes,
which will be called right processes, whereas the former processes may be called left processes.
It is clear that if xt is a left process, then x−1
is a right process, and vice versa.
t
Let
∫
∑
∑
1∑
aij (t)ξir ξjr −
βi (t)ξir + m(t, dy −1 )[ly − 1 −
ϕi (y)ξir ].
(7.45)
L∗ (t) =
2 i,j
G
i
i
This is just the operator L(t) in (6.44) when all ξi are replaced by −ξir , ry by ly and m(t, dy)
by m(t, dy −1 ) = Jm(t, dy), where J: y 7→ y −1 is the inverse map on G.
Because the coordinate functions ϕi are exponential, ϕi (x−1 ) = −ϕi (x) for x near e. By
suitably modifying ϕi off a neighborhood of e, we may assume this holds for all x ∈ G.
Because such a modification may be offset by a change in bt , without loss of generality, we
may assume ϕi (x−1 ) = −ϕi (x) for all x ∈ G in the proof of Lemma 7.19. Then it is easy to
show that for f ∈ Cc∞ (G),
L(t)(f ◦ J) = [L∗ (t)f ] ◦ J.
Thus, if xt is a left process with generator L(t), then x−1
is a right process with generator
t
∫t ∗
−1
∗
−1
L (t), that is, f (xt ) − 0 L (s)f (xs )ds is a martingale under Ftx for f ∈ Cc∞ (G).
Let Lr (t) be the operator L(t), given in (6.44), when all ξi are replaced by ξir and ry is
replaced by ly . That is,
Lr (t) =
∫
∑
∑
1∑
ϕi (y)ξir ].
βi (t)ξir + m(t, dy)[ly − 1 −
aij (t)ξir ξjr +
2 i,j
G
i
i
231
(7.46)
Then Lr (t) is just L∗ (t) when βi (t) and m(t, dy) are replaced by −βi (t) and m(t, dy −1 )
respectively, and hence is the generator of a right process.
Let T > 0 be fixed. The time-reversed operator L′ (t) = Lr (T − t) is also the generator
of a right process yt with y0 = e for 0 ≤ t ≤ T . Let gt,T = yT −t . Because for f ∈ Cc∞ (G),
∫
f (yt ) − 0t Lr (T − s)f (ys )ds is a martingale under Fty , replacing t by T − t, it then follows
that for f ∈ Cc∞ (G),
Mt,T = f (gt,T ) −
∫
T
t
Lr (s)f (gs,T )ds
(7.47)
g
g
is a time reversed martingale, that is, for 0 ≤ s < t ≤ T , E[Ms,T | Ft,T
] = Mt,T , where Ft,T
is
the σ-algebra generated by gr,T , t ≤ r ≤ T . We will say gt,T is a time-reversed right process
with generator Lr (t). Note that t 7→ gt,T have left continuous paths with right limits, and
gT,T = y0 = e.
A random variable taking values in [0, T ] will be called a stopping time under the timeg
g
reversed filtration Ft,T
if for t ∈ [0, T ], [τ ≥ t] ∈ Ft,T
. The standard optional stopping for
martingales holds also for a time-reversed martingale. After all, this is just the standard
result in the reversed time. In particular, E(Mτ,T ) = f (e).
For f ∈ Cc∞ (G), let
u(t, x) = E[f (xgt,T )],
0 ≤ t ≤ T, x ∈ G.
(7.48)
Then u(t, x) is bounded in (t, x) with u(T, x) = f (x).
Lemma 7.20 Let xt be a rcll process in G with generator L(t) and x0 = e. Then u(t, xt ) is
a martingale under Ftx .
Proof For a bounded smooth function f (x, g) on G × G, we will write Lx (t)f (x, g) for
[L(t)f (·, g)](x) and Lg (t)f (x, g) for [Lr (t)f (x, ·)](g). Similarly, for ξ ∈ g and y ∈ G, we
will write ξ x f (x, g), ξ g f (x, g) and lyg f (x, g) for [ξf (·, g)](x), [ξ r f (x, ·)](g), and [ly f (x, ·)](g) =
f (x, yg) respectively. In the sequel, for f ∈ Cc∞ (G), these operations will be applied to
f˜(x, g) = f (xg), but to simplify the notation, we will write f (xg) for f˜(x, g), and thus for
example, Lx (t)f˜(x, g) is written as Lx (t)f (xg). However, L(t)f (xg) is [L(t)f ](xg).
Let U be a relatively compact open neighborhood of e. By a direct computation, it is
easy to show that for f ∈ Cc∞ (G),
Lx (t)f (xg) = Lg (t)f (xg),
(7.49)
which is smooth in (x, g), and is bounded for (t, x, g) ∈ [0, T ] × U × G or for (t, x, g) ∈
[0, T ] × G × U . For the smoothness, see Lemma 2.2. To check the boundedness, note
ξ x f (xg) = ξ g f (xg) = [Ad(x)ξ]r f (xg) = [Ad(g −1 )ξ]f (xg),
232
(7.50)
which is bounded if x ∈ U or g ∈ U .
As ξ g and lxg commute with the operator Lx (s), ξ g Lx (s)f (xg) = Lx (s)(ξ g f )(xg) and
lyg Lx (s)f (xg) = Lx (s)(lyg f )(xg), it then follows that
Lg (t)Lx (s)f (xg) = Lx (s)Lg (t)f (xg).
(7.51)
Moreover, by (7.50), it can be shown that the expression in (7.51) is bounded for (s, t, x, g) ∈
[0, T ] × [0, T ] × U × U . To prove these claims in more details require some rather tedious
computation. A part of the proof is to show Itg Isx f (xg) = Isx Itg f (xg), where Isx f (xg) and
Itg f (xg) are respectively the integral terms in Lx (s)f (xg) and Lg (t)f (xg). We will provide
below some details on how this is done.
x
For a Borel subset V of G, let Is,V
f (xg) be the integral Isx f (xg) restricted to V . Similarly
g
x
for Is,V
f (xg). By Lemma 2.2 (with η being the restriction of m(s, ·) to V ), Is,V
f (xg) and
g
Ls,V f (xg) are smooth in (x, g), and by (7.50), they are bounded for (x, g) ∈ U × G or for
(x, g) ∈ G × U . For a small
∑ neighborhood V of e, by Taylor’s expansion of ψ(u) = f (xu g)
u p ϕp (y)ξp
at u = 0, where xu = xe
with y ∈ V ,
x
Is,V
f (xg)
= (1/2)
∑∫
i,j
V
ϕi (y)ϕj (y)ξi ξj f (xe
v
∑
p
ϕp (y)ξi
g)m(s, dy)
x
for some v ∈ [0, T ]. Then for (x, g) ∈ U × G or (x, g) ∈ G × U , |Is,V
f (xg)| ≤ c m(s, ∥ϕ∥2 1V )
x
for some constant c > 0. It follows that Is,V
f (xg) → 0 as V ↓ {e} uniformly for (x, g) ∈ U ×G
g
f (xg).
or for (x, g) ∈ G × U . The same holds for Is,V
Let h be a bounded smooth function on G. The above argument using Taylor’s expansion
x
may be applied to ψ(u) = h(xu g), then we obtain Is,V
h(xg) → 0 as V ↓ {e} uniformly for
g
g
Isx f (xg) → 0 as V ↓ {e}.
(x, g) ∈ U × U . The same holds for Is,V h(xg). In particular, It,V
x
For two small neighborhoods
V and W of e, by Taylor’s expansion of ψ(u) = Is,V
f (xgu )
∑
at u = 0, where gu = eu
g
x
It,W
Is,V
f (xg)
=
1∑
2 i,j
∫
W
p
ϕp (y)ξp
g with y ∈ W ,
∑
1 ∑∫
v
g g x
=
ϕi (y)ϕj (y)ξi ξj (Is,V f )(xe p ϕp (y)ξp g)m(t, dy)
2 i,j W
x
ϕi (y)ϕj (y)Is,V
(ξig ξjg f )(xev
∑
p
ϕp (y)ξp
g)m(t, dy)
∑
∑
1 ∑ ∫ ∫
w
ϕp (z)ξp v
x x g g
p
=
e p ϕp (y)ξp g)m(s, dz)m(t, dy)
ϕi (y)ϕj (y)ϕk (z)ϕl (z)ξk ξl ξi ξj f (xe
4 i,j,k,l W V
g
x
for some v, w ∈ [0, T ]. Then for (x, g) ∈ U × G or (x, g) ∈ G × U , |It,W
f (xg)| ≤
Is,V
g
x
c m(t, ∥ϕ∥2 1W )m(s, ∥ϕ∥2 1V ) for some constant c > 0, and hence It,W Is,V
f (xg) → 0 as W ↓
{e} uniformly in V , or as V ↓ {e} unformly in W .
x
f (xg) → 0 as V ↓ {e} uniformly in (x, g) ∈ U × G. Because ξig commutes with
Recall Is,V
x
x
x
(ξig f )(xg) → 0 as V ↓ {e} uniformly for (x, g) ∈ G × U . Then for
f (xg) = Is,V
, ξig Is,V
Is,V
233
g
g x
g
x
x
(x, g) ∈ U × U , It,W
c Is,V f (xg) → 0 as V ↓ {e}. It follows that It Is,V f (xg) = It,W Is,V f (xg) +
g
g
g
g x
x
x
x
It,W
c Is,V f (xg) → 0 as V ↓ {e}, and hence It,V c Is,V f (xg) = It Is,V f (xg) − It,V Is,V f (xg) → 0
as V ↓ {e}. We have, for (x, g) ∈ U × U ,
g
g
g
x
x
x
Itg Isx f (xg) = It,V
c Is,V c f (xg) + It,V c Is,V f (xg) + It,V Is f (xg)
g
g
x
x
x g
= lim V ↓{e} It,V
c Is,V c f (xg) = lim V ↓{e} Is,V c It,V c f (xg) = Is It f (xg).
g
x
c
The above interchange of It,V
c and Is,V c is easily justified because m(T, V ) < ∞. This shows
Itg Isx f (xg) = Isx Itg f (xg), and completes our proof of (7.51).
Let gt,T be the time reversed right process defined earlier with generator Lr (t). Assume it
is independent of process xt . Let V be a relatively compact open neighborhood of e such that
ϕi = 0 on V c . Write L(t) = LV (t) + L̃V (t), where LV (t) is the L(t) with the measure m(t, ·)
∫
restricted to V and L̃V (t) = V c [ry − 1]m(t, dy). Because for w ∈ [0, T ], LV (w)f ∈ Cc∞ (G),
∫
LxV (w)f (xgt,T ) − tT Lg (s)LxV (w)f (xgs,T )ds is a time reversed martingale in t. However, this
time-reversed martingale property may not hold when LxU is replaced by Lx because L̃V (r)f
may not have a compact support. We will show below that the time-reversed martingale
property also holds for Lx when gt,T is suitably stopped in the reversed time.
Let
τUg = sup{t ∈ [0, T ); gt,T ∈ U c or gt+,T ∈ U c } (sup ∅ = 0).
g
}-stopping time and gt,T ∈ U for t > τUg . We will write τ for τUg in
Then τUg is a {Ft,T
∫
this paragraph. Because f (xgt,T ) − tT Lg (s)f (xgs,T )ds is a time-reversed martingale, so is
∫T
the stopped process f (xgt∨τ,T ) − t∨τ
Lg (s)f (xgs,T )ds. Note that Lg (s)f (xgs,T ) is bounded
for x ∈ G and s > τ because (7.49) is bounded for g ∈ U . Replacing x by xy in the
stopped process and then integrating over y ∈ V c by m(t, dy) show that L̃xV (w)f (xgt∨τ,T ) −
∫T
g
x
t∨τ L (s)L̃V (w)f (xgs,T )ds is a time-reversed martingale. This implies that
∫
Mtg,w (x) = Lx (w)f (xgt∨τ,T ) −
T
t∨τ
Lg (s)Lx (w)f (xgs,T )ds
(7.52)
is a time-reversed martingale, where τ = τUg .
Similarly, let
τUx = inf{t ∈ (0, T ]; xt ∈ U c or xt− ∈ U c }
(inf ∅ = T ).
Then τ = τUx is a stopping time under {Ftx }, and
Mtx,w (g) = Lg (w)f (xt∧τ g) −
∫
t∧τ
0
Lx (s)Lg (w)f (xs g)ds
(7.53)
is a martingale, where τ = τUx .
For 0 < s < t < T , let α = s ∨ τUg and β = (t ∧ τUx ) ∨ α. Then α ≤ β. Because processes
xt and gt,T are independent, α and β may be regarded as stopping times under both Ftx and
234
g
Ft,T
. In the sequel, we will write P x for the probability distribution of process xt and E x
for the expectation taken under Px . Similarly P g and E g are the probability distribution
and the expectation of process gt,T . Note that almost surely, α = s and β = t when U is
sufficiently large, and hence
u(t, xt ) = lim E g [f (xβ gβ,T )]
U ↑G
u(s, xs ) = lim E g [f (xα gα,T ].
and
U ↑G
(7.54)
We have, for 0 ≤ s < t < T ,
E x {E g [f (xβ gβ,T )] − E g [f (xα gα,T )] | Fsx }
(7.55)
= E x {E g [f (xβ gβ,T ) − f (xβ gα,T ) + f (xβ gα,T ) − f (xα gα,T )] | Fsx }
= E x {E g [−
∫
β
α
∫
Lg (w)f (xβ gw,T )dw] | Fsx } + E g {E x [
β
α
Lx (w)f (xw gα,T )dw] | Fsx ]}
(by the time-reversed martingale property for gt,T and the martingale property for xt , noting
∫
that if α < w < β, then τUg < w < τUx , and hence the integrands of αβ above are bounded)
= −E g {E x {
∫
+E x {E g {
β
[Lg (w)f (xβ gw,T ) − Lg (w)f (xw gw,T )]dw | Fsx }}
α
∫
β
α
[Lx (w)f (xw gα,T ) − Lx (w)f (xw gw,T )]dw} | Fsx },
(7.56)
because Lx (w)f (xg) = Lg (w)f (xg). By (7.53), noting β ≤ τUx unless when β = α, the first
term in (7.56) is
−E g {E x {
∫
β
α
∫
β
w
∫
Lx (r)Lg (w)f (xr gw,T )drdw +
β
α
[Mβx,w (gw,T ) − Mwx,w (gw,T )]dw | Fsx }}.
Because under Px , α is a constant ≥ s,
E x{
∫
β
α
∫
[Mβx,w (gw,T )−Mwx,w (gw,T )]dw | Fsx } = E x {
T
α
x,w
[Mβx,w (gw,T )−Mw∧β
(gw,T )]dw | Fsx } = 0.
This shows that the first term in (7.56) is equal to
∫
−E g {E x [
β
∫
α
β
w
Lx (r)Lg (w)f (xr gw,T )drdw | Fsx ]}.
(7.57)
Let β ′ = t ∧ τUx and α′ = (s ∨ τUg ) ∧ β ′ . Then α′ = α and β ′ = β if either α < β or
∫
∫ ′
α′ < β ′ . Therefore, the integral αβ in (7.56) may be understood as αβ′ . Since under P g , β ′
is a constant, the argument used for the first term in (7.56) can be easily modified to show
the second term is equal to
∫
E {E [
g
x
β
α
∫
w
α
Lg (r)Lx (w)f (xw gr,T )drdw | Fsx ]}.
235
(7.58)
∫ ∫
∫ ∫
Because Lx (r)Lg (w)f (xg) = Lg (w)Lx (r)f (xg), the two double integrals αβ wβ and αβ αw in
(7.57) and (7.58) are equal after a simple change of integration order. It follows that the two
terms in (7.56) cancel out, and hence,
E x {E g [f (xβ gβ,T )] − Eg [f (xα gα,T )] | Fsx } = 0.
By (7.54), let U ↑ G, we obtain Ex [u(t, xt ) − u(s, xs ) | Fsx ] = 0. This proves that u(t, xt ) is
a martingale under Ftx . 2
Let xt and x′t be two rcll processes in G, x0 = x′0 = e, represented by the same (b, A, η).
Then after the time change by ψ in (6.42), they have the same generator L(t). By Lemma 7.20, for f ∈ Cc∞ (G) and any T > 0,
E[f (xT )] = E[u(T, xT )] = u(0, e) = E[u(T, x′T )] = E[f (x′T )].
This shows that the two processes xt and x′t have the same one-dimensional distributions.
We will employ a rather standard method to derive the same distribution from the same onedimensional distribution, under a martingale property, as in the proof of [22, 4.Theorem 4.2].
We will show
E[f1 (xt1 )f2 (xt2 ) · · · fn (xtn )] = E[f1 (x′t1 )f2 (x′t2 ) · · · fn (x′tn )]
(7.59)
for any t1 < t2 < · · · < tn and f1 , f2 , . . . , fn ∈ Cc∞ (G). Using induction, we assume it holds
for n = m and proceed to show that it holds for n = m + 1. We may assume fi (x) ≥ 0.
Define two probability measures Q and Q′ by
Q(ξ) = P [ξ
m
∏
fi (xti )]/P [
i=1
m
∏
fi (xti )]
and
Q′ (ξ) = P [ξ
i=1
m
∏
fi (x′ti )]/P [
i=1
m
∏
fi (x′ti )]
i=1
∏
∏
m
′
for any bounded real valued random variable ξ, assuming P [ m
i=1 fi (xti )] = P [ i=1 fi (xti )] >
0. Because xt has the martingale property with generator L(t), for s < t, f ∈ Cc∞ (G) and
an Ftxm +s -measurable bounded random variable η,
P {[f (xtm +t ) − f (xtm +s ) −
∫
t
s
L(tm + u)f (xtm +u )du]η
m
∏
fi (xti )} = 0.
i=1
We may replaced f (x) above by f ′ (x) = f (x−1
tm x). Because the operator L(t) is left invariant,
it follows that
P {[f (x−1
tm xtm +t )
−
f (x−1
tm xtm +s )
−
∫
t
s
L(tm +
u)f (x−1
tm xtm +u )du]η
m
∏
fi (xti )} = 0.
i=1
This implies that x̃t = x−1
tm xtm +t has the martingale property with generator L̃(t) = L(tm + t)
′
under the probability measure Q. Similarly, x̃′t = x−1
tm xtm +t has the martingale property with
236
the same generator L̃(t) under the probability measure Q′ . Then x̃t under Q and x̃′t under
Q′ have the same one-dimensional distribution, and hence
P[
m
∏
fi (xti )fm+1 (x−1
tm xtm+1 )] = Q[fm+1 (x̃tm+1 )]P [
i=1
m
∏
fi (xti )] = Q′ [fm+1 (x̃′tm+1 )]P [
i=1
= P[
m
∏
m
∏
fi (x′ti )]
i=1
′
fi (x′ti )fm+1 (x′−1
tm xtm+1 )].
i=1
∏m+1
∏m+1
This implies P [ i=1 fi (xti )] = P [ i=1 fi (x′ti )], and hence proves the uniqueness of the
process xt in distribution, and hence completes the proof of Lemma 7.19.
Proof of Theorem 6.9: Let xt be a stochastically continuous inhomogeneous Lévy process
in G with x0 = e. By Lemma 7.10, it is represented by an extended Lévy triple (b, A, η) with η
being its jump intensity measure. The stochastic continuity implies that η(t, ·) is continuous
in t, so (b, A, η) is a Lévy triple. By Lemma 6.32, the Lévy triple (b, A, η) is unique. This
proves the first half of Theorem 6.9.
Now let (b, A, η) be a Lévy triple. By Lemma 7.11, there is an inhomogeneous Lévy process
xt with x0 = e represented by (b, A, η). By Lemma 6.32, (b, A, η) is unique for xt , then by
Lemma 7.10, η is the jump intensity measure of xt . This implies that xt is stochastically
continuous. Let xt = zt bt . By Lemma 6.19, zt has the (b̄, Ā, η̄, δˆe ))-martingale property. This
means that zt is represented by the Lévy triple (b̄, Ā, η̄). Because b̄t has a finite variation,
by Lemma 7.19, the rcll process represented by (b̄, Ā, η̄) is unique in distribution. Therefore,
xt is unique in distribution as a rcll processes in G with x0 = e represented by (b, A, η).
Theorem 6.9 is proved. 2
7.5
Proof of Theorem 6.11
Recall that for any neighborhood U of e, and a stopping time σ, the U c -displacement time
τUσ of process xt from time σ is defined by (7.21). The following lemma generalizes the weak
compactness criterion in Lemma 7.8.
Lemma 7.21 Let xnt be a sequence of rcll processes in G as in Lemma 7.8 and un > 0.
For each n, let xn,m
be the process obtained from xnt when possible fixed jumps at times
t
u1 , u2 , . . . , um are removed according to (6.25). If for any T > 0, η > 0 and neighborhood U
of e, there are δ > 0 and an integer m ≥ 1 such that
lim sup P (τUσ,n,m < δ) ≤ η,
n→∞ σ≤T
(7.60)
where supσ≤T is taken over all stopping times σ ≤ T and τUσ,n,m is τUσ for process xn,m
, and
t
n
if (7.23) also holds, then a subsequence of xt converges weakly in D(G).
237
Proof Using the notation in the proof of Lemma 7.8, let τiε,n,m be the ε-displacement times
τiε for process xn,m
. By (7.29) and (7.60), for any η and ε > 0, there are δ and m such that
t
ε,n,m
limn→∞ P [min{τi+1
− τiε,n,m ; τiε,n,m < T } ≤ δ] ≤ η.
i≥0
Then by (7.27), limn→∞ P [w′ (xn,m , δ, T ) > 2ε] ≤ η.
The computation of w′ (x, δ, T ) in (7.24) is based on the oscillation of x(t), called xoscillations, over partitions that cover [0, T ] with spacing > δ, called δ-partitions. Let
Jm = {u1 , . . . , um }. We may assume
δ < min{u; u ∈ Jm } and δ <
1
min{|u − v|; u, v ∈ Jm with u ̸= v},
2
and either T ∈ Jm or |T − u| > δ for u ∈ Jm . Suppose there is a δ-partition with xn,m oscillation ≤ 4ε. If an interval of this δ-partition contains more than one point in Jm , then
by the condition imposed on δ, we may divide this interval into two or more intervals so that
the resulting partition is still a δ-partition with xn,m -oscillation ≤ 4ε, and any interval of
this δ-partition contains at most one point in Jm . Because adding more partition points will
not increase oscillation, adding Jm to the δ-partition together with the midpoints of those
intervals that do not intercept Jm , and then suitably combining some intervals, we obtain
a (δ/2)-partition containing Jm , with xn,m -oscillation ≤ 8ε. However, for this partition,
the xn,m -oscillation is just the xn -oscillation. This shows that if [w′ (xn , δ/2, T ) > 8ε], then
[w′ (xn,m , δ, T ) ≥ 4ε] ⊂ [w′ (xn,m , δ, T ) > 2ε]. It follows that limn→∞ P [w′ (xn , δ/2, T ) > 8ε] ≤
η. This verifies (7.26).
It remains to verify (7.25). If w′ (x, δ, T ) < ε, then there is a δ-partition {ti } with
oscillation < ε. This implies that τ1ε ̸∈ [0, t1 ) and τiε belong to different intervals of the
partition. Thus
ε
ε
w′ (x, δ, T ) < ε =⇒ min{τi+1
− τi−1
; τiε < T } > δ
i≥0
ε
(setting τ−1
= 0).
ε,n
ε,n
By (7.26) and above, limn→∞ P (An ) → 0 as δ → 0, where An = [mini≥0 {τi+1
− τi−1
; τiε,n <
T } < δ]. The rest of proof is very similar to the last part of the proof of Lemma 7.8. 2
The following lemma generalizes Lemma 7.9. It may be used to verify the condition
(7.60) in Lemma 7.21, just like Lemma 7.9 is used to verify (7.22) in Lemma 7.8.
Lemma 7.22 For any integers n, m ≥ 1 and f ∈ Cc∞ (G), let (t, x) 7→ Ftn,m (f, x) be a real
valued function on R+ × D(G) such that it is rcll in t for fixed x ∈ D(G) and is measurable
under σ{x(u); 0 ≤ u ≤ t} for fixed t ∈ R+ . Assume ∀ε > 0, ∃δ > 0 and m ≥ 1 such that
limn→∞ |Ftn,m (f, x) − Fsn,m (f, x)| < ε
238
for all s, t ∈ [0, T ] with |t − s| < δ and x ∈ D(G). Let xnt and xn,m
be as in Lemma 7.21
t
n,m
n,m
n,m
∞
such that f (xt ) − Ft (f, x· ) is an martingale for any f ∈ Cc (G). Then the condition
(7.60) in Lemma 7.21 holds.
Proof The lemma is proved in the same way as Lemma 7.9. 2
Let zt be an inhomogeneous Lévy process in G having the (b, A, η, ν)-martingale property
and let J = {u > 0; νu ̸= 0} = {u1 , u2 , . . .}. As in the proof of Lemma 6.31, we will write
the martingale in (6.28) as
Mt f = f (zt ) − St f (z; b, A, η) −
∑∫
u≤t G
[f (zu− x) − f (zu− )]νu (dx),
∫
∫
(7.61)
∫ ∫
where St f (z; b, A, η) is a sum of integrals 0t (· · ·)dbj (s), 0t (· · ·)dAjk (s) and 0t G (· · ·)η(ds, dx)
with bounded integrands. Moreover, the integrand of η(ds, dx)-integral is also controlled by
∥ϕ(x)∥2 for x near e. Because bi (t), Ajk (t) and η(t, ·) are continuous in t, one can show that
St f (z; b, A, η) is a bounded continuous function of z ∈ D(G) under the Skorohod metric.
Let Ss,t f (z; b, A, η) = St f (z; b, A, η) − Ss f (z; b, A, η) for s < t, and let zt1 be the process
zt when its jump at time u ∈ J is removed. Then for t ≥ v ≥ u,
Mt f − Mv f = f (zt ) − f (zv ) − Sv,t f (z; b, A, η) −
∑ ∫
v<s≤t G
[f (zs− x) − f (zs− )]νs (dx).
Replacing f (x) by f ′ (x) = f (zu− zu−1 x) in the above yields
Mt f ′ − Mv f ′ = f (zt1 ) − f (zv1 ) − Sv,t f (z 1 ; b, A, η) −
∑
1
1
[f (zs−
x) − f (zs−
)]νs (dx).
v<s≤t
Because E[Mt f ′ − Mv f ′ | Fvz ] = 0, it is then easy to show that Mt f in (7.61) is still a
martingale when zt and J are replaced by zt1 and J − {u} respectively.
For p < q, the jumps of zt at up+1 , up+2 , . . . , uq may be successively removed to obtain
a new process ztp,q such that Mt f is still a martingale with zt and J replaced by ztp,q and
J p,q = J − {up+1 , up+2 , . . . , uq }, that is, ztp,q has the (b, A, η, ν p,q )-martingale property with
ν p,q = {νu ; u = u1 , . . . , up , uq+1 , uq+2 , . . .}. Let Jp = {u1 , u2 , . . . , up }.
Lemma 7.23 The family of processes, ztp,q for p < q, are relatively weak compact in D(G).
Moreover, along a subsequence of n → ∞, ztp,n converges weakly to an inhomogeneous
Lévy process ztp that has the (b, A, η, ν p )-martingale property with ν p = {νu ; u ∈ Jp }.
Proof Let ztn = ztpn ,qn , where pn < qn and qn → ∞ as n → ∞. Recall St f (z; b, A, η) in
(7.61) is a bounded continuous function of z on D(G) under the Skorohod metric. We can
verify the condition (7.60) in Lemma 7.21 for xnt = ztn using Lemma 7.22 with
Ftn,m (f, x) = St f (z; b, A, η) +
∑ ∫
′
u≤t
G
239
[f (x(u−)y) − f (x(u−))]νu (dy),
∑
where the summation ′u≤t is taken over all u ∈ Jp,q − Jm with u ≤ t, p = pn and q = qn ,
which can be arbitrarily small when m is large. Because the jumps of ztn are those of zt ,
n −1 n
−1
−1
sup sup P [(zσ−
) zσ ∈ K c ] ≤ sup P [zσ−
zσ ∈ K c ] ≤ P [zt−
zt ∈ K c for some t ≤ T ] ↓ 0
n
σ≤T
σ≤T
as compact K ↑ G by the rcll property of zt . This verifies condition (7.23). Now Lemma 7.21
may be applied to show the weak convergence of a subsequence of ztn in Skorohod metric to
a rcll process zt′ in G. This proves that the family of the processes ztp,q , p < q, are relatively
weak compact in D(G).
Let ztp be the weak limit of ztp,n as n → ∞ (along a subsequence). By Lemma 7.14, ztp
is an inhomogeneous Lévy process. We now show that for u ∈ Jp , νu is the distribution of
p −1 p
(zu−
) zu . For ε > 0, choose ϕε ∈ Cb (R) such that
0 ≤ ϕε ≤ 1,
ϕε (u) = 1 and ϕε (t) = 0 for |t − u| > ε.
For any f ∈ Cb (G) with 0 ≤ f ≤ 1 and f = 0 near e, let
Fε (x) = [
∑
ϕε (t)f (x(t−)−1 x(t))] ∧ 2
t>0
for x ∈ D(G). Then Fε (x) is a bounded continuous function on D(G), and so E[Fε (z·p,n )] →
p,n −1 p,n
−1
E[Fε (z· )] as n → ∞. We have E[Fε (z· )] → E[f (zu−
) zu )] =
zu )] as ε → 0. As E[f ((zu−
νu (f ), it suffices to show
p,n −1 p,n
) zu )] → 0 as ε → 0 uniformly in n.
r(n, ε) = E[Fε (z·p,n )] − E[f ((zu−
Because zt have independent fixed jumps and rcll paths, and supp(f ) does not contain
∑
e, by Borel-Cantelli Lemma, |t−u|<ε νt (f ) < ∞. Because |Fε (x) − f (x(u−)−1 x(u))| ≤
∑
∑
−1
|t−u|<ε, t̸=u f (x(t−) x(t)) for x ∈ D(G), it follows that |r(n, ε)| ≤
t̸=u, |t−u|<ε νt (f ) → 0
p −1 p
as ε → 0. We have proved that νu is the distribution of (zu− ) zu for u ∈ Jp .
We now prove that ztp has the (b, A, η, ν p )-martingale property, that is,
Mtp f = f (ztp ) − St f (z p ; b, A, η) +
∑
u≤t
∫
(p)
G
p
p
[f (zu−
y) − f (zu−
)]νu (dy)
∑(p)
is a martingale for any f ∈ Cc∞ (G), where the summation u≤t is taken over all u ∈ Jp
∑(p)
with u ≤ t. Let Mtp,n f be Mtp f when ztp and u≤t are replaced, respectively, by ztp,n
∑(p,n)
and the summation u≤t over all u ∈ Jp ∪ {un+1 , un+2 , . . .} with u ≤ t. Because ztp,n
has the (b, A, η, ν p,n )-martingale property, Mtp,n f is a martingale. To prove that Mtp f is a
martingale, it suffices to show that for any t1 < t2 < · · · < tk ≤ s < t and h ∈ Cc∞ (Gk ), with
F (x) = h(x(t1 ), x(t2 ), . . . , x(tk )) for x ∈ D(G), E[F (z·p )(Mtp f − Msp f )] = 0. By the right
240
continuity of paths of ztp , we just have to prove this under the assumption that ti , 1 ≤ i ≤ k,
are not fixed jump times of ztp . It suffices to show
E[F (z·p,n )(Mtp,n f − Msp,n f )] → E[F (z·p )(Mtp f − Msp f )] as n → ∞.
When ti are not fixed jump times of ztp , under the probability distribution of the process
ztp on D(G), almost all x in D(G) are continuous points of F (x). As mentioned earlier, St f (z; b, A, η) in (7.61) is a bounded continuous function on D(G). By Lemma 7.15,
E[F (z·p,n )Ss,t f (z n,p ; b, A, η)] → E[F (z·p )Ss,t f (z p ; b, A, η)] as n → ∞. Because by (6.27),
∫
∑
u≤t, u∈J(>n)
G
p,n
p,n
[f (zu−
y) − f (zu−
)]νu (dy) → 0
uniformly as n → ∞, where J(> n) = {un+1 , un+2 , . . .}, it suffices to show that for any
u ∈ Jp with u > s, as n → ∞,
∫
p,n
p,n
E{F (z·p,n ) [f (zu−
y) − f (zu−
)]νu (dy)} → E{F (z·p )
∫
p
p
[f (zu−
y) − f (zu−
)]νu (dy)}. (7.62)
We may assume F and f in (7.62) are bounded by 1 in absolute values. Choose ϕε ∈ Cb (R)
as before, and choose ψ ∈ Cb (G) with 0 ≤ ψ ≤ 1 on G and ψ = 0 near e. Let
Φε (x) = F (x)
∑
ϕε (t)[f (x(t)) − f (x(t−))]ψ(x(t−)−1 x(t))
t>0
for x ∈ D(G), and let Φ̃ε (x) = [Φε (x) ∧ 3] ∨ (−3). Then Φ̃ε (x) is a bounded continuous
function on D(G), and Φ̃ε (x) → F (x)[f (x(u)) − f (x(u−))]ψ(x(u−)−1 x(u)) as ε → 0 for
x ∈ D(G). We have E[Φ̃ε (z·p,n )] → E[Φ̃ε (z·p )] as n → ∞,
p −1 p
p
) zu )} (as ε → 0)
)]ψ((zu−
E[Φ̃ε (z·p )] → E{F (z·p )[f (zup ) − f (zu−
∫
=
E{F (z·p )
and the above holds also when
z·p
G
p
p
)]ψ(y)νu (dy)},
y) − f (zu−
[f (zu−
is replaced by z·p,n . Because for x ∈ D(G),
|Φ̃ε (x) − F (x)[f (x(u)) − f (x(u−))]ψ(x(u−)−1 x(u))|
∑
≤
|f (x(t)) − f (x(t−))|ψ(x(t−)−1 x(t)) ≤ 2
0<|t−u|<ε
∑
ψ(x(t−)−1 x(t)),
0<|t−u|<ε
and hence
p,n
p,n −1 p,n
|E[Φ̃ε (z·p,n )] − E{F (z·p,n )[f (zup,n ) − f (zu−
)]ψ((zu−
) zu )| ≤ 2
∑
νt (ψ),
0<|t−u|<ε
which converges to 0 as ε → 0 uniformly in n, it follows that
∫
E{F (z·p,n )
G
p,n
p,n
[f (zu−
y) − f (zu−
))]ψ(y)νu (dy)}
→
∫
E{F (z·p )
G
p
p
[f (zu−
y) − f (zu−
))]ψ(y)νu (dy)}
as n → ∞. For any neighborhood U of e, we may choose ψ ∈ Cb (G) with 0 ≤ ψ ≤ 1 on G such
∫
p,n
p,n
that ψ = 1 on U c and ψ = 0 near e. Then |E{F (z·p,n ) U [f (zu−
y) − f (zu−
))]ψ(y)νu (dy)}| ≤
2νu (U − {e}) → 0 as U ↓ {e} uniformly in n. This implies (7.62). 2
241
Lemma 7.24 The distribution of an inhomogeneous Lévy process zt in G with z0 = e having
the martingale property under a given quadruple (b, A, η, ν) is unique.
Proof Because ztp in Lemma 7.23 has only finitely many fixed jumps, after removing these
fixed jumps, it becomes stochastically continuous. By Lemma 7.19, the distribution of ztp is
completely determined by (b, A, η, ν p ), and hence also by (b, A, η, ν).
By Lemma 7.23, the processes xp,q
t , p < q, are weakly compact on D(G). Then by [22,
3.Remark 7.3], for any ε > 0, there is a compact Kε ⊂ G such that
∀ p < q,
P (zsp,q ∈ Kε for s ≤ t) ≥ 1 − ε.
(7.63)
Let f ∈ Cc∞ (G) with |f | ≤ 1. We will show that for any ε > 0, there exist a constant
cε = c(ε, f, t) > 0 and a small neighborhood U of e such that
∀ p < q,
|E[f (zt )] − E[f (ztp,q )]| ≤ cε
q
d
∑
∑
[
|νui (ϕj )| + νui (∥ϕ∥2 ) + νui (U c )] + 2ε. (7.64)
i=p+1 j=1
Recall ztp,q is obtained from zt after the fixed jumps at times up+1 , up+2 , . . . , uq are removed. Let v1 < v2 < · · · < vq−p be the ordered values of these time points and let m ≤ q − p
be the largest integer such that vm ≤ t. Let σi = zv−1
z and for u < v, let zu,v = zu−1 zv−
i − vi
∏
p,q
p,q
p,q
and zu,v
= (zup,q )−1 zv−
. For xi ∈ G, let ni=1 xi = x1 x2 · · · xn . Set v0 = 0 and zu,u = zu,u
= e,
p,q
m
and let H = ∩i=1 [z0,vi ∈ Kε ]. By (7.63), P (H) ≥ 1 − ε. For simplicity, assume zt = zt− .
m
∑
p,q
p,q
z )]
σ z ) − f (z0,v
[f (z0,v
i vi ,t
i i vi ,t
p,q
p,q
z )=
σ z ) − f (z0,v
f (zt ) − f (ztp,q ) = f (z0,v
m vm ,t
1 1 v1 ,t
i=1
=
m
∑
[f (z ′ σz) − f (z ′ z)]
i=1
=
m
∑
[f (z ′ e
i=1
=
=
m
∑
{
j
ϕj (σ)ξj
z) − f (z ′ z)]1[σ∈U ] +
m
∑
[f (z ′ σz) − f (z ′ z)]1[σ∈U c ]
i=1
∑
[f (e
i=1
m
∑
∑
p,q
(z ′ = z0,v
, z = zvi ,t and σ = σi )
i
j,k
[Ad(z ′ )]kj ϕj (σ)ξk ′
z z) − f (z ′ z)]1[σ∈U ] +
m
∑
[f (z ′ σz) − f (z ′ z)]1[σ∈U c ]
i=1
d
∑
ξkr f (z ′ z)[Ad(z ′ )]kj ϕj (σ) + O(∥ϕ(σ)∥2 )}1[σ∈U ] +
i=1 j,k=1
m
∑
[f (z ′ σz) − f (z ′ z)]1[σ∈U c ]
i=1
by Taylor’s formula, where ξ r f (z) = dtd f (etξ z) |t=0 . Note that ztp,q and H are independent of
σi , and on H, z ′ ∈ Kε and hence [Ad(z ′ )]kj are bounded. It follows that
|E[f (zt ) − f (ztp,q )]| ≤ |E{[f (zt ) − f (ztp,q )]1H }| + 2ε
≤ c′ε
m ∑
∑
[
i=1
|νvi (ϕj 1U )| + O(∥ϕ∥2 ) + νvi (U c )] + 2ε
j
for some constant c′ε > 0. This proves (7.64) because |νu (ϕj 1U )| ≤ |νu (ϕj )|+(supG |ϕj |)νu (U c ).
242
By (6.27), letting first q → ∞ and then p → ∞ in (7.64) yields E[f (zt )] − E[f (ztp )] → 0.
Because E[f (ztp )] is determined by (b, A, η, ν), so is E[f (zt )]. This proves the uniqueness of
the one-dimensional distributions of an inhomogeneous Lévy process zt having the (b, A, η, ν)martingale property. To complete the proof, one just need to repeated the argument that
proves (7.59), to show that the same one-dimensional distributions and the martingale property imply the same of multi-dimensional distributions, noting that we may assume z0 has
an arbitrary distribution because z0 and z0−1 zt are independent. 2
By Lemmas 6.19 and 7.24, we obtain the unique distribution for the process xt in the
second half of Theorem 6.11.
Corollary 7.25 The distribution of an inhomogeneous Lévy process xt in G with x0 = e
represented by a possibly improper extended Lévy triple (b, A, η) is unique.
We are now ready to prove that given a quadruple (b, A, η, ν), there is an inhomogeneous Lévy process zt in G, with z0 = e, having the (b, A, η, ν)-martingale property. By
Theorem 6.9, there is a stochastically continuous inhomogeneous Lévy process zt′ represented
by the Lévy triple (b, A, η), and then by Theorem 6.27, whose proof does not depend on
Theorem 6.11, zt′ has the (b, A, η, 0)-martingale property. This means that for f ∈ Cc∞ (G),
f (zt′ ) − St f (z ′ ; b, A, η) is a martingale under the natural filtration of zt′ .
Let J = {u1 , u2 , u3 , . . .} be the set of time points u > 0 such that νu ̸= 0, and let {vu ,
u ∈ J} be a family of G-valued random variables with distributions νu such that they are
mutually independent, and independent of process zt′ . For u = u1 , and let zt1 be the process
obtained from zt′ when the fixed jump vu is added at time u, that is, zt1 = zt′ for t < u and
′
zt1 = zu−
vu zu′−1 zt′ for t ≥ u. Then it is easy to show that
f (zt1 ) − St f (z 1 ; b, A, η) − 1[u≤t]
∫
G
1
1
[f (zu−
x) − f (zu−
)]νu (dx)
is a martingale under the natural filtration of zt1 .
Let ztn be the process obtained from zt′ when the fixed jumps vu1 , vu2 , . . . , vun are successively added at times u1 , u2 , . . . , un . Then
f (ztn ) − St f (z n ; b, A, η) −
∑
u≤t, u∈Jn
∫
G
n
n
)]νu (dx)
x) − f (zu−
[f (zu−
(7.65)
is a martingale under the natural filtration of ztn , where Jn = {u1 , u2 , . . . , un }.
As in the proof of Lemma 7.23, we may use Lemmas 7.21 and 7.22 to show that a
subsequence of ztn converge weakly to a rcll process zt in D(G). The argument used in the
proof of Lemma 7.23 to show that Mtp f is a martingale can be suitably modified to prove
(7.65) is still a martingale when ztn and Jn are replaced by zt and J. This shows that zt has
243
the (b, A, η, ν)-martingale property. Because ztn is obtained by adding n independent fixed
jumps to the inhomogeneous Lévy process zt′ , so ztn is an inhomogeneous Lévy process. The
same is true for zt as the weak limit of ztn . By the uniqueness of distribution for the process
zt given in Lemma 7.24, the whole sequence converges weakly to zt . To summarized, we have
obtained the following result.
Lemma 7.26 Given a quadruple (b, A, η, ν), there is an inhomogeneous Lévy process zt in
G, with z0 = e, having the (b, A, η, ν)-martingale property. Moreover, it is the weak limit in
D(G) of the processes ztn defined above.
Proof of Theorem 6.11: The first half of Theorem 6.11, except the uniqueness of the
extended Lévy triple (b, A, η), is given Lemma 7.10. The uniqueness of (b, A, η) is given in
Lemma 6.32. The uniqueness in distribution for an inhomogeneous Lévy process represented
by an extended Lévy triple (b, A, η), possibly improper, is given in Corollary 7.25.
It remains to prove that given a possibly improper extended Lévy triple (b, A, η), there
is an inhomogeneous Lévy process xt in G represented by (b, A, η). By Lemma 6.19, with
xt = zt bt , it is enough to show that zt has the (b̄, Ā, η̄, ν̄)-martingale property. This follows
from Lemma 7.26. Theorem 6.11 is now completely proved.
244
Chapter 8
Inhomogenous Lévy processes in
homogeneous spaces
The martingale representation of inhomogeneous Lévy processes in a Lie group G in terms
of a Lévy triple, or an extended Lévy triple, (b, A, η), obtained in Chapter 6, is extended
to a homogeneous space X = G/K in §8.1. The results are similar, but require a careful
formulation of a product and certain K-invariant structures on G/K so that the formulae
obtained on G, and their proofs (in §8.4), may be carried over to G/K. In §8.2, two special
cases are considered. The first case is when the drift, or the extended drift, bt on G/K
has a finite variation, then a direct martingale representation as in Theorem 6.27 on G may
be obtained. The second case is an irreducible G/K, such as a sphere, the representation
takes an especially simple form because then the drift bt is trivial and the covariance matrix
function A(t) is proportional to the identity matrix.
8.1
Martingale representation on homogeneous spaces
Let K be a compact subgroup of a Lie group G. As Before, let e be the identity element
of G, π: G → G/K be the natural projection, and o = π(e). Recall that a G-invariant
inhomogeneous Markov process xt in the homogeneous space X = G/K, with rcll paths
and an infinite life time, is called an inhomogeneous Lévy process. By Theorem 1.24, a rcll
process in X is an inhomogeneous Lévy process if and only if it has independent increments
−1
x
in the sense that for s < t, x−1
s xt = S(xs ) xt is independent of Fs , and its distribution µs,t
does not depend on the choice of the section map S. Recall Ftx is the natural filtration of
xt , and a section map S is a Borel map: X → G such that π ◦ S = idX .
A point b ∈ X is called K-invariant if kb = b for all k ∈ K. For x ∈ X and K-invariant
b, the product xb = S(x)b ∈ X is well defined, not dependent on the section map S. As in
245
§3.1, let p be an Ad(K)-invariant subspace of the Lie algebra g of G that is complementary
to the Lie algebra k of K. Let pK = {ξ ∈ p; Ad(k)ξ = ξ for k ∈ K}, a subspace of p.
Proposition 8.1 Let N be the set of g ∈ G such that go = π(g) are K-invariant. Then N
is the normalizer of K, that is, N = {g ∈ G; gKg −1 = K}, which is a closed subgroup of G.
Moreover, the Lie algebra of N is k ⊕ pK .
Proof Note that g ∈ G with go being K-invariant is characterized by g −1 Kg ⊂ K. This
implies that g −1 Kg and K have the same identity component K0 = g −1 K0 g. Because K is
compact, its coset decomposition K = K0 ∪ [∪pi=1 (ki K0 )] is a finite union. Since g −1 Kg =
K0 ∪ [∪pi=1 (g −1 ki gK0 )] is the coset decomposition of g −1 Kg, it follows that g −1 Kg = K.
To show the Lie algebra n of N is k ⊕ pK , we first check that k ⊕ pK is a Lie algebra.
For η ∈ k and ζ ∈ pK , [η, ζ] = (d/dt)Ad(etη )ζ |t=0 = 0 because Ad(K)-invariance of ζ. For
ζ ′ ∈ pK , [ζ, ζ ′ ] is Ad(K)-invariant, and so is its p-component. This shows [ζ, ζ ′ ] ∈ k ⊕ pK ,
and proves that k ⊕ pK is a Lie sub-algebra of g.
It is easy to see that k ⊕ pK is a subspace of n. Let ξ ∈ n with ξ = η + ζ for η ∈ k and
ζ ∈ p. For k ∈ K and t close to 0, etAd(k)ξ = ketξ k −1 = etξ kt for some kt ∈ K. Because k0 = e
and kt is smooth in t, kt = etθ+o(t) for some θ ∈ k. Because Ad(k)η ∈ k and Ad(k)ζ ∈ p, and
etξ etθ+o(t) = et(ξ+θ)+o(t) , it follows that Ad(k)ζ = ζ, and hence ξ ∈ k ⊕ pK . 2
By Proposition 8.1, the set Λ of K-invariant points in X has a natural group structure
with product bb′ = S(b)b′ and inverse b−1 = S(b)−1 o, not dependent on the choice of the
section map S. In fact, S(b) here may be replaced by any g ∈ G with go = b. Note that
Λ is naturally identified with the quotient Lie group N/K, which is naturally embedded in
X = G/K, and its Lie algebra may be identified with pK with Lie bracket given by [ξ, ζ]p
(p-component of [ξ, ζ]) for ξ, ζ ∈ pK .
A measure function η(t, ·), t ≥ 0, on X is defined just as a measure function on G, that
is, as a nondecreasing and right continuous family of σ-finite measures on X. It is called
K-invariant if η(t, ·) is K-invariant for each t ≥ 0. As on G, a measure function η(t, ·) on
X may be regarded as a measure η on R+ × X, with σ-finite η(t, ·) = η([0, t] × ·) for all
t ∈ R+ , and it may be written as a sum of its continuous part η c (t, ·) and its discontinuous
∑
part η d (t, ·) = s≤t η({s} × ·) as in (6.12).
Let xt be an inhomogeneous Lévy process in X with J being the set of its fixed jump
times. Let ∆n : 0 = tn0 < tn1 < tn2 < · · · < tni ↑ ∞ as i ↑ ∞ be a sequence of partitions of
R+ with mesh ∥∆n ∥ → 0 as n → ∞, and assume ∆n contains J as n → ∞ in the sense that
any u ∈ J belongs to ∆n when n is sufficiently large. Let µni be the distribution of x−1
tn i−1 xtni
for i ≥ 1. Define
∑
µni ,
(8.1)
ηn (t, ·) =
tni ≤t
246
setting ηn (0, ·) = 0. Then ηn is a K-invariant measure function on X.
−1
The proof of Proposition 6.5 may be repeated on X, regarding x−1
s xt as S(xs ) xt with
the choice of a section map S. Note that with a G-invariant metric r on X, r(x, y) =
r(o, S −1 (x)y), and hence S(xs )−1 xt ∈ U c for a neighborhood U of o if and only if r(xs , xt ) > δ
for some δ > 0. Moreover, for a K-invariant U , the event [S(xs )−1 xt ∈ U c ] does not depend
on the choice of S. Then for any t > 0 and any neighborhood U of o,
ηn (t, U c ) is bounded in n and ηn (t, U c ) ↓ 0 as U ↑ X uniformly in n.
(8.2)
∫
For f ∈ Cb (X), let fˆ = K dk(f ◦ k), where dk is the normalized Haar measure on K.
Then fˆ is K-invariant, that is, fˆ ◦ k = fˆ for k ∈ K, and for a K-invariant measure µ on
X, µ(f ) = µ(fˆ). We note that to show the weak convergence of a sequence of K-invariant
measures µn on K, it suffices to show the convergence of µn (f ) for K-invariant f ∈ Cb (X).
Let f ∈ Cb (X) be K-invariant and vanish near o. Then the function (x, y) 7→ f (S(x)−1 y)
is not dependent on the choice of the section map S, and is continuous because S can be
chosen to be continuous near any point. We may write f (x−1 y) for f ((S(x)−1 y). The proof
of Proposition 6.7 may be repeated on X to show that as n → ∞,
ηn (t, f ) =
∑
tni ≤t
∑
E[f (x−1
tn i−1 xtni )] → E[
f (x−1
s− xs )].
s≤t
This shows that η(t, ·) defined by η(0, ·) = 0, η(t, {o}) = 0 and
∑
η(t, f ) = η(t, fˆ) = E[
fˆ(x−1
s− xs )]
for f ∈ Cb (X) vanishing near o,
(8.3)
s≤t
is a well defined K-invariant measure function on X, which will be called the jump intensity
measure of the inhomogeneous Lévy process xt . The following result is now obvious.
Proposition 8.2 η = 0 if and only if xt is continuous, and η(t, ·) is continuous in t if and
only if xt is stochastically continuous. Moreover, for any t > 0 and f ∈ Cb (X) vanishing
near o, ηn (t, f ) → η(t, f ) as n → ∞.
As in §3.1, let {ξ1 , . . . , ξn } be a basis of To X, the tangent space of X at the base point
o. By the identification of p with To X via Dπ, it may also be regarded as a basis of p.
Choose exponential coordinate functions ϕ1 , . . . , ϕn on X, which are functions in Bb ((X)
such that ϕi vanish outside a compact subset of X and are smooth near o, and satisfy (3.9)
∑
∑
near o and (3.10) on X, that is, x = exp[ i ϕi (x)ξi ]o for x near o, and ni=1 ϕi (x)[Ad(k)ξi ] =
∑n
i=1 ϕi (kx)ξi for x ∈ X and k ∈ K.
Similar to the definitions given on G, the local mean of a random variable x in X or its
distribution µ is defined as
b = exp[
n
∑
µ(ϕi )ξi ]o.
i=1
247
(8.4)
By (3.10), if µ is K-invariant, then so is b. We will call x or µ moderate if b has coordinates
µ(ϕi ), that is, ϕi (b) = µ(ϕi ) for 1 ≤ i ≤ n. This is the case when µ is sufficiently concentrated
near o.
Recall that an n × n real symmetric matrix aij is called Ad(K)-invariant if aij =
p,q apq [Ad(k)]ip [Ad(k)]jq , where [Ad(k)]ij is the matrix representing Ad(k): p → p under
∑
basis of {ξ1 , . . . , ξn }, that is, Ad(k)ξj = i [Ad(k)]ij ξi .
∑
A drift or an extended drift b, a covariance matrix function A, and a Lévy measure
function η or an extended Lévy measure function on X = G/K, are defined just as on G (see
§6.3), in terms of the coordinate functions ϕi on X, with the additional requirement that for
each t, bt and η(t, ·) are K-invariant, and A(t) is Ad(K)-invariant. With these modifications,
Lévy triples and extended Lévy triples are defined exactly as on G. An extended Lévy triple
∑n
(b, A, η) is called proper if b−1
t− bt = ht for t > 0, where ht = exp[ i=1 νt (ϕi )ξi ]o is the local
mean of the K-invariant probability measure
νt = η({t} × ·) + [1 − η({t} × X)]δo .
(8.5)
Otherwise, if b−1
t− bt = ht does not hold for some t > 0, then (b, A, η) is called improper.
However, we will always assume extended Lévy triples are proper unless when explicitly
stated otherwise. A Lévy triple (b, A, η) is clearly an extended Lévy triple.
Let (b, A, η) be an extended Lévy triple on X, possibly improper. For a rcll path zt in X
and f ∈ Cc∞ (X), let
∫
Tt f (z; b, A, η) =
∫
t∫
+
0
+
X
∑∫
s≤t X
t
0
n
1 ∑
[Ad(bs )ξj ][Ad(bs )ξk ]f (zs )dAjk (s)
2 j,k=1
{f (zs bs xb−1
s )
− f (zs ) −
n
∑
ϕj (x)[Ad(bs )ξj ]f (zs )}η c (ds, dx)
j=1
−1
[f (zs− bs− xh−1
s bs− ) − f (zs− )]νu (dx).
(8.6)
This is the same as Tt f (z; b, A, η) on G defined by (6.22) but with G replaced by X. However,
the expression in (8.6) requires some careful explanation. For simplicity, we use the same Tt
to denote an operator on G or on X, its meaning should be clear from the context.
The basic principle is to use a section map to lift points from X to G to perform some
group operations, such as multiplication and invariant differentiation, see the product convention on X introduced in §1.3 and G-invariant operators on X discussed in §3.1. One may
use only one section map to lift all points in X, or several section maps for different points
in X. It turns out that many useful expressions, such as the one in (8.6), are not dependent
on the choice of section maps. More details are given below.
248
The first integral in (8.6) should be understood as
∫
t
n
1 ∑
[Ad(bs )ξj ][Ad(bs )ξk ]f (zs )dAjk (s)
2 j,k=1
t
n
1 ∑
[Ad(S(bs ))ξj ][Ad(S(bs ))ξk ](f ◦ π)(S(zs ))dAjk (s)
2 j,k=1
0
∫
=
0
(8.7)
with the choice of a section map S, but by the Ad(K)-invariance of A(t) and the K-invariance
of bt , this integral does not depend on S. Note that for a K-invariant b ∈ X and any k ∈ K,
kS(b) = S(b)k ′ for some k ′ ∈ K because S(b) belongs to the normalizer of K.
The η c -integral integral in (8.6) should be understood as
∫
t
∫
0
∫
X
t
∫
=
0
−
X
n
∑
{f (zs bs xb−1
s ) − f (zs ) −
n
∑
ϕj (x)[Ad(bs )ξj ]f (zs )}η c (ds, dx)
j=1
{f (S(zs )S(bs )S(x)S(bs )−1 o) − f (zs )
ϕj (x)[Ad(S(bs ))ξj ](f ◦ π)(S(zs ))}η c (ds, dx).
(8.8)
j=1
By the K-invariance of η c (t, ·) and bt , and (3.10), this integral does not depend on the choice
for the section map S. We provide a little more details here. Suppose S ′ is another section
map. Then S ′ (zs ) = S(zs )k1 and S ′ (bs ) = S(bs )k2 for some k1 , k2 ∈ K. Because S(bs )
normalizes K, k1 S(bs ) = S(bs )k1′ for some k1′ ∈ K. We have
f (S ′ (zs )S ′ (bs )S ′ (x)S ′ (bs )−1 o) = f (S(zs )S(bs )S(k1′ k2 x)S(bs )−1 o),
n
∑
=
=
j=1
n
∑
j=1
n
∑
ϕj (x)[Ad(S ′ (bs ))ξj ](f ◦ π)(S ′ (zs )) =
n
∑
and
ϕj (x)[Ad(S(bs )k2 )ξj ](f ◦ π)(S(zs )k1 )
j=1
ϕj (x)[Ad(k1 S(bs )k2 )ξj ](f ◦ π)(S(zs )) =
n
∑
ϕj (x)[Ad(S(bs )k1′ k2 )ξj ](f ◦ π)(S(zs ))
j=1
ϕj (k1′ k2 x)[Ad(S(bs ))ξj ](f ◦ π)(S(zs )) (by (3.10)).
j=1
It then follows from the K-invariance of η c (t, ·) that the η c -integral does not depend on the
choice of the section map S. To see the integrability of this integral, note that when x is
contained in a small neighborhood U of o, f (S(zs )S(bs )S(x)S(bs )−1 o) is
∑
f (S(zs )S(bs )e
i
ϕi (x)ξi
S(bs )−1 o) = f (S(zs )e
∑
∑
i
ϕi (x)Ad(S(bs ))ξi
o).
By a Taylor expansion of ψ(t) = f (S(zs )et i ϕi (x)Ad(S(bs ))ξi o) at t = 0, we see that the
integrand of the η c -integral in (8.6) on U is bounded in absolute value by c∥ϕ(x)∥2 for some
constant c > 0, and hence it is absolutely integrable.
249
The νs -integrals in (8.6) should be understood as
∫
∫
X
=
X
−1
[f (zs− bs− xh−1
s bs− ) − f (zs− )]νs (dx)
[f (S(zs− )S(bs− )S(x)S(hs )−1 S(bs− )−1 o) − f (zs− )]νs (dx).
(8.9)
By the K-invariance of νs and bt , these integrals do not depend on the section map S. To
see the sum ∑
of these integrals is absolutely convergent, we may assume all νs are moderate.
Then hs = e i ϕi (hs )ξi o. By a Taylor expansion of
ψ(t) = f (S(zs− )S(bs− )et
∑
i
[ϕi (x)−ϕi (hs )]ξi +
∑
i
ϕi (hs )ξi
S(hs )−1 S(bs− )−1 o)
at t = 0, we see that the integrand of the νs -integral, on a small neighborhood of o, is
bounded in absolute value by c∥ϕ(x) − ϕ(hs )∥2 for some constant c > 0. This implies that
the sum of these νs -integrals is absolutely convergent.
To summarize, we record the following result.
Proposition 8.3 The expression in (8.6) is well defined as explained above, and for each
t ∈ R+ , it is bounded in absolute value by a finite constant for all rcll paths zt in X.
A rcll process xt in X = G/K is said to be represented by an extended Lévy triple (b, A, η)
on X, if xt = zt bt and for any f ∈ Cc∞ (X),
Mt f = f (zt ) − Tt f (z; b, A, η)
(8.10)
is a martingale under the natural filtration Ftx of xt .
Theorem 8.4 Let xt be an inhomogeneous Lévy process in X = G/K with x0 = o. Then
there is a unique extended Lévy triple (b, A, η) on X such that xt is represented by (b, A, η)
as defined above. Moreover, η(t, ·) is the jump intensity measure of process xt given by (8.3).
Consequently, xt is stochastically continuous if and only if (b, A, η) is a Lévy triple.
Conversely, given an extended Lévy triple (b, A, η) on X, there is an inhomogeneous
Lévy process xt in X with x0 = o, unique in distribution, represented by (b, A, η). Moreover, for a Lévy triple (b, A, η), the unique distribution holds among all rcll processes in X.
Theorem 8.4 may be proved in large part by essentially repeating the proof of Theorem 6.11 for the corresponding results on G, interpreting a product xy and an inverse x−1
on X = G/K as S(x)y and S(x)−1 with the choice of a section map S, and taking various
functions and sets to be K-invariant. See §8.4 for more details.
250
Remark 8.5 As in Remark 6.12 for the representation on G, as the jump intensity measure,
η(t, ·) in the first half of Theorem 8.4 does nor depend on the choice for the basis {ξj } of p
and coordinate functions ϕj on X = G/K, and by Proposition 6.33, which holds also on X,
∑
A(t) does not depend on {ϕj }, and so the operator nj,k=1 A(t)ξj ξk does not depend on {ξj }
and ϕj . However, η(t, ·) may depend on the choice for the origin o in X.
Recall from §1.4 that an inhomogeneous Lévy process gt in G is called K-conjugate invariant if its transition semigroup Ps,t is K-conjugate invariant. If g0 = e, then this is also
equivalent to the equality in distribution of the two processes gt and kgt k −1 for any k ∈ K.
The following result is an extension of Theorem 3.10 from Lévy processes to inhomogeneous
Lévy processes, its first half is proved as Theorem 1.29 and the second half will be proved
later.
Theorem 8.6 Let gt be a K-conjugate invariant inhomogeneous Lévy process in G with
g0 = e. Then xt = gt o is an inhomogeneous Lévy process in X = G/K. Conversely, if xt
is an inhomogeneous Lévy process in X with x0 = o, then there is a K-conjugate invariant
inhomogeneous Lévy process gt in G such that processes xt and gt o are equal in distribution.
See Proposition 8.21 and Remark 8.22 for how the representing triples for gt and xt = gt o
are related.
8.2
Finite variation and irreducibility
Recall that an Ad(K)-invariant ξ ∈ g may be regarded as a vector field on X = G/K defined
by ξf (x) = ξ(f ◦ π)(S(x)) for f ∈ C ∞ (X) and x ∈ X, with the choice of a section map S.
This definition does not depend on S.
Let b1 (t), . . . , bn (t) be real-valued continuous functions of finite variation and assume
∑n
∑
i=1 bi (t)ξi is Ad(K)-invariant in p, which implies exp( j bj (t)ξj )o is a K-invariant point of
∑
X, for any t ≥ 0. Then for b ∈ X and f ∈ Cc∞ (X), nj=1 bj (t)ξj f (b) is well defined, and so
is the integral
∫
∫
t
0
∑
t
ξj f (bs )dbj (s) =
j
0
∑
ξj (f ◦ π)(S(bs ))dbj (s)
(8.11)
j
for any rcll path bt in X, not dependent on the choice of the section map S.
Proposition 8.7 Let bt be an extended drift in X = G/K with finite variation. Then there
is a unique set of real-valued continuous functions bi (t) with bi (0) = 0 and finite variation,
∑
1 ≤ i ≤ n, such that for any t ≥ 0, i bi (t)ξi is Ad(K)-invariant, and for f ∈ Cc∞ (X),
∫
f (bt ) = f (o) +
n
t∑
0 i=1
ξi f (bs )dbi (s) +
∑
[f (bs ) − f (bs− )].
s≤t
251
(8.12)
Conversely, given a set of real-valued continuous functions bi (t) with finite variation and
∑
Ad(K)-invariant i bi (t)ξi , 1 ≤ i ≤ n, there is a unique drift bt in X of finite variation,
∑
such that (8.12) holds for f ∈ Cc∞ (G) (with s≤t [· · ·] = 0).
Proof: Recall the set Λ of all K-invariant points in X is a Lie group, which may be
naturally identified with the quotient Lie group N/K, and its Lie algebra is pK = {ξ ∈ p:
Ad(k)ξ = ξ for k ∈ K} with Lie bracket [ξ, ζ]p . Let {ξ1′ , . . . , ξp′ } be a basis of pK . By
Proposition 6.14, if bt is a rcll path in Λ of finite variation, then there is a unique set of
real valued continuous functions b′1 (t), . . . , b′p (t) with b′i (0) = 0 and finite variation such that
∫ ∑
∑
f (bt ) = f (b0 ) + 0t pi=1 ξi′ f (bs )db′i (s) + u≤t [f (bu ) − f (bu− )] for any f ∈ Cc∞ (Λ). Conversely,
given b′i (t), 1 ≤ i ≤ p, as above, and b0 ∈ Λ, there is a unique continuous path bt in Λ for
∑
∑
which the above equation holds for any f ∈ Cc∞ (Λ). Note that pi=1 b′i (t)ξi′ = nj=1 bj (t)ξj
for a unique set of continuous real valued functions bj (t) with bj (0) = 0 and finite variation,
1 ≤ j ≤ n. Because an extended drift bt in X is a rcll path in Λ with b0 = o, a drift is a
continuous path in Λ, and f ∈ Cc∞ (X) may be restricted to be a function in Cc∞ (Λ), the
results of the present proposition follows from Proposition 6.14. 2
For an extended drift bt of finite variation in X, the associated bj (t) in Proposition 8.7
determine a unique drift bct which will be called the continuous part of bt . The bj (t) will be
called the components of this continuous part of bt under the basis {ξ1 , . . . , ξn } of p. For
simplicity, bj (t) may also be called the components of bt . Note that an extended drift bt
of finite variation in X, as part of an extended Lévy triple (b, A, η), is determined by its
components as its jumps are determined by the discontinuous part of η.
Considering an extended drift bt in X of finite variation as a rcll path in the Lie group
Λ. By Proposition 6.16, its continuous part bct may be obtained from bt after its jumps are
removed. Conversely, for a drift bt in X of finite variation, jumps hj ∈ Λ at distinct times
uj > 0 may be added to bt to obtain an extended drift in X of finite variation, provided that
∑
for any finite t > 0 and any neighborhood U of o, uj ≤t ∥ϕ(hj )∥ < ∞, and hj ∈ U for all
uj ≤ t except finitely many, where ϕ(h) = (ϕ1 (h), . . . , ϕn (h)) and ∥ · ∥ is the Euclidean norm.
As in §6.5, we may introduce an admissible quadruple (b, A, η, ν) on X consisting of
a Lévy triple (b, A, η) with bt of finite variation, and a family ν = {νt ; t ∈ R+ } of probability measures νt satisfying (6.27) on X. For f ∈ Cc∞ (G) and a rcll path zt in G, let
St f (z; b, A, η, ν) be defined as in (6.29) but with G replaced by X. The proof of Proposition 6.17 can be carried out on X to show that for any t ∈ R+ , |St f (z; b, A, η, ν)| is bounded
for all rcll paths zt in X.
All the results in sections 6.5, 6.6, 6.7 and 6.8 for admissible quadruples and the associated martingale property have versions on X = G/K. We will first state the versions of
Proposition 6.18 and Lemma 6.19 on X below with some adjustments.
252
For any measure ν on X, let ν ∗ denotes the measure obtained from ν when its charge at
point o is removed, and for a family ν of measures νt , t ∈ R+ , let ν ∗ = {νt∗ ; t ∈ R+ }. The
version of Proposition 6.18 on X can be stated as follows.
Proposition 8.8 (a) Let (b, A, η, ν) be an admissible quadruple on X = G/K. Then ν ∗
∑
is an extended Lévy measure function on X with ν ∗d = ν ∗ and u≤t ∥ϕ(hu )∥ < ∞ for any
finite t > 0, where hu is the local mean of νu . Moreover, if b′t is the extended drift obtained
from bt by adding all local means hu of νu as jumps at times u (as in Proposition 6.16 (b)),
then (b′ , A, η + ν ∗ ) is an extended Lévy triple with b′t of finite variation.
(b) If (b, A, η) is an extended Lévy triple with bt of finite variation, then (bc , A, η c , ν) is an
admissible quadruple, where bc is the continuous part of b and ν = {νt ; t ∈ R+ } is the family
of jump measures of η as defined by (8.5).
To convert Lemma 6.19 into a result on X, the left translation lg on G in this lemma
should be understood as the G-action on X, so for x, y ∈ X, ly x is understood as yx = S(y)x,
but the result of the lemma does not depend on the choice for the section map S. The right
translation ry may also be defined on X when y is K-invariant. Precisely, for b ∈ Λ and
x ∈ X, let rb x = xb = S(x)b, which does not depend on the section map S. For any
ξ ∈ g, let ξp and ξk be respectively the projections of ξ to p and k with respect to the direct
sum g = p ⊕ k. For g ∈ G, let [Adp (g)] = {[Adp (g)]ij } be the n × n matrix representing
the linear map Adp (g): p → p, given by ξ 7→ [Ad(g)ξ]p , under the basis {ξ1 , . . . , ξn } of p,
∑
that is, [Adp (g)ξj ]p = i [Ad(g)p ]ij ξi . Because for b ∈ Λ, Ad(b)k ⊂ k, it is easy to see that
Adp (b1 )Adp (b2 ) = Adp (b1 b2 ) on p for b1 , b2 ∈ Λ, and hence Adp (b) on p is a linear action of
the group Λ on p. Note that for f ∈ Cc∞ (G),
[Ad(g)ξi ][Ad(g)ξj ](f ◦ π) = [Ad(g)ξi ]p [Adξj ]p (f ◦ π) + [Ad(g)ξi ]k [Ad(g)ξj ]p (f ◦ π)
= [Ad(g)ξi ]p [Ad(g)ξj ]p (f ◦ π) + [[Ad(g)ξi ]k , [Ad(g)ξj ]p ](f ◦ π).
For ξ, ζ ∈ g, let [ξ, ζ]i , 1 ≤ i ≤ n, be the coordinates of [ξ, ζ]p under the basis {ξ1 , . . . , ξn },
∑
that is, [ξ, ζ]p = ni=1 [ξ, ζ]i ξi . With these preparation, the version of Lemma 6.19 on X can
be stated as follows, noting that for b ∈ Λ, Ad(b) is understood as Ad(S(b)) but the result
does not depend on the section map S.
Lemma 8.9 Let (b, A, η) be an extended Lévy triple on X, possibly improper. Then a rcll
process xt in X, with x0 = o, is represented by (b, A, η) if and only if, with xt = zt bt , zt has
the (b̄, Ā, η̄, ν̄)-martingale property, where (b̄, Ā(t), η̄, ν̄) are determined by
dĀ(t) = [Ad(bt )p ]dA(t)[Ad(bt )p ]′ ,
∫
db̄i (t) =
X
{ϕi (bt xb−1
t )−
∑
ϕp (x)[Ad(bt )]ip }η c (dt, dx)
p
253
+
n
∑
[[Ad(bt )ξp ]k , [Ad(bt )ξq ]p ]i dApq (t)
p,q=1
c
η̄(dt, ·) = cbt η (dt, ·)
(components of b̄t ),
(cb = lb ◦ rb−1 ),
ν̄t (·) = (cbt− ◦ rh−1
)νt ,
t
where η c is the continuous part of η and νt is given in (8.5) with local mean ht , noting ν̄t = δe
if η(t, ·) is continuous at time t, and ν̄t = (lbt− ◦ rb−1
)νt if (b, A, η) is proper.
t
The the martingale property transformation rule, Proposition 6.26, also has a version on
X that requires a similar modification as in Lemma8.9, but we will not state it as it will not
be needed.
Lemmas 6.20 and 6.21 hold on X with essentially the same proof, therefore, if zt is
−1
a rcll process in X having a (b, A, η, ν)-martingale property, then for any t > 0, zt−
zt is
z
independent of Ft− and has distribution νt , and if xt is a rcll process in X represented by
x
an extended Lévy triple (b, A, η), then for any t > 0, x−1
t− xt is independent of Ft− and has
distribution νt .
The following theorem provides a direct martingale representation of an inhomogeneous
Lévy process xt in X, represented by an extended Lévy triple (b, A, η) with bt of finite variation. It takes exactly the same form as its counterpart on G, given by Theorem 6.27, and has
essentially the same proof. More precisely, Lemmas 6.24 and 6.25 hold on X, with essentially
the same proofs, and the next theorem can be derived from these two lemmas in the same
way as Theorem 6.27 is proved.
Theorem 8.10 Let (b, A, η) be an extended Lévy triple on G with bt of finite variation. Then
a rcll process xt in G with x0 = e is represented by (b, A, η) if and only if
f (xt ) − S̃t f (x; b, A, η)
(8.13)
is a martingale under Ftx for any f ∈ Cc∞ (G), where S̃t f (x; b, A, η) is defined by (6.38) with
G replaced by X.
Remark 8.11 As in Remark 6.29, bt is of finite variation if and only if xt is a semimartingale
in X.
Lemma 6.31 holds also on X with the same proof, and Lemma 6.32 on X can be derived
from Lemmas 6.21, 6.19, 6.25 and 6.31 on X in the same way as Lemma 6.32 is proved.
Therefore, a rcll process in X can have the martingale property for at most one admissible
quadruple, and it can be represented by at most one extended Lévy triple as in Theorem 8.4.
The representation, in the reduced form, of inhomogeneous Lévy processes in a Lie group
G, as discussed in §6.9, can also be developed on X = G/K in the same fashion. Let (b, A, η)
254
be an extended Lévy triple on X. Assume η c has a finite first moment, that is, η c (t, ∥ϕ∥) < ∞
∑
for any t ∈ R+ , where ∥ϕ∥ = ( ni=1 ϕ2i )1/2 . Let Tt′ f (z; b, A, η) be defined as Tt f (z; b, A, η)
∑
in (8.6) but with the term − nj=1 ϕj (x)ξj f (zs ) removed from the η c -integral. A rcll process
xt in X is said to be represented by (b, A, η) in the reduced form if with xt = zt bt , for any
f ∈ Cc∞ (X), f (zt ) − Tt′ f (z; b, A, η) is a martingale under Ftx .
All the results in §6.9 for the representation on G in the reduced form holds also on X.
In particular, we have the following two results.
Theorem 8.12 Let (b, A, η) be an extended Lévy triple on X = G/K with bt of finite variation and η c having a finite first moment. Then a rcll process xt with x0 = o is represented
by (b, A, η) in the reduced form if and only if for any f ∈ Cc∞ (X), f (xt ) − S̃t′ f (x; b, A, η) is a
martingale under Ftx , where S̃t′ f (x; b, A, η) is defined as S̃t f (x; b, A, η) but without the term
∑
− j ϕj (τ )ξj f (xs ) in the η c -integral.
Theorem 8.13 Let xt be a rcll process in X = G/K with x0 = o, represented by an extended
Lévy triple (b, A, η), and assume η c has a finite first moment. Then there is an extended
drift brt such that (br , A, η) is an extended Lévy triple, and xt is represented by (br , A, η) in
the reduced form. Moreover, the extended Lévy triple (br , A, η), with η c having a finite first
moment, representing a given rcll process in the reduced form, is unique.
Conversely, given an extended Lévy triple (br , A, η) on X with η c having a finite first
moment, there is an inhomogeneous Lévy process xt in X with x0 = o, unique in distribution,
represented by (br , A, η) in the reduced form. Moreover, xt is represented by the extended
Lévy triple (b̃, A, η) for some b̃. Furthermore, if (br , A, η) is a Lévy triple, then the above
uniqueness in distribution holds among all rcll processes in X starting at o.
Recall that a homogeneous space X = G/K is called irreducible if the induced K-action
on the tangent space To X at o is irreducible. This is equivalent to the irreducibility of the
Ad(K)-action on p. When dim(X) > 1, the irreducibility implies that there is no nonzero
Ad(K)-invariant vector in p. Because π ◦ exp: p → X is diffeomorphic from a neighborhood
V of 0 in p onto a neighborhood U of o in X, and for ξ ∈ p and k ∈ K, keξ o = eAd(k)ξ o,
it follows that if X is irreducible and dim(X) > 1, then there is no K-invariant point in U
except o.
Let X be irreducible with dim(X) > 1 and let V above be convex. The exponential
∑
coordinate functions ∑
ϕi on X may be chosen so that ni=1 ϕi (x)ξi ∈ V for all x ∈ X. Then
the local mean h = e i µ(ϕi )ξi o of any distribution µ on X belongs to U . If µ is K-invariant,
then by (3.10), so is h and hence h = o. It follows that for any extended Lévy triple (b, A, η),
because it aways assumed to be proper, bt = o for all t ≥ 0. Moreover, the covariance matrix
function is A(t) = a(t)I for some continuous nondecreasing function a(t), where I is the
255
n × n identity matrix. This follows from Lemma 3.6 and the Ad(K)-invariance of A(t). To
summarize, we record the following.
Proposition 8.14 On an irreducible X = G/K, any covariance matrix function A(t) is
given by A(t) = a(t)I for some nondecreasing continuous function a(t) with a(0) = 0, where
I is the identity matrix. Moreover, if dim(X) > 1 and the coordinate functions ϕi are chosen
as above, then for any extended Lévy triple (b, A, η), bt = o for all t ≥ 0.
For an extended Lévy triple (b, A, η), by Proposition 8.14, bt = o and At = a(t)I, and
hence zt = xt in Theorem 8.4. By (3.10), for a K-invariant neighborhood U of o and a
∫
∑
K-invariant measure µ on X, U c µ(dx) nj=1 ϕj (x)ξj ∈ p is K-invariant and hence is 0 by
∫ ∫
the irreducibility of X = G/K. It follows that the integral 0t X (· · ·)η c (ds, dx) in (8.6) is
∫ ∫
∑
limU ↓{o} 0t U c [f (xs τ )−f (xs )]η c (ds, dτ ). Therefore, this integral combined with the sum u≤t
∫ ∫
in (8.6) may be written as 0t X [f (xs− τ ) − f (xs− )]η(ds, dτ ), where the integral is understood
∫ ∫
as the principal value, that is, the limit of 0t U c [· · ·]η(ds, dτ ) as a K-invariant neighborhood
U of o shrinks to o. We obtain the following simple form of a martingale representation on
an irreducible X = G/K, which does not directly involve the coordinate functions ϕi .
Theorem 8.15 Let X = G/K be irreducible with dim(X) > 1. For an inhomogeneous
Lévy process xt in X with x0 = o, there is a unique pair (a, η) of a continuous real-valued
nondecreasing function a(t) with a(0) = 0 and an extended Lévy measure function η(t, ·) on
X such that for any f ∈ Cc∞ (X),
f (xt ) −
∫
0
t
∫ t∫
n
1∑
ξj ξj f (xs )da(s) −
[f (xs− τ ) − f (xs− )]η(ds, dτ )
2 j=1
0 X
(8.14)
∫ ∫
is a martingale under Ftx , where 0t X [· · ·]η(ds, dτ ) is the principal value as described above
and is bounded for fixed t ∈ R+ . Moreover, η is the jump intensity measure of xt . Consequently, xt is stochastically continuous if and only if η is a Lévy measure function.
Conversely, given a pair (a, η) as above, there is an inhomogeneous Lévy process xt in
X with x0 = o, unique in distribution, such that (8.14) is a martingale under Ftx for any
f ∈ Cc∞ (X). Moreover, if η is a Lévy measure function, then the unique distribution holds
among rcll processes in X.
Note: By Theorem 8.15, an inhomogeneous Lévy process xt in an irreducible X = G/K,
with dim(X) > 1 and x0 = o, is completely determined in distribution by the operator
∑
D = a(t) ni=1 ξi ξi and the Lévy measure function η(t, ·), which do not depend on the choice
of the basis {ξ1 , . . . , ξn } of p (see Remark 8.5).
256
Remark 8.16 As mentioned above, when X = G/K is irreducible with dim(X) > 1, there
is no nonzero Ad(K)-invariant vector in p. It follows that in a neighborhood U of o, there
is no K-invariant points except o, and any extended Lévy triple (b, A, η) on X has bt = o
for all t ≥ 0. In fact, we just have to assume that p has no nonzero Ad(K)-invariant vector,
then in the first half of Theorem 8.4, the martingale representation (8.10) for an extended
Lévy triple takes the following simpler form:
f (xt ) −
where
8.3
∫
0
t
∫ t∫
n
1 ∑
[f (xs− τ ) − f (xs− )]η(ds, dτ ),
ξj ξj f (xs )dAij (s) −
2 i,j=1
0 X
∫t∫
0 X [· · ·]η(ds, dτ )
(8.15)
is the principal value and is bounded for fixed t > 0.
Additional properties and proof of Theorem 3.3
The following four simple propositions for inhomogeneous Lévy processes on X = G/K correspond to their counterparts for inhomogeneous Lévy processes in G, namely Propositions
6.38, 6.39, 6.40 and 6.41. They can be proved in essentially the same way, with the understanding that a product on X may depend on the choice for a section map S, but its
distribution as a random variable is not.
Proposition 8.17 Let xt be an inhomogeneous Lévy process in X = G/K with x0 = o
represented by an extended Lévy triple (b, A, η). Then for a fixed v > 0, the time shifted
′
′ ′
process x′t = x−1
v xv+t is represented by the extended Lévy triple (b , A , η ), given by
b′t = b−1
v bv+t ,
A′ (t) = A(v + t) − A(v),
η ′ (t, ·) = η(v + t, ·) − η(v, ·).
−1
Note that x−1
v xv+t = S(xv ) xv+t is defined with a choice of a section map S, but its distribution as a process in t does not depend on S. Also b−1
v bv+t is well defined as bt is K-invariant.
Proposition 8.18 Let x1t and x2t be two indepdent inhomogeneous Lévy processes in X with
x10 = x20 = o, represented by extended Lévy triples (b1 , A1 , η 1 ) and (b2 , A2 , η 2 ) respectively. Fix
s > 0 and a section map S. Let xt = x1t for t ≤ s and xt = S(xs )x2t−s for t ≥ s. Then xt is
an inhomogeneous Lévy process in X represented by the extended Lévy triple (b, A, η), whose
distribution does not depend on the choice for S, where bt = b1t , A(t) = A1 (t) and η(t, ·) =
η 1 (t, ·) for t ≤ s, and bt = b1s b2t−s , A(t) = A1 (s) + A2 (t − s) and η(t, ·) = η 1 (s, ·) + η 2 (t − s, ·)
for t ≥ s.
Proposition 8.19 Let x1t and x2t be two inhomogeneous Lévy processes in X with x10 = x20 =
o, represented by extended Lévy triples (b1 , A1 , η 1 ) and (b2 , A2 , η 2 ) respectively. Fix a > 0.
Then the two processes x1t and x2t have the same distribution on [0, a] if and only if the two
triples (b1 , A1 , η 1 ) and (b2 , A2 , η 2 ) are equal for t ∈ [0, a]. This conclusion holds also when
[0, a] is replaced by [0, a).
257
To state the next proposition, recall Λ is the group of K-invariant points in X, and for
b ∈ Λ, Adp (b): p → p is defined before Lemma 8.9. For x ∈ X and b ∈ Λ, let rb (x) = S(x)b,
which does not depend on the choice of the section map S.
Proposition 8.20 Let xt be an inhomogeneous Lévy process in X with x0 = o, represented
by an extended Lévy triple (b, A, η), and let b′t be an extended drift on X. Let S be a section
map. Then x̂t = S(xt )b′t is an inhomogeneous Lévy process in X represented by the extended
Lévy triple (b̂, Â, η̂) on X, where b̂t is an extended drift on X,
′−1
dÂ(t) = [Adp (b′−1
t )]dA(t)[Adp (bt )]
and
η̂(dt, ·) = (S(b′−1
t− ) ◦ rb′t )η(dt, ·).
(8.16)
Note that by the K-invariance of η(t, ·), the above does not depend on the choice of S.
The next proposition is an extension of Proposition 3.8 from Lévy processes to inhomogeneous Lévy processes, and is a supplement to Theorem 8.6. See also Remark 8.22.
As before, let {ξ1 , . . . , ξn } be a basis of p, and let ϕ1 , . . . , ϕn be the associated exponential
∑
∑
coordinate functions on X satisfying (3.10) on X, that is, ni=1 ϕi (x)Ad(k)ξi = ni=1 ϕi (kx)ξi
for k ∈ K and x ∈ X. As in §3.2, extend {ξ1 , . . . , ξn } to become a basis {ξ1 , . . . , ξd } of g by
adding a basis {ξn+1 , . . . , ξd } of k, and let ψ1 , . . . , ψd be associated exponential coordinate
∑
∑
functions on G such that (3.17) holds on G, that is, di=1 ψi (kgk −1 )ξi = di=1 ψi (g)[Ad(k)ξi ]
for k ∈ K and g ∈ G.
Recall that for k ∈ K, [Ad(k)] denotes the n × n matrix representing the linear map
Ad(k): p → p under the basis {ξ1 , . . . , ξn }, and an n × n matrix A is called Ad(K)-invariant
if A = [Ad(k)]A[Ad(k)]′ for any k ∈ K. We will also use [Ad(k)] to denote the matrix
representing Ad(k): g → g under the basis {ξ1 , . . . , ξd }. A d × d matrix A will be called
Ad(K)-invariant if A = [Ad(k)]A[Ad(k)]′ for any k ∈ K.
Proposition 8.21 Assume the bases of p and g, and the associated coordinate functions
are chosen as above. Let gt be an inhomogeneous Lévy process in G represented by an extended Lévy triple (b, A, η). Then gt is K-conjugate invariant if and only if bt and A(t) are
Ad(K)-invariant, and η(t, ·) is K-conjugate invariant, for all t ∈ R+ . Moreover, the induced
inhomogeneous Lévy process xt = gt o in X = G/K is represented by an extended Lévy triple
(b̃, Ã, η̃) with
Ãij (t) = Aij (t) for i, j = 1, 2, . . . , n
and η̃(t, ·) = πη(t, ·) for all t ∈ R+ .
Proof: Let gt = zt bt . Then for f ∈ Cc∞ (G), f (zt ) − Tt f (z; b, A, η) is a martingale. For
k ∈ K, let ztk = kzt k −1 , bkt = kbt k −1 , Ak (t) = [Ad(k)]A(t)[Ad(k)]′ and η k (t, ·) = ck η(t, ·).
Substituting f ◦ ck for f yields that f (ztk ) − Tt (f ◦ ck )(z; b, A, η) is a martingale. If we can
show
Tt (f ◦ ck )(z; b, A, η) = Tt f (z k ; bk , Ak , η k ),
(8.17)
258
then gt is represented by the extended Lévy triple (bk , Ak , η k ). Because kgt k −1 = ztk bkt , by
the uniqueness of the triple and the uniqueness in distribution in Theorem 8.4, this proves
the first part of Proposition 8.21, that is, gt is K-conjugate invariant if and only if bkt = bt ,
Ak (t) = A(t) and ck η(t, ·) = η(t, ·) for all k ∈ K and t ∈ R+ .
It is easy to see that
[Ad(b)ξi ](f ◦ ck )(z) =
d
d
f (kzeu[Ad(b)ξi ] k −1 ) |u=0 =
f (z k eu[Ad(kb)ξi ] ) |u=0 = [Ad(bk k)ξi ]f (z k ).
du
du
Then
d
∑
=
i,j=1
n
∑
Aij (t)[Ad(b)ξi ][Ad(b)ξj ](f ◦ ck )(z) =
d
∑
Aij (t)[Ad(bk k)ξi ][Ad(bk k)ξj ]f (z k )
i,j=1
Akij (t)[Ad(b)ξi ][Ad(b)ξj ]f (z k ),
i,j=1
and by (3.17),
n
∑
ψi (y)[Ad(b)ξi ](f ◦ ck )(z) =
i=1
n
∑
k
k
ψi (y)[Ad(b k)ξi ]f (z ) =
i=1
n
∑
ψ ◦ ck (y)[Ad(b)ξi ]f (z k ).
i=1
It is now easy to see (8.17) holds. This proves the first part of Proposition 8.21.
Let xt = gt o be represented by an extended Lévy triple (b̃, Ã, η̃). Because η and η̃ are
respectively the jump intensity measures of gt and xt = π(gt ), for any K-invariant f ∈ Cb (X)
vanishing in a neighborhood of o, by (8.3),
∑
η̃(t, f ) = E[
∑
f (x−1
s xs )] = E[
∑
f (S(xs− )−1 xt )] = E[
s≤t
s≤t
s≤t
−1
f (gs−
gt o)]
because S(xs− ) = gs− ks for some ks ∈ K. It follows that η̃(t, f ) = η(t, f ◦ π), and hence
η̃(t, ·) = πη(t, ·). It remains to show à = A0 , where A0 is the covariance matrix function on
X defined by A0ij (t) = Aij (t) for i, j = 1, 2, . . . , n.
We have xt = z̄t b̄t = S(z̄t )b̄t with b̄t = bt o and z̄t = zt o, where S is a section map. For
f ∈ Cc∞ (X),
f (z̄t ) − Tt (f ◦ π)(z; b, A, η) = f ◦ π(zt ) − Tt (f ◦ π)(z; b, A, η)
(8.18)
is a martingale. Because bt is K-conjugate invariant, bt kb−1
= k for any k ∈ K, and hence
t
Ad(bt )ξ = ξ for any ξ ∈ k. It then follows
Tt (f ◦ π)(z; b, A, η) =
∫
t
∫
+
0
G
{f (z̄s bs yb−1
s )
∫
0
t
d
1 ∑
[Ad(bs )ξi ][Ad(bs )ξj ]f (z̄s )dAij (s)
2 i,j=1
− f (z̄s ) −
d
∑
i=1
259
ψi (y)[Ad(bs )ξi ]f (z̄s )}η c (ds, dy)
∑∫
+
u≤t G
∫
= ζt f +
∫
t
0
t∫
+
0
G
n
1 ∑
[Ad(bs )ξi ][Ad(bs )ξj ]f (z̄s )dAij (s)
2 i,j=1
[f (z̄s bs yb−1
s ) − f (z̄s ) −
n
∑
c
ϕ(π(bs yb−1
s ))ξi f (z̄s )]η (ds, dy)
i=1
∑∫
+
−1
[f (z̄u− bu− yh−1
u bu− ) − f (z̄u− )]νu (dy)
u≤t G
−1
[f (z̄u− bu− yh−1
u bu− ) − f (z̄u− )]νu (dy)
(8.19)
where
∫
ζt f =
t
0
∫
d
n
∑
1 ∑
Ad(bs )[ξi , ξj ]f (z̄s )dAij (s)
2 i=n+1 j=1
t
∫
n
∑
n
∑
G i=1
i=1
+
0
{
{ϕi (π(bs yb−1
s ))ξi f (z̄s ) −
ψi (y)[Ad(bs )ξi ]f (z̄s )}η c (ds, dy).
In the proof of Proposition 3.8, it is shown that ϕi (π(y)) − ψi (y) = O(∥ψ(y)∥2 ). Because
∑d
∑d
−1
c
i=1 ψi (bs ybs )ξi =
i=1 ψi (y)[Ad(bs )ξi ] for y near e, it follows that the η -integral in ζt f
is absolutely integrable. Note that for k ∈ K, Ad(k) preserves p and k. Then by the
Ad(K)-invariance of A(t) and the K-conjugate invariance of bt , it is easy to show that
∑n
j=1 Ad(bs )[ξi , ξj ]Aij (s) is an Ad(K)-invariant element of g. Similarly, by the K-conjugate
invariance of bt and η c (t, ·), and (3.10) and (3.17), it is easy to show that
∫
0
t
∫
{
n
∑
G i=1
ϕi (π(bs yb−1
s ))ξi −
n
∑
ψi (y)[Ad(bs )ξi ]}η c (ds, dy)
i=1
is an Ad(K)-invariant element of g. Therefore, their projections to p are Ad(K)-invariant.
Then there are continuous real valued functions b′i (t) of finite variation, 1 ≤ i ≤ n, such that
∫ ∑
ζt f = 0t ni=1 ξi f (z̄s )db′i (s). Then by (8.18) and (8.19), z̄t has the (b̄′ , Ā, η̄, ν̄)-martingale
property with
dĀ(t) = [Adp (b̄t )]dA0 (t)[Adp (b̄t )]′ ,
and suitable expressions for b̄′ , η̄ and ν̄.
∞
Let z̃t = xt b̃−1
t . Then for any f ∈ Cc (X), f (z̃t ) − Tt f (z̃; b̃, Ã, η̃) is a martingale. Let
ut = b̃t b̄−1
t . Then z̄t = z̃t ut , and ut is an extended drift on X of finite variation. Let ui (t) be
the components of ut . By the version of Lemma 6.25 on X, for any f ∈ Cc∞ (X),
f (z̄t ) − Tt f (z̄; b̄, Ã, η̃) −
∫
0
t
(· · ·)dui (s) −
= f (z̃t ut ) − Tt f (z̃u; u−1 b̃, Ã, η̃) −
260
∫
0
t
∑∫
s≤t G
(· · ·)νs (dx)
(· · ·)dui (s) −
∑∫
s≤t G
(· · ·)νs (dx)
is a martingale. It follows that the covariance matrix function Ā(t) in the admissible quadruple for the martingale property of z̄t should be given by
dĀ(t) = [Adp (b̄t )]dÃ(t)[Adp (b̄t )]′ .
Comparing this with an earlier expression for dĀ(t) yields
[Adp (b̄t )]dÃ(t)[Adp (b̄t )]′ = [Adp (b̄t )]dA0 (t)[Adp (b̄t )]′ .
This proves Ã(t) = A0 (t) by the uniqueness of the admissible quadruple. 2
Remark 8.22 In Proposition 8.21, if bt has a finite variation with components bi (t) for
1 ≤ i ≤ d, then b̃t also has a finite variation with components
b̃i (t) = bi (t) +
d
n
∑
1 ∑
ci Ajk (t) +
2 j=n+1 k=1 jk
∫
G
[ϕi (π(y)) − ψi (y)]η c (t, dy),
1 ≤ i ≤ n,
∑
where cijk are the Lie algebra structure constants given by [ξj , ξk ] = di=1 cijk ξi , and the
η c -integral above is absolutely integrable. The proof is based on a computation of (8.13),
similar to the computation related to ξ˜0 in the proof of Proposition 3.8.
Proof of Theorem 3.3: Theorem 3.3 for the generator of a Lévy process in X = G/K
can be proved in exactly the same way as its counterpart Theorem 2.1 for the generator of
a Lévy process in G. First, Lemma 6.43 can be extended from G to X = G/K with the
same proof, using Proposition 8.17 and Theorem 8.4, but noting ξ0 ∈ p and aij are Ad(K)invariant, and η1 is K-invariant. Next, Theorem 3.3, which is stated for a Feller transition
semigroup on X, can be stated equivalently in terms of a Lévy process in X. Therefore, we
have to prove that if xt is a Lévy process in X, then as a Feller process, the domain D(L) of
its generator L contains Cc∞ (X), and for f ∈ Cc∞ (X), Lf is given by (3.11), and conversely,
given such an operator L, there is a Lévy process xt in G with x0 = o, unique in distribution,
with L as generator. This can be done in the same way as in the proof of Theorem 2.1 given
in §6.10. 2
8.4
Proof of Theorems 8.4 and 8.6
The martingale representation on X = G/K, Theorem 8.4, may be proved in large part
by essentially repeating the proofs of Theorems 6.9 and 6.11 for the corresponding results
on G, interpreting a product xy and an inverse x−1 on X = G/K as S(x)y and S(x)−1 by
choosing a section map S, and taking various functions and sets to be K-invariant. Note
that a product on X, such as xyz, may have several interpretations. For example, xyz
261
may be regarded either as S(x)S(y)z or as S(S(x)y)z, but when x, y, z are independent and
K-invariant random variables, the distribution of xyz does not depend on the choice of the
interpretation and the section map S. We will outline the main steps and provide details for
the necessary modifications.
The computations in §7.1 and §7.2 may be suitably modified to work on X = G/K to
show that any inhomogeneous Lévy process xt in X is represented by an extended Lévy triple
(b, A, η) with η being the jump intensity measure of xt . Note that because µni = µtn i−1 ,tni
and νt (·) are K-invariant, by (3.10), their local means bni and ht are K-invariant, and so is
bnt . Moreover, An (t, f ) is Ad(K)-invariant for K-invariant f ∈ Cb (G), and (7.7) holds on X
with A(t) being a covariance matrix function on X.
To prove the existence of an inhomogeneous Lévy process in X represented by a given
extended Lévy tripe (b, A, η), we will construct a triple (b̄, Ā, η̄) on G from (b, A, η), and
will show that if gt is an inhomogeneous Lévy process in G with g0 = e represented by
(b̄, Ā, η̄) (whose existence is guaranteed by Theorem 6.11), then xt = gt o is an inhomogeneous
Lévy process in X represented by (b, A, η). The proof will also show that gt is K-conjugate
invariant, and hence provides a proof for Theorem 8.6.
As before, let {ξ1 , . . . , ξn } be a basis of p with associated exponential coordinate ϕ1 , . . . , ϕn
∑
∑
satisfying (3.10) on X, that is, ni=1 ϕi (x)Ad(k)ξi = ni=1 ϕi (kx)ξi for k ∈ K and x ∈ X.
Extend {ξ1 , . . . , ξn } to become a basis {ξ1 , . . . , ξd } of g by adding a basis {ξn+1 , . . . , ξd } of
k, and let ψ1 , . . . , ψd be associated exponential coordinate functions on G such that (3.17)
∑
∑
holds on G, that is, di=1 ψi (kgk −1 )ξi = di=1 ψi (g)[Ad(k)ξi ] for k ∈ K and g ∈ G.∑
n
As in the proof of Theorem 3.10, let S be a section map on X such that S(x) = e i=1 ϕi (x)ξi
for x in a K-invariant neighborhood U of o. By (3.10),
kS(x)k −1 = S(kx),
∀k ∈ K and x ∈ U,
(8.20)
and
∀x ∈ X,
ψi ◦ S(x) = ϕi (x), 1 ≤ i ≤ n,
and
∑n
ψi ◦ S(x) = 0, n + 1 ≤ i ≤ d.
(8.21)
Let h̄t = e j=1 νt (ϕj )ξj . Then π(h̄t ) = ht = b−1
t− bt . By (3.10) and the K-invariance of νt ,
h̄t is K-conjugate invariant. Let W be a K-invariant neighborhood of o such that its closure
is contained in U . There is a partition of R+ : 0 = t0 < t1 < t2 < · · · < tn ↑ ∞, such that
b−1
ti bt ∈ W for t ∈ [ti , ti+1 ). Define
b̄t = S(bt ), t < t1 , and inductively b̄t = b̄ti − h̄ti S(b−1
ti bt ), ti ≤ t < ti+1 , i ≥ 1.
(8.22)
Then b̄−1
t− b̄t = h̄t for t = t1 , t2 , t3 , . . ., and by (8.20), b̄t is a K-conjugate invariant extended
drift in G with bt = π(b̄t ) = b̄t o. Let Ā(t) be the d × d matrix given by
Āij (t) = Aij (t) for i, j ≤ n
and
262
Āij (t) = 0 otherwise.
(8.23)
Then Ā(t) is a covariance matrix function on G that is Ad(K)-invariant in the sense that
[Ad(k)]Ā(t)[Ad(k)]′ = Ā(t) for k ∈ K, where [Ad(k)] is the matrix representing the linear
map Ad(k): g → g under the basis {ξ1 , . . . , ξd }. For f ∈ B+ (G), let
∫
∫
f (kS(x)k −1 )dkη(t, dx).
η̄(t, f ) =
X
(8.24)
K
Then η̄(t, ·) is a K-conjugate invariant measure function with η(t, ·) = π η̄(t, ·). Its continuous
∫∫
part η̄ c is given by η̄ c (t, f ) =
f (kS(x)k −1 )dkη c (t, dx). For each t > 0, let ν̄t = η̄({t}, ·) +
∫∫
[1 − η̄({t}, G)]δe . Then ν̄t (f ) =
f (kS(x)k −1 )dkνt (dx).
By (3.10), (3.17) and (8.21),
d
∑
ν̄t (ψj )ξj =
j=1
=
d ∫
∑
j=1 X
n ∫
∑
j=1 X
∫
[ψj (kS(x)k −1 )ξj ]dkνt (dx) =
K
∫
{
K
ϕj (x)[Ad(k)]ξj }dkνt (dx) =
d ∫
∑
j=1 X
n
∑
∫
K
{ψj (S(x))[Ad(k)]ξj }dkνt (dx)
νt (ϕj )ξj .
j=1
This shows that ν̄t (ψj ) = νt (ϕj ) for 1 ≤ j ≤ n and ν̄t (ψj ) = 0 for n < j ≤ d, and hence h̄t is
the local mean of ν̄t on G as defined in §6.3.
Let V = π −1 (U ). Then V is a K-conjugate invariant neighborhood of e, and η̄(t, V c ) =
η(t, U c ) < ∞. We now show that η̄ is an extended Lévy measure function on G by checking
η̄ c (t, ∥ψ∥2 ) < ∞
and
∑
ν̄u (∥ψ(·) − ψ(h̄u )∥2 ) < ∞,
(8.25)
u≤t
∑
where ∥ψ∥ = [ di=1 ψi2 ]1/2 is the Euclidean norm of ψ = (ψ1 , . . . , ψd ). By (8.20) and (8.21),
the first condition in (8.25) follows from η̄ c (t, ∥ψ∥2 1V ) = η c (t, ∥ϕ∥2 1U ) < ∞. To verify the
second condition, we may assume all ν̄t are moderate in the sense as defined after (8.4), then
ψi (h̄t ) = ν̄t (ψi ), and this condition may be written as
∑∫
∫
ν̄u (dx)∥
u≤t G
Because
∑
u≤t
ν̄u (V c ) =
∑
u≤t
νu (U c ) < ∞, this is equivalent to each of the following relations,
∑∫
u≤t V
∑∫
u≤t U
∑∫
u≤t G
∑
ν̄u (dy)[ψ(x) − ψ(y)]∥2 < ∞.
G
∫
ν̄u (dx)∥
V
∫
νu (dx)∥
U
∫
νu (dx)∥
G
ν̄u (dy)[ψ(x) − ψ(y)]∥2 < ∞;
νu (dy)[ϕ(x) − ϕ(y)]∥2 < ∞;
νu (dy)[ϕ(x) − ϕ(y)]∥2 < ∞;
νu (∥ϕ − ϕ(hu )∥2 ) < ∞.
u≤t
The last relation above holds because νt satisfies the version of (6.20) on X.
263
We have proved that (b̄, Ā, η̄) is an extended Lévy triple on G with π(b̄t ) = bt and
π η̄(t, ·) = η(t, ·), but it may be improper because b̄−1
t− b̄t = h̄t is known to hold only for
t = ti . By Theorem 6.11, there is an inhomogeneous Lévy process gt in G with g0 = e
represented by (b̄, Ā, η̄). Write gt = z̄t b̄t and for f ∈ Cc∞ (G), let M̄t f (z̄· ) be the martingale given in (6.23) with (b, A, η) and ϕi replaced by (b̄, Ā, η̄) and ψi . By the K-conjugate
invariance of b̄ and η̄, the Ad(K)-invariance of Ā, and (3.17), it is easy to show that for
k ∈ K, M̄t (f ◦ ck )(z̄· ) = M̄t f (kz̄· k −1 ). This means that kgt k −1 = kz̄t k −1 b̄t is also represented by (b̄, Ā, η̄). By the unique distribution in Theorem 6.11, gt and kgt k −1 have the same
distribution as processes. This shows that the process gt is K-conjugate invariant.
Let xt = gt o. By the first part of Theorem 8.6 which has been proved, xt is an inhomogeneous Lévy process in X. We may write xt = zt bt with zt = z̄t o. By the properties of (b, A, η)
and (b̄, Ā, η̄), and various K-invariance, K-conjugate invariance and Ad(K)-invariance,
d ∫
∑
j,k=1 0
∫
t
0
∫
∫
G
t
∫
=
0
∫
G
G
t
[Ad(b̄s )ξj ][Ad(b̄s )ξk ](f ◦ π)(z̄s )dĀjk (s) =
n
∑
[Ad(bs )ξj ][Ad(bs )ξk ]f (zs )dAjk (s),
j,k=1
{(f ◦ π)(z̄s b̄s x̄b̄−1
s ) − (f ◦ π)(z̄s ) −
d
∑
ψi (x̄)[Ad(b̄s )ξi ](f ◦ π)(z̄s )}η̄(ds, dx̄)
i=1
{f (zs bs xb−1
s )
− f (zs ) −
n
∑
ϕi (x)[Ad(bs )ξi ]f (zs )}η(ds, dx), and
i=1
−1
[f (z̄u− b̄u− x̄h̄−1
u b̄u− ) − f (z̄u− )]ν̄u (dx̄) =
∫
G
−1
[f (z− bu− xh−1
u bu− ) − f (zu− )]νu (dx),
for f ∈ Cc∞ (X). This shows that M̄t (f ◦ π)(z̄· ) is the martingale Mt f in (8.10), and hence
xt is represented by (b, A, η).
It remains to show the uniqueness of the extended Lévy triple (b, A, η) in the first half of
Theorem 8.4 and the unique distribution in the second half.
In Chapter 6, the above mentioned uniqueness on G is derived from Lemma 7.19. In the
same way, the uniqueness on X may be derived from the following Lemma.
Lemma 8.23 Let (b, A, η) be a Lévy triple on X with bt of finite variation, and let xt be a
rcll process in X with x0 = o represented by (b, A, η). Then the distribution of the process xt
is uniquely determined by (b, A, η). Moreover, the triple (b, A, η) is also uniquely determined
by the distribution of process xt .
By the construction of (b̄, Ā, η̄) from (b, A, η) via (8.22), (8.23) and (8.24), it is clear that
if (b, A, η) is a Lévy triple on X, then (b̄, Ā, η̄) is a Lévy triple on G, and hence there is a
K-conjugate invariant stochastically continuous inhomogeneous Lévy process gt in G with
g0 = e, represented by (b̄, Ā, η̄), such that xt = gt o is an inhomogeneous Lévy process in X
264
represented by (b, A, η). Because bt is of finite variation and νt = 0, by Theorem 8.10,
f (xt ) −
−
∫
t∫
0
G
∫
t
0
∑
ξj f (xs )dbj (s) −
j
{f (xs τ ) − f (xs ) −
∑
∫
t
0
1∑
ξj ξk f (xs ) dAjk (s)
2 j,k
ϕj (τ )ξj f (xs )}η(ds, dτ )
(8.26)
j
is a martingale under Ftx for any f ∈ Cc∞ (X).
After performing a non-random time change as in §7.2, we may assume that A(t), bi (t)
and η(t, ·) have t-densities a(t), βi (t) and m(t, ·) with respect to the Lebesgue measure on
R+ such that
∫
∑
Tr[a(t)] +
|βi (t)| +
f0 (x)m(t, dx) ≤ 1
(8.27)
X
i
for some f0 ∈ Cc∞ (X) satisfying (6.41) with G, e and d replaced by X, o and n. Because
∑
A(t) and i bi (t)ξi are Ad(K)-invariant, and η(t, ·) are K-invariant, we may assume their
∑
densities a(t), i βi (t)ξi and m(t, ·) have the same invariance property.
Then for f ∈ Cc∞ (X),
f (xt ) −
∫
0
t
Lx (s)f (xs )ds
is a martingale, where Lx (t) is the operator on X defined by
∫
n
n
n
∑
∑
1 ∑
L (t) =
aij (t)ξi ξj +
βi (t)ξi +
m(t, dy)[ry − 1 −
ϕi (y)ξi ],
2 i,j=1
X
i=1
i=1
x
where ry is the operator defined by ry f (z) = f (S(z)y) for f ∈ Cc∞ (X) with choice of a section
map S. Because of the K-invariance of m(t, ·), the definition of Lx (t) does not depend on
the choice of S. The operator Lx (t) on X here will play a similar role as the operator Lx (t)
on G in the proof of Lemma 7.20.
Fix a constant T > 0. Let Lg (t) be the operator on G defined by
Lg (t) =
∫
n
n
d
∑
∑
1 ∑
aij (t)ξir ξjr +
βi (t)ξir + m̄(t, dy)[ly − 1 −
ψi (y)ξir ],
2 i,j=1
G
i=1
i=1
∫
where m̄(t, ·) is the density of η̄(t, ·) given by m̄(t, f ) = K f (kS(x)k −1 )dk m(t, dx) for f ∈
Cc∞ (G). Note that Lg (t) is similar in form to Lr (t) given by (7.46) in §7.4, thus is the
generator of a time-reversed right process gt,T in G (as defined in the proof of Lemma 7.20),
∫
with gT,T = e, in the sense that for h ∈ Cc∞ (G), Mt,T = h(gt,T ) − tT Lg (s)h(gs,T )ds is a
g
g
time-reversed martingale, that is, E[Ms,T | Ft,T
] = Mt,T for s < t, where Ft,T
= σ{gr,T ;
t ≤ r ≤ T }.
The process gt,T will be assumed to be independent of xt . Note that gt,T is K-conjugate
invariant because its inverse is a K-conjugate invariant inhomogeneous Lévy process in G.
265
For f ∈ Cc∞ (X), x ∈ X and g ∈ G, we will write f (xg) = f (S(x)go) with the choice of a
section map S. This notation will be very convenient in the computation. Note that although
f (xg) depends on the choice of S, when g is a K-conjugate invariant random variable, its
distribution does not depend on S.
For ξ ∈ g, let ξ x f (xg) = (d/dt)f (S(x)etξ go) |t=0 and Lx (t)f (xg) be respectively the
expressions obtained when ξ and Lx (t) are applied to f (xg) as a function of x, and let ξ g f (xg)
and Lg (t)f (xg) be respectively the expressions obtained when ξ r and Lg (t) are applied to
f (xg) as a function of g. Although these expressions depend on the choice of a section map
S, the distributions of Lx (t)f (xg) and Lg (t)f (xg) do not when g is a K-conjugate invariant
random variable. It is easy to see that ξ x f (xg) = ξ g f (xg). By (3.10), (3.17) and (8.21), for
any K-invariant neighborhood U of o with H = π −1 (U ),
∫
Hc
m̄(t, dy)
d
∑
∫
ψi (y)ξi =
i=1
Uc
m(t, dy)
n
∑
ϕi (y)ξi .
i=1
It follows from this relation that Lx (t)f (xg) = Lg (t)f (xg).
Now assume U is a relatively compact K-invariant neighborhood of o. It can be shown
as in the proof of Lemma 7.20 that Lx (t)f (xg) = Lg (t)f (xg) is bounded for (t, x, g) ∈
[0, T ] × U × G or for (t, x, g) ∈ [0, T ] × X × π −1 (U ), and Lg (t)Lx (s)f (xg) = Lx (s)Lg (t)f (xg)
is bounded for (s, t, x, g) ∈ [0, T ]×[0, T ]×U ×π −1 (U ). Moreover, letting u(t, x) = E[f (xgt,T )]
for f ∈ Cc∞ (X), then u(t, xt ) is a martingale. From this, the rest of the proof of Lemma 8.23 is
essentially the same as the proof of Lemma 7.19. After Lemma 8.23 is proved, the uniqueness
of the properly extended Lévy triple (b, A, η) and the unique distribution of the process xt
in Theorem 8.4 are derived from Lemma 8.23 in the same way as the corresponding claims
in Theorem 6.11 are derived from Lemmas 7.11 and 7.19.
266
Chapter 9
Decomposition of Markov processes
As an application of the representation of inhomogeneous Lévy processes in homogeneous
spaces in the previous chapter, we present in §9.1 a decomposition of an invariant Markov
process into radial and angular parts, as described in §1.5 and §1.6, and in §9.2, for continuous
processes and irreducible orbits, a skew-product decomposition into a radial motion and an
independent angular motion with a time change, which extends the well known skew product
of Brownian motion in Rn .
9.1
Decomposition into radial and angular parts
We will consider an invariant Markov process in a manifold under the non-transitive action
of a Lie group. As discussed in §1.5, by suitably restricting the process, it may be possible to
assume the setup described as the beginning of §1.6. We will assume this setup throughout
this chapter, but replace the (continuous) action of a topological group on a topological space
by the (smooth) action of a Lie group on a manifold. Thus, let X = Y × Z be a product
manifold, and let Z = G/K for a Lie group G and a compact subgroup K. Let G act on
X = Y × Z via its natural action on Z = G/K.
Let J1 : X → Y and J2 : X → Z be, respectively the projections x 7→ y and x 7→ z for
x = (y, z). Let xt be a G-invariant rcll Markov process in X. By the discussion in §1.5 and
§1.6, its radial part yt = J1 (xt ) is a Markov process in Y , and its angular part zt = J2 (xt )
is an inhomogeneous Lévy process in Z = G/K under the conditional distribution given the
radial process yt .
By Theorem 8.4, the conditional distribution of zt , as an inhomogeneous Lévy process,
can be characterized by an extended Lévy triple on Z that depends on y(·) ∈ D′ (Y ), the
space of rcll paths in Y with possibly finite life time. To state this result precisely, we need
to introduce some definitions. The canonical path space of the radial process yt is D′ (Y ). As
in §1.6, let θtY be the time shift on D′ (Y ) defined by θtY y(s) = y(s + t) for y(·) ∈ D′ (Y ), and
267
let ζ be the life time on D′ (Y ). Note that ζ is a stopping time under the natural filtration
of the radial process yt regarded as the coordinate process on D′ (Y ). For x ∈ X and y ∈ Y ,
let Px and Qy be respectively the distributions of xt with x0 = x and yt with y0 = y on their
canonical path spaces D′ (X) and D′ (Y ).
We have used {FtY } in §1.6 to denote the natural filtration of yt on D′ (Y ), we will now let
{FtY } to denote the larger augmented filtration of yt with respect to all initial distributions
as defined in Appendix A.3, and will let FtY,0 be the natural filtration of yt .
A real-valued rcll process at ≥ 0 on D′ (Y ), with a0 = 0, defined for t ∈ [0, ζ), is called an
additive functional on D′ (Y ), or an AF on D′ (Y ) for short, if it is adapted to the filtration
FtY (that is, at restricted to [t < ζ] is FtY -measurable for any t ≥ 0), and
∀s, t ∈ R+ with s + t < ζ,
as+t = at + as ◦ θtY .
(9.1)
If t 7→ at is continuous on [0, ζ), then at will be called a continuous AF.
We note that the definition of AFs depends on the distribution of the radial process yt
because the augmented filtration FtY of yt is used. We may slightly weaken this definition
by requiring (9.1) to hold not identically on D′ (Y ), but for Qy -almost all y(·) ∈ D′ (Y ) and
for all y ∈ Y . If a(t) is an AF defined by this weaker property, then Qy (Λ) = 0 for all y ∈ Y ,
where Λ is the set on which (9.1) does not hold. We may modify a(t) on Λ so that (9.1)
holds identically on D′ (Y ). Because Qµ (Λ) = 0 for any probability measure µ on Y , the
modified a(t) will still be adapted to FtY , and so is an AF as defined in the last paragraph.
Let n = dim(Z). An n × n symmetric matrix valued process A(t) on D′ (Y ), defined for
t ∈ [0, ζ), will be called a matrix valued AF on D′ (Y ) if it is adapted to FtY , t 7→ A(t) is rcll,
A(0) = 0 and A(t) − A(s) ≥ 0 (nonnegative definite) for s ≤ t < ζ, and (9.1) holds with at
replaced by A(t). If t 7→ A(t) is continuous on [0, ζ), then A(t) will be called a continuous
matrix-valued AF. In this case, for each y(·) ∈ D′ (Y ), A(t) is a covariance matrix function
on Z for t ∈ [0, ζ), and so A(t) will also be called a covariant matrix AF.
A measure valued process η(t, ·) on D′ (Y ), where η(t, ·) is a measure on Z for t ∈ [0, ζ),
will be called a measure-valued AF on D′ (Y ) if it is adapted to FtY (that is, η(t, B) is
adapted to FtY for any B ∈ B(Y )), η(0, ·) = 0, η(s, ·) ≤ η(t, ·) for s ≤ t < ζ, η(t, ·) ↓ η(s, ·)
as t ↓ s < ζ, and (9.1) holds with at replaced by η(t, ·). If η(s, ·) ↑ η(t, ·) as s ↑ t < ζ, then
η(t, ·) will be called a continuous measure-valued AF. It is clear that a measure-valued AF
on D′ (Y ) is a measure function on Z at any y(·) ∈ D′ (Y ), and if it is a continuous measurevalued AF, then it is a continuous measure-valued function at any y(·) ∈ D′ (F ). A measurevalued AF η(t, ·) will be called an (extended) Lévy measure AF if at any y(·) ∈ D′ (Y ), it is
an (extended) Lévy measure function.
Recall the set Λ of K-invariant points in Z = G/K form a group under the product
xy = S(x)y and inverse x−1 = S(x)−1 o, which does not depend on the choice of the section
268
map S. A Λ-valued process bt on D′ (Y ), defined for t ∈ [0, ζ), will be called a Z-valued
multiplicative functional on D′ (Y ), or a Z-valued MF on D′ (Y ) for short, if it is adapted to
FtY , b0 = o, t 7→ bt is rcll on [0, ζ), and
∀s, t ∈ R+ with s + t < ζ,
bs+t = bt (bs ◦ θtY ).
(9.2)
It is clear that for each y(·) ∈ D′ (Y ), bt is an extended drift on Z for t < ζ. If t 7→ bt is
continuous on [0, ζ), then bt will be called a continuous Z-valued MF on D′ (Y ), and for each
y(·) ∈ D′ (Y ), bt is a drift in Z for t < ζ.
As in §8.1, let p be an Ad(K)-invariant subspace of the Lie algebra g of G that is
complementary to the Lie algebra k of K, let {ξ1 , . . . , ξn } of a basis of p, and let ϕ1 , . . . , ϕn
be associated exponential coordinate functions on Z = G/K. Lévy triples and extended
Lévy triples on Z may be defined under this setup.
A triple (b, A, η) of a continuous Z-valued MF bt , a covariance matrix AF A(t) and a
Lévy measure AF η(t, ·) on D′ (Y ), will be called a functional Lévy triple on D′ (Y ). More
generally, a triple (b, A, η) of a Z-valued MF bt , a covariance matrix AF A(t) and an extended
Lévy measure AF η(t, ·) such that at each y(·) ∈ D′ (Y ), (b, A, η) is an extended Lévy triple
for t ∈ [0, ζ), will be called a functional extended Lévy triple.
For z ∈ Z and y(·) ∈ D′ (Y ), let Rzy(·) be the conditional distribution of the angular
process zt on D′ (Z) given the radial path y(·) and z0 = z. By Theorem 1.39, for any y ∈ Y
and Qy -almost all y(·) ∈ D′ (Y ), zt is an inhomogeneous Lévy process in Z = G/K given
y(·), that is, zt is an inhomogeneous Lévy process in Z under Rzy(·) for any z ∈ Z. Note that
Rzy(·) is supported by the closed subset Dζ′ (Z) of D′ (Z) consisting of all paths in D′ (Z) with
the same constant life time ζ, the life time of y(·).
As in §1.6, the projection maps J1 : X → Y and J2 : X → Z will also be used to denote
the induced projections J1 : D′ (X) → D′ (Y ) and J2 : D′ (X) → D′ (Z), given by J1 x(·) = y(·)
and J2 x(·) = z(·) for x(·) = (y(·), z(·)). Then Qy = J1 Px for x ∈ X and y = J1 (x), and by
(1.36), for any measurable F ⊂ D′ (Z) and z = J2 (x),
Y
Rzy(·) (F ) = Px [J2−1 (F ) | J1−1 (F∞
)] for Qy -almost all y(·).
By Theorem 8.4, an inhomogeneous Lévy process zt in Z with z0 = o is represented by
a unique extended Lévy triple (b, A, η). This representation extends naturally to processes
zt in Z defined for t in a finite time interval. An inhomogeneous Lévy process zt in Z, with
z0 not necessarily equal to o, will be called represented by (b, A, η) if so is z0−1 zt . Recall
z0−1 zt = S(z0 )−1 zt with the choice of a section map S, and its distribution does not depend
on S.
By the following result, a G-invariant rcll Markov process in X is completely characterized
in distribution by a rcll Markov process in Y and a functional extended Lévy triple on D′ (Y ).
269
Recall that a Markov process in X, without a specified initial distribution, refers to a family
of processes, one for each starting point x ∈ X with distribution Px on D′ (X). Two Markov
processes will be called equal in distribution if they have the same Px for all x ∈ X.
Theorem 9.1 Let xt be a G-invariant rcll Markov process in X with radial part yt and the
angular part zt as above. Then yt is a rcll Markov process in Y , and there is a functional
extended Lévy triple (b, A, η) on D′ (Y ) such that for any y ∈ Y and Qy -almost all y(·) ∈
D′ (Y ), given y(·), zt for t < ζ is an inhomogeneous Lévy process in Z represented by (b, A, η).
Moreover, the (b, A, η) is unique in the sense that if (b̃, Ã, η̃) is another functional extended
Lévy triple on D′ (Y ) such that for any y ∈ Y and Qy -almost all y(·), zt is represented by
(b̃, Ã, η̃), then for any y ∈ Y and Qy -almost all y(·), (b, A, η) = (b̃, Ã, η̃).
Conversely, given a rcll Markov process yt in Y and a functional extended Lévy triple
(b, A, η) on D′ (Y ), there is a G-invariant rcll Markov process xt in X, unique in distribution,
such that its radial part is equal to yt in distribution, and for any y ∈ Y and Qy -almost all
y(·) ∈ D′ (Y ), its angular part, given y(·), is an inhomogeneous Lévy process in Z represented
by (b, A, η).
Proof: Assume xt is a G-invariant rcll Markov process in Z. By Theorem 1.31, the radial
process yt is a rcll Markov process in Y . By Theorem 1.39, for any y ∈ Y and Qy -almost
all y(·) ∈ D′ (Y ), given y(·), the angular process zt is an inhomogeneous Lévy process in Z
Y,0
for t < ζ, and its associated convolution semigroup µs,t is Fs,t
-measurable for s < t, where
Y,0
Fs,t = σ{zu ; s ≤ u ≤ t}, and has the time shift property (1.37), that is, µs,t = µ0,t−s ◦ θs .
Note that given y(·), ζ may be regarded as a constant. By Theorem 8.4, zt is represented
by an extended Lévy triple (b, A, η). By the proof of Theorem 8.4, the triple (b, A, η) on
X is obtained in the same way as on G. Recall its construction is based on a sequence of
partitions ∆n : 0 = tn0 < tn1 < tn2 < · · · < tni ↑ ∞ (as i ↑ ∞) of R+ with mesh ∥∆n ∥ → 0 as
n → ∞, and assume ∆n contains the set J of fixed jump times of zt as n → ∞ in the sense
that any u ∈ J belongs to ∆n when n is sufficiently large. Because fixed jump times of zt are
random for yt , we have to replace the non-random time points tni by suitable {FtY }-stopping
times τni .
Let 0 = sn0 < sn1 < · · · < sni ↑ ∞ as i ↑ ∞ be a sequence of partitions of R+ with mesh
→ 0 as n → ∞. It is possible to choose {FtY }-stopping times τni with τn0 = 0 such that
τni < τn i+1 on [τn i+1 < ∞], and the time points τni 1[τni <∞] , as n and i > 0 vary, are precisely
the fixed jump times of zt . Now let αn0 = 0, and for each n, let αn1 < αn2 < · · · < αni < · · ·
be the ordered values from the set {sni and τni 1[τni <∞] ; i > 0}. Then αni are {FtY }-stopping
times, and they form a sequence of (random) partitions of R+ with mesh → 0 and contain J
∑
as n → ∞. Now let µni = µαn i−1 ,αni and define ηn (t, ·) = tni ≤t µni . Then ηn (t, f ) → η(t, f )
as n → ∞ for any f ∈ Cb (X) vanishing in a neighborhood of o. Note that η(t, ·) does not
depend on the sequence of partitions as long as the required properties are satisfied.
270
Any s > 0 may be included in the partition as an si , then by the time-shift property of
µs,t , for t > s, η(t, ·) = η(s, ·) + η(t − s, ·) ◦ θsY . Thus, to prove that η(t, ·) is an extended
Lévy measure functional, it remains to show that η(t, ·) is adapted to {FtY }. It suffices to
show ηn (t, ·) is adapted to {FtY }. We just have to show that for any two {FtY }-stopping
times σ and τ , µσ,τ restricted on [σ < τ ≤ t] is FtY -measurable. It is easy to see that this is
true if σ and τ are discrete. For general σ and τ , we may approximate them by decreasing
discrete {FtY }-stopping times σn ↓ σ and τn ↓ τ , then by the right continuity of µs,t in (s, t),
Y
µσ,τ 1[σ<τ ≤t] is Ft+ε
-measurable for any ε > 0. Because {FtY } is right continuous, it follows
that µσ,τ 1[σ<τ ≤t] is FtY -measurable. We have proved that η(t, ·) is an extended Lévy measure
functional. In a similar fashion, we may prove A(t) is a covariance matrix functional and bt
is a Z-valued MF, and hence (b, A, η) is a functional extended Lévy triple.
The uniqueness of (b, A, η) as a functional extended Lévy triple on D′ (Y ) follows from its
uniqueness, for Qy -almost all y(·) ∈ D′ (Y ), as an extended Lévy triple given in Theorem 8.4.
This proves the first half of Theorem 9.1.
Let yt be a rcll Markov process in Y and let (b, A, η) be a functional extended Lévy triple
on D′ (Y ). For each y(·) ∈ D′ (Y ), (b, A, η) is an extended Lévy triple on Z. Let zt be
an inhomogeneous Lévy process in Z represented by (b, A, η). Let µs,t be the convolution
semigroup associated to zt . Suppose µs,t is FtY -measurable for any s < t. For a fixed
v > 0, µ′s,t = µv+s,v+t is the convolution semigroup associated to the time shifted process
zt′ = zv−1 zv+t , which is represented by the time shifted triple (b′ , A′ , η ′ ) as in Proposition 8.17.
Because b′t = bt ◦θvY , A′t = At ◦θvY and η ′ (t, ·) = (η ◦θvY )(t, ·), by the uniqueness of distribution
for a given triple, µ′s,t = µs,t ◦ θvY . This shows that µs,t has the time shift property (1.37).
Now the second half of Theorem 9.1 follows from Theorem 1.40. Note that in Theorem 1.40,
FtY is the natural filtration FtY,0 of yt , but its proof is still valid when FtY is the augmented
filtration of yt with respect to all initial distributions, as assumed here.
It remains to prove that µs,t is FtY -measurable under the assumption that (b, A, η) is
adapted to FtY , that is, for any f ∈ Bb (Z), µs,t (f ) is FtY -measurable under the assumption
that Bt ,A(t) and η(t, ·) are adapted to {Ft }. This is given by the following Lemma.
Lemma 9.2 Let (b, A, η) be an extended Lévy triple on Z = G/K for each y(·) ∈ D′ (Y ),
and assume it is adapted to FtY . Let µs,t be the convolution semigroup associated to the
inhomogeneous Lévy process in Z represented by (b, A, η). Then µs,t (f ) is FtY -measurable for
any s < t and f ∈ Bb (Z).
Proof: Because µs,t is µ0,t−s for the time shifted process x′t = x−1
s xs+t , which is represented
′
′ ′
by the time shifted triple (b , A , η ) in Proposition 8.17, it follows that we just have to
prove that µ0,t is FtY -measurable. In the proof of Theorem 8.4 given in §8.4, to show the
existence of an inhomogeneous Lévy process xt in G/K represented by (b, A, η), we first
271
construct an extended Lévy triple (b̄, Ā, η̄) on G, possibly improper, by (8.22), (8.23) and
(8.24), and then let xt = gt o, where gt is the inhomogeneous Lévy process represented by
(b̄, Ā, η̄). Therefore, to prove the FtY -measurability of µ0,t , we may assume that µ0,t is the
distribution of an inhomogeneous Lévy process xt in G represented by a possibly improper
extended Lévy triple (b, A, η) on G that is adapted to FtY .
By Lemma 6.19, xt = zt bt and zt has the (b̄, Ā, η̄, ν̄)-martingale property, where (b̄, Ā, η̄) is
a Lévy triple in G with b̄t of finite variation, and ν̄ is a family of probability measures ν̄t on G
with ν̄t = δe except for countably many t > 0. Note that zt here is a process in G, and is not
the same zt in the statement of Theorem 9.1. The ν̄ may be regarded as a measure function
∑
ν̄(t, ·) = u≤t ν̄u . From the explicit expressions given in Lemma 6.19, we can see that b̄, Ā,
η̄ and ν̄ are adapted to FtY . Because for f ∈ Cc∞ (G), µ0,t (f ) = E[f (xt )] = E[f (zt b−1
t )], it
Y
∞
is easy to see that µ0,t (f ) is Ft -measurable for any f ∈ Cc (G) if and only if E[f (zt )] is
FtY -measurable for any f ∈ Cc (G). Therefore, it suffices to show that if zt has a (b, A, η, ν)martingale property for some quadruple (b, A, η, ν) as described in §6.5, and if (b, A, η, ν) is
adapted to FtY , then E[f (zt )] is FtY -measurable for any f ∈ Cc∞ (G).
Let zt′ be the inhomogeneous Lévy process in G represented by the Lévy triple (b, A, η),
and for any integer n > 0, let ztn be the process obtained when independent fixed jumps with
distributions νu1 , νu2 , . . . , νun are added at times u1 , u2 , . . . , un . Suppose for f ∈ Cc∞ (G),
E[f (zt′ )] is FtY -measurable, then it is easy to show that so is E[f (ztn )]. By Lemma 7.26,
the processes ztn converge weakly to zt in D(G). Thus, if t is not a fixed jump time of zt ,
then E[f (ztn )] → E[f (zt )], and hence E[f (zt )] is FtY -measurable. This holds in fact for any
t ≥ 0 by the right continuity of zt and FtY in t. Therefore, it suffices to prove that if xt is
an inhomogeneous Lévy process in G represented by a Lévy triple (b, A, η) that is adapted
to FtY , then E[f (xt )] is FtY -measurable for any t ≥ 0 and f ∈ Cc∞ (G).
As in §7.3, let (bn , An , η n ) be a piecewise linearized Lévy triples that converge to (b, A, η)
as defined in Lemma 7.12 (b). Let Vn ↓ {e} be a sequence of open sets. We may replace
η n (t, ·) by its restriction on Vnc , which will still be denoted as η n (t, ·), then it is easy to see
that (bn , An , η n ) → (b, A, η) still holds, and now η n (t, ·) is a finite measure on G. By the
construction of (bn , An , η n ) from (b, A, η), it is easy to see that for any fixed ε > 0, when n
Y
. Let xnt be inhomogeneous Lévy processes
is sufficiently large, (bn , An , η n ) is adapted to Ft+ε
in G represented by (bn , An , η n ). By Theorem 7.17, xnt converge to xt weakly in D(G), and
Y
-measurable,
hence E[f (xnt )] → E[f (xt )] as n → ∞ for any f ∈ Cc∞ (G). If E[f (xnt )] is Ft+ε
Y
Y
then by the right continuity of Ft , E[f (xt )] is Ft -measurable. By the discussion in §7.3, xnt
is piecewise a Lévy process. More precisely, xnt is a Lévy process for t ∈ (tnk , tn k+1 ], where
0 = tn0 < tn1 < tn2 < · · · < tnk ↑ ∞ as k ↑ ∞ is a sequence of partitions of R+ with
mesh → 0 as n → ∞, and its characteristics (ξ0nk , ank , η nk ) are FtYn k+1 -measurable. Note
that η nk is a finite measure on G because η n (t, ·) is finite. If E[f (xnt )] is FtYn k+1 -measurable
272
Y
for t ∈ (tnk , tn k+1 ], then E[f (xnt )] is Ft+ε
-measurable for large n. Therefore, it suffices to
prove that if xt is a Lévy process in G with characteristics (ξ, a, η) such that (ξ, a, η) are
G-measurable for some σ-algebra on D′ (Y ) and η is finite, then E[f (xt )] is G-measurable for
any t ≥ 0 and f ∈ Cc∞ (G).
The ξ and a may be regarded as elements of Euclidean spaces, and η is an element
of the space Mf (G) of finite measures on G. There is a metric on Mf (G) under which
the convergence is just the weak convergence, and hence the function F (µ) = µ(f ) on
Mf (G), for f ∈ Cb (G), is continuous. The Lévy process xt with characteristics (ξ, a, η)
may be regarded as an inhomogeneous Lévy process represented by the Lévy triple (b, A, η̄),
where bt = exp(tξ), A(t) = ta and η̄(t, ·) = tη. Then the convergence of (ξ, a, η) implies
the convergence of the Lévy triple (b, A, η̄). It follows from Theorem 7.17 that E[f (xt )]
is continuous in (ξ, a, η), and hence is G-measurable. This proves Lemma 9.2, and hence
completes the proof of Theorem 9.1. 2
Example 9.3 As a simple application of Theorem 9.1, we identify a large class of Ginvariant Markov processes in X with a given radial process yt in Y by explicitly constructing
the associated functional Lévy triples (b, A, η) as follows. Let a(y) be a Borel function on
Y that takes values in the space of nonnegative definite symmetric n × n matrices, and let
m(y, ·) be a kernel from Y to Z, such that a(y) and m(y, ∥ϕ∥∧1) are bounded for y contained
in any compact subset of Y , where ϕ = (ϕ1 , . . . , ϕn ) are coordinate functions on Z = G/K
as in the definition of Lévy measure functions. Let ξ(y) be a smooth function on Y with
values in pK = {ξ ∈ p; Ad(k)ξ = ξ for k ∈ K}. Recall pK may be identified with the Lie
algebra of the Lie group Λ of K-invariant points in Z. Now define
∫
t
A(t) =
0
a(ys )ds and η(t, ·) =
∫
t
0
m(ys , ·)ds,
and let bt be the unique solution of the ordinary differential equation (dbt /dt) = bt ξ(yt ) on
Λ with b0 = o. Here, for b ∈ Λ and ξ ∈ pK , bξ denotes Dlb (ξ) ∈ Tb Λ. Then it is easy to show
that (b, A, η) is a functional Lévy triple.
When Z = G/K is irreducible, by the discussion in §8.2, any covariance matrix function
A(t) is a(t)I for some continuous real-valued nondecreasing function a(t) with a(0) = 0.
Moreover, when dim(Z) > 1, with a proper choice of coordinate functions on Z, any extended drift bt is trivial, that is, bt = o for all t ≥ 0. In this case, any inhomogeneous
Lévy process zt in Z has a simpler representation by a pair (a, η) of a(t) as above and an extended Lévy measure function η(t, ·) as in Theorem 8.15, and hence Theorem 9.1 has a more
convenient form as in the following corollary. Note that in this corollary, as in Theorem 9.1,
the angular part is defined for t < ζ, but for simplicity, this qualification is omitted in the
statement.
273
Corollary 9.4 Assume Z = G/K is irreducible and dim(Z) > 1. Let xt be a G-invariant
rcll Markov process in X. Then its radial part yt is a rcll Markov process in Y , and there is a
pair (a, η) of a continuous AF a(t) and an extended Lévy measure functional η(t, ·) on D′ (Y )
such that for any y ∈ Y and Qy -almost all y(·) ∈ D′ (Y ), given y(·), the angular part zt is
an inhomogeneous Lévy process in Z represented by (a, η) as in Theorem 8.15. Moreover,
the (a, η) is unique in the sense that if (ã, η̃) is another pair of a continuous AF and an
extended Lévy measure functional on D′ (Y ) such that for any y ∈ Y and Qy -almost all y(·),
zt is represented by (ã, η̃), then for any y ∈ Y and Qy -almost all y(·), (a, η) = (ã, η̃).
Conversely, given a rcll Markov process yt in Y and a pair (a, η) as above, there is a
G-invariant rcll Markov process xt in X, unique in distribution, such that its radial part is
equal to yt in distribution, and for any y ∈ Y and Qy -almost all y(·) ∈ D′ (Y ), its angular
part zt , given y(·), is an inhomogeneous Lévy process in Z represented by (a, η).
9.2
A skew-product decomposition
Let X be Rn without the origin. Assume n ≥ 2. Let Y = (0, ∞) and Z be the unit sphere
S n−1 in Rn . Then X is the product manifold Y × Z by the map (r, z) 7→ rz from Y × Z to X.
Under the action of the orthogonal group G = O(n), Z may be regarded as the homogeneous
G/K with K = O(n − 1). Let Bt be the standard Brownian motion in Rn , and for any point
x0 in Rn that is not the origin, let xt = x0 + Bt . It is well known that a Brownian motion
in Rn with n ≥ 2 will not hit any single point (see [46, Theorem 18.6]), so xt will not hit
the origin , and may be regarded as a G-invariant Markov process in X. Let rt = |xt | be its
radial part. It is well known that its angular part, zt = xt /rt , is a time changed spherical
Brownian motion in Z = S n−1 . More precisely, zt = z̃a(t) , where z̃t is a Riemannian Brownian
motion in S n−1 (under the induced Riemannian metric from Rn ), that is independent of the
∫
radial process rt , and a(t) = 0t ds/rs2 . This is called the skew-product decomposition of
the Brownian motion in Rn (see Itô-McKean [43, §7.15]). This decomposition was extended
in Galmarino [28] to a general O(n)-invariant diffusion process in Rn that does not hit the
origin for n ≥ 3. See also Pauwels-Rogers [73] for a skew-product of Brownian motion in a
Riemannian manifold. Theorem 9.5 below provides an extension under a more general space
setting, obtained in [57].
We now return to the general setup of the last section. Let p be given an Ad(K)invariant inner product ⟨·, ·⟩, which induces a G-invariant Riemannian metric on Z = G/K,
under which the Riemannian Brownian motion in Z is defined as a diffusion process in Z
(see Appendix 5) with generator (1/2)∆Z , where ∆Z is the Laplace-Beltrami operator on
Z. Let the basis {ξ1 , . . . , ξn } of p be chosen to be orthonormal under ⟨·, ·⟩, which may be
identified with a basis in the tangent space To Z that is orthonormal with respect to the
274
G-invariant Riemannian metric. Note that when Z = G/K is irreducible and dim(Z) > 1,
∑
by Proposition 3.23, ∆Z = ni=1 ξi ξi .
Theorem 9.5 Assume Z = G/K is irreducible and dim(Z) > 1. Let xt be a G-invariant
continuous Markov process in X with radial part yt and angular part zt . Then there are a
continuous AF a(t) on D′ (Y ), and a Riemannian Brownian motion z̃t in Z independent of
the process yt , such that the processes xt = (yt , zt ) and (yt , z̃a(t) ) are equal in distribution.
Proof: Because zt is continuous, by Corollary 9.4, given y(·), zt as an inhomogeneous
Lévy process in Z is represented by (a, 0) for some continuous AF a(t) on D′ (Y ). Equivalently, this means that under the conditional distribution given y(·),
f (zt ) − (1/2)
∫
n
t∑
0 i=1
ξi ξi f (zs )da(s)
is a martingale for any f ∈ C ∞ (Z). Let z̃t be a Riemannian Brownian motion in Z
∑
independent of the process yt . Because z̃t has generator (1/2)∆Z = (1/2) ni=1 ξi ξi , so
∫ ∑
∫ a(t) ∑n
f (z̃t ) − (1/2) 0t ni=1 ξi ξi f (z̃s )ds is a martingale. Then f (z̃a(t) ) − (1/2) 0
i=1 ξi ξi f (z̃s )ds
is a martingale, but after a simple substitution of integrating variable, this expression be∫ ∑
comes f (z̃a(t) ) − (1/2) 0t ni=1 ξi ξi f (z̃a(s) )da(s). This shows that z̃a(t) is also represented by
(a, 0), and hence has the same distribution as zt given y(·). It follows that the processes
xt = (yt , zt ) and (yt , z̃a(t) ) are equal in distribution. 2
Remark 9.6 If in Theorem 9.5, limt→∞ a(t) = ∞ almost surely, then the result of this
theorem can be strengthened in the following form: There is a Riemannian Brownian motion
z̄t in Z, independent of the process yt , such that zt = z̄a(t) . The proof will be given in the
next two paragraphs.
If a(t) is strictly increasing, then its inverse may be expressed as a−1 (s) = inf{t > 0;
a(t) > s}. In general, it may not have an inverse in the usual sense, but a−1 (t) given here
is still a well defined rcll function on R+ satisfying a ◦ a−1 (s) = s for all s ≥ 0. However,
a−1 ◦ a(t) may not be equal to t. If a(·) is constant on [t, b] and a(u) > a(b) for u > b, then
a−1 ◦ a(t) = b. From this, it is easy to see that for any function f (t) on R+ , if f (t) is constant
on any interval on which a(t) is a constant, then f (a−1 ◦ a(t)) = f (t).
Because given the process yt , a(t) is non-random, and the two processes zt and z̃a(t) are
equal in distribution, by the rcll path property, it follows that given the process yt , zt is a
constant on any interval on which a(t) is a constant almost surely, and the two processes
za−1 (t) and z̃t are equal in distribution. This implies that z̄t = za−1 (t) is a Riemannian
Brownian motion in Z independent of the process yt , and zt = za−1 ◦a(t) = z̄a(t) .
275
Remark 9.7 The condition dim(Z) > 1 in Theorem 9.5 cannot be removed, because when
dim(Z) = 1, the angular process may have a drift, and then the skew-product does not hold.
In particular, an O(2)-invariant Markov process in R2 that does not hit the origin may not
have a skew-product decomposition as dim(S 1 ) = 1. See also Evans-Hening-Wayman [24]
for more examples.
Remark 9.8 A main reason for the skew-product in Theorem 9.5 to hold is that on an
irreducible homogeneous space, a covariance matrix function must be proportional to the
identity matrix, see the proof of Corollary 9.4. However, there is no such restriction on a
Lévy measure function, so the skew-product typically does not hold when the angular process
is discontinuous. It may hold under some special circumstances, see [59] for such an example.
When xt is a diffusion process in X as defined in Appendix 5, the skew product in
Theorem 9.5 can be established more directly and has a more explicit form as in Theorem 9.9
below. In the sequal, a differential operator T on Y or Z may be regarded as a differential
operator on X given by T f (y, z) = [T f (·, z)](y) or T f (y, z) = [T f (y, ·)](z) for f ∈ Cc∞ (X).
Theorem 9.9 Let xt be a G-invariant diffusion process in X = Y × Z with generator L,
and let yt and zt be respectively its radial and angular parts. Let Z be equipped with a
G-invariant Riemannian metric so that the Laplace-Beltrami operator ∆Z is defined on Z.
Assume Z = G/K is irreducible and dim(Z) > 1. Then
L = L1 + (b/2)∆Z ,
(9.3)
where L1 is a diffusion generator on Y (as defined in Appendix A.5) and b ≥ 0 is in C ∞ (Y ).
Moreover, yt is an L1 -diffusion process in Y , and the processes xt = (yt , zt ) and (yt , z̃a(t) )
are equal in distribution, where z̃t is a Riemannian Brownian motion in Z, independent of the
∫
process yt , and a(t) = 0t b(ys )ds. Consequently, with respect to an orthonormal basis at To Z,
under the conditional distribution given the process yt , zt is an inhomogeneous Lévy process
in Z represented by the Lévy triple (o, aI, 0), where I is the identity matrix and o denotes
the trivial drift in Z.
Note: If limt→∞ a(t) = ∞ almost surely, then Theorem 9.9 (b) may be strengthened as in
Remark 9.6.
Proof: The proof given below will not depend on Theorem 9.5. Let p = dim(Y ) and q =
dim(Z). By the definition of a diffusion process in Appendix A.5, for any x = (y, z) ∈ Y × Z,
there are open coordinate neighborhoods U of y in Y and V of z in Z such that L, restricted
276
on U × V , has the following form under local coordinates y1 , . . . , yp on U and z1 , . . . , zq on
V,
p
p
p ∑
q
q
q
∑
∑
∑
1 ∑
1 ∑
y z
y y
y
z z
γil ∂i ∂l +
L=
αij ∂i ∂j +
µi ∂i +
βlm ∂l ∂m +
νl ∂lz ,
2 i,j=1
2 l,m=1
i=1
i=1 l=1
l=1
(9.4)
where αij , βlm , γil , µi , νl ∈ C ∞ (U × V ), ∂iy = ∂/∂yi and ∂lz = ∂/∂zl . Moreover, the n × n
symmetric matrix


α
γ

Λ= ′
γ β
is ≥ 0 (nonnegative definite), where α = {αij }, γ = {γil } and β = {βlm }. This implies that
α ≥ 0 and β ≥ 0.
Any smooth function f on Y may be naturally extended to be a smooth function on
X = Y × Z by setting f¯(y, z) = f (y). Similarly, for any smooth function f on Z, we
let f˜(y, z) = f (z), a smooth function on X. By the G-invariance of L, for f ∈ C ∞ (Y ),
Lf¯(y, z) = Lf¯(y, gz) for any g ∈ G. This implies that Lf¯ does not depend on z, and so is
a smooth function on Y . Therefore, letting L1 f = Lf¯ defines an operator L1 on Y . From
∑
∑
(9.4), it is clear that L1 = (1/2) pi,j=1 αij ∂iy ∂jy + pi=1 µi ∂iy with αij , µi ∈ C ∞ (Y ) in local
coordinates y1 , . . . , yp . Because α ≥ 0, L1 is a diffusion generator on Y .
For fixed y ∈ Y , let Ly be the operator on Z defined by Ly f (z) = Lf˜(y, z) for f ∈ C ∞ (Z)
and z ∈ Z. The G-invariance of L implies that Ly is also G-invariant. By (9.4), it is clear that
∑
∑
z
Ly = (1/2) ql,m=1 βlm (y, z)∂lz ∂m
+ ql=1 νl (y, z)∂lz in local coordinates z1 , . . . , zq . Therefore,
Ly is a G-invariant second order linear differential operator on Z = G/K with Ly 1 = 0. By
Proposition 5.5, Ly = (b(y)/2)∆Z for some function b(y). By the above local expression of
Ly , its coefficients are smooth in (y, z) and so b(y) is smooth in y. Because β ≥ 0, b(y) ≥ 0.
For f ∈ C ∞ (Y ×Z), we may now write Lf (y, z) = L1 [f (·, z)](y)+(b(y)/2)∆Z [f (y, ·)](z)+
L̂f (y, z) for some expression L̂f (y, z). It remains to show L̂f (y, z) = 0. By (9.4), it is clear
∑
that L̂f (y, z) = i,l γil ∂iy ∂lz f (y, z). It suffices to show L̂(f1 f2 ) = 0 for f1 ∈ C ∞ (Y ) and
f2 ∈ C ∞ (Z). For fixed f1 , the map f2 7→ L̂(f1 f2 ) is a G-invariant vector field on Z. By
Proposition 3.2, any G-invariant vector field is identified with an Ad(K)-invariant vector in
p, which has to be zero by the irreducibility of Z = G/K and dim(Z) > 1. We have proved
(9.3).
Let J: X = Y × Z → Y be the natural projection. Then for f ∈ Cc∞ (Y ), L(f ◦ J) =
(L1 f ) ◦ J, it follows that yt = J(xt ) is an L1 -diffusion process in Y .
∑
By Propositions 3.22 and 3.23, ∆Z = qi=1 ξi ξi , where ξ1 , . . . , ξq is an orthonormal basis
of p with respect to the Ad(K)-invariant inner product that induces the G-invariant Rieman∑
nian metric on Z. Note that qi=1 ξi ξi may also be regarded as a left invariant differential
operator on G, which will be denoted as T . Then for f ∈ Cc∞ (Z), T (f ◦ π) = (∆Z f ) ◦ π,
where π: G → Z = G/K is the natural projection.
277
Any y ∈ Y has an open neighborhood U and smooth vector fields ζ0 , ζ1 , . . . , ζp such that
∑
L = (1/2) pi=1 ζi ζi + ζ0 on U . Then the radial process yt may be obtained as the solution
of the following Stratonovich sde on U ,
1
dyt =
p
∑
ζi (yt ) ◦ dBti + ζ0 (yt )dt,
(9.5)
i=1
where Bt = (Bt1 , . . . , Btp ) is a standard Brownian motion of dim p. We may extend yt beyond
the time τ when it reaches the boundary of U by solving another sde driven by the same
Brownian motion Bt time shifted by τ , and thus we may assume the process yt is adapted
to the completed natural filtration of Bt .
By the standard theory of sde’s as reviewed in Appendix A.5, a diffusion process in Y ×G
with generator L1 + (b/2)T may be obtained as the solution of a system of sde’s consisting
of the sde (9.5) on Y and the following sde on G
dgt =
q
∑
[b(yt )1/2 ξi (gt )] ◦ dBt′i ,
(9.6)
i=1
where Bt′ = (Bt′1 , . . . , b′q
t ) is a standard Brownian motion of dim q, independent of Bt . Because
b(y) ≥ 0 is smooth, b(y)1/2 is locally Lipschitz continuous by [42, IV.Proposition 6.2], so the
system of sde’s on Y × G has a unique solution (yt , gt ) for any given initial point. Because
T (f ◦ π) = (∆Z f ) ◦ π for f ∈ Cc∞ (Z), it follows that (yt , zt ) = (yt , π(gt )) is a diffusion process
in X = Y × Z with generator L0 = L1 + (b/2)∆Z .
By (9.6), for f ∈ Cc∞ (Z),
f (zt ) = f ◦ π(gt ) =
q ∫
∑
i=1 0
t
[b(ys )1/2 ξi (gs )]dBs′i +
∫
0
t
[b(ys )/2]T (f ◦ π)(gs )ds,
it follows that under the conditional distribution given the process yt ,
f (zt ) −
∫
t
0
[b(ys )/2]∆Z f (zs )ds =
q ∫
∑
i=1 0
t
[b(ys )1/2 ξi (gs )]dBs′i
∑
is a martingale. Recall ∆Z = qi=1 ξi ξi . By the above martingale property, we see that zt is
an inhomogeneous Lévy process in Z represented by the Lévy triple (o, aI, 0).
Let z̃t be a Riemannian Brownian motion in Z, independent of the process yt . Then
∫
for f ∈ Cc∞ (Z), Mt = f (z̃t ) − 0t (1/2)∆Z f (z̃s )ds is a martingale. Let zt′ = z̃a(t) , where
∫
a(t) = 0t b(ys )ds. Then given the process yt ,
f (zt′ ) −
∫
t
0
= f (z̃a(t) ) −
[b(ys )/2]∆Z f (zs′ )ds = f (z̃a(t) ) −
∫
0
a(t)
(1/2)∆Z f (z̃s )ds = Ma(t)
278
∫
0
t
[b(ys )/2]∆Z f (z̃a(s) )ds
is a martingale, and hence zt′ is an inhomogeneous Lévy process in Z represented by the
Lévy triple (o, aI, 0). It follows that given the process yt , the two processes zt and zt′ are
equal in distribution. This implies that the two processes (yt , zt ) and (yt , z̃a(t) ) are equal in
distribution. 2
As at the beginning of this section, let X be Rn with the origin removed, n ≥ 2, and
X = Y × Z with Y = (0, ∞) and Z = S n−1 , the unit sphere in Rn . A Brownian motion xt
in Rn with x0 ∈ X may be regarded as a diffusion process in X, invariant under G = O(n),
with generator (1/2)∆. It is well known that the Laplace operator ∆ on Rn has the following
decomposition under spherical polar coordinates.
∆=
n−1 ∂
∂2
1
+
∆Z ,
+
∂r2
r ∂r r2
(9.7)
where ∆Z is the Laplace-Beltrami operator on Z = S n−1 . By Theorem 9.9, the radial part
rt = |xt | is a diffusion process in (0, ∞) with generator
1 ∂2
n−1 ∂
L1 = [ 2 +
],
2 ∂r
r ∂r
(9.8)
which is known as a Bessel process, and the angular part zt = xt /rt is given by zt = z̃a(t) ,
where z̃t is a Riemannian Brownian motion in Z = S n−1 , independent of the process rt ,
∫
and a(t) = 0t ds/rs2 . Consequently, given the process rt , zt is a continuous inhomogeneous
Lévy process in Z = S n−1 with covariance matrix function a(t)I (with respect to any orthonormal basis at a chosen base point in Z).
279
280
Appendices
Basic definitions and facts about Lie groups are recalled in the first two appendices. The
reader is referred to the standard textbooks such as Warner [90] or the first two chapters of
Helgason [33] for more details. Appendices 3, 4 and 5 contain a brief summary of stochastic
processes and stochastic analysis, including Markov processes, Feller processes, Brownian
motion, martingales, Itô and Stratonovich stochastic integrals, stochastic differential equations, and Poisson random measures. The reader is referred to standard textbooks such as
those by Kallenberg [46] or Revuz and Yor [79] for more details.
A.1
Lie groups
A Lie group G is a group and a manifold such that both the product map G × G ∋ (g, h) 7→
gh ∈ G and the inverse map G ∋ g 7→ g −1 ∈ G are smooth. Let e be the identity element
of G, the unique element of G satisfying eg = ge = g for any g ∈ G. It is well known that
a Lie group is in fact analytic in the sense that the underlying manifold structure together
with the product and inverse maps are analytic.
The Lie algebra g of a Lie group G is the tangent space Te G of G at the identity element
e of G. For g ∈ G, let lg and rg : G → G be respectively the left and right translations
on G. A vector field X̃ on G is called left invariant if it is lg -related to itself, that is, if
Dlg (X̃) = X̃, for any g ∈ G, where Dlg is the differential of lg . Similarly, we define a right
invariant vector field on G using rg . Any left or right invariant vector field X̃ is determined
by its value at e, X ∈ g, and may be written as X̃(g) = Dlg (X) or X̃(g) = Drg (X) for
g ∈ G. Such a left or right invariant vector field will be denoted by X l or X r respectively.
The Lie bracket of two vector fields Z1 and Z2 on a manifold is defined to be the vector field
[Z1 , Z2 ] = Z1 Z2 − Z2 Z1 . For any two elements X and Y of g, [X l , Y l ] is left invariant. The
Lie bracket [X, Y ] is defined to be the element of g corresponding to this left invariant vector
field, that is, [X, Y ]l = [X l , Y l ]. It satisfies the following properties: For any X, Y, Z ∈ g,
(a) [aX + bY, Z] = a[X, Z] + b[Y, Z] for a, b ∈ R,
(b) [X, Y ] = −[Y, X], and
281
(c) [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0.
The identity (c) is called the Jacobi identity.
In general, any linear space g equipped with a Lie bracket satisfying (a), (b) and (c) is
called a Lie algebra. For example, the space of all the vector fields on a manifold is a Lie
algebra with Lie bracket of vector fields.
The exponential map exp: g → G is defined by exp(X) = ψ(1), where ψ(t) is the unique
solution of the ordinary differential equation (d/dt)ψ(t) = X(ψ(t)) satisfying the initial
condition ψ(0) = e. It is a diffeomorphism of an open neighborhood of 0 in g onto an open
neighborhood of e in G. We may write eX for exp(X).
A group homomorphism F : G → G′ between two Lie groups G and G′ is called a Lie
group homomorphism (resp. isomorphism) if it is a smooth (resp. diffeomorphic) map. A
linear map f : g → g′ between two Lie algebras g and g′ is called a Lie algebra homomorphism if [f (X), f (Y )] = f ([X, Y ]) for X, Y ∈ g. Such an f will be called a Lie algebra
isomorphism if it is a bijection. Let g and g′ be respectively the Lie algebras of G and G′ .
If F : G → G′ is a Lie group homomorphim (resp. isomorphism), then DF : g → g′ is a Lie
algebra homomorphism (resp. isomorphism) and F (eX ) = eDF (X) for any X ∈ g. Two Lie
groups (resp. Lie algebras) are called isomorphic if there is a Lie group (resp. Lie algebra)
isomorphism between them. A Lie group (resp. Lie algebra) isomorphism from G (resp. g)
onto itself will be called a Lie group (resp. Lie algebra) automorphism on G (resp. g).
For g ∈ G, the map cg : G ∋ h 7→ ghg −1 ∈ G is a Lie group automorphism on G,
called a conjugation map on G. Its differential map Ad(g): g → g, given by Ad(g)X =
Dlg ◦ Drg−1 (X), is a Lie algebra automorphism on g and satisfies geX g −1 = eAd(g)X for any
X ∈ g. It can be shown that for X, Y ∈ g,
d
Ad(etX )Y |t=0 .
(A.1)
dt
A Lie group G is called abelian if any two elements of G commute, that is, if gh = hg
for g, h ∈ G. A Lie algebra g is called abelian if [X, Y ] = 0 for X, Y ∈ g. It is easy to show
that the Lie algebra of an abelian Lie group is abelian and that a connected Lie group with
an abelian Lie algebra is abelian.
A subset K of G is called a Lie subgroup of G if it is a subgroup and a sub-manifold of G
such that both the product and inverse maps are smooth under the sub-manifold structure.
It is well known that if K is a closed subgroup of G, then there is a unique manifold structure
on K under which K is a Lie subgroup and a topological subspace of G. It is easy to show
that the identity component of G, that is, the connected component of G containing e, is an
open and closed Lie subgroup of G.
For any two subsets h and k of a Lie algebra g, let [h, k] denote the subspace of g spanned
by [X, Y ] for X ∈ h and Y ∈ k. A linear subspace h of a Lie algebra g is called a Lie
[X, Y ] =
282
sub-algebra, or simply a sub-algebra, of g, if [h, h] ⊂ h. If G is a Lie group with Lie algebra
g, then the Lie algebra k of a Lie subgroup K of G is a sub-algebra of g. Conversely, given
any sub-algebra k of g, there is a unique connected Lie subgroup K whose Lie algebra is k,
called the Lie subgroup generated by k. A sub-algebra k of g is called an ideal if [g, k] ⊂ k. In
this case, the Lie subgroup K generated by k is a normal subgroup of G, that is, gKg −1 = K
for g ∈ G. For Γ ⊂ g, the Lie algebra generated by Γ is defined to be the smallest Lie
sub-algebra of g containing Γ and is denoted by Lie(Γ).
The center of G is Z = {h ∈ G; hg = gh for any g ∈ G}. This is a closed normal
subgroup of G with Lie algebra z = {X ∈ g; [X, Y ] = 0 for any Y ∈ g}, which is called the
center of g.
Let G1 and G2 be two Lie groups. Then G = G1 × G2 is a Lie group with the product
group and product manifold structures, called the product Lie group of G1 and G2 . Its Lie
algebra is isomorphic to g = g1 ⊕ g2 , where gi is the Lie algebra of Gi for i = 1, 2 with
[g1 ⊕ {0}, {0} ⊕ g2 ] = {0}. Therefore, each gi can be regarded as an ideal of g and each Gi
as a normal subgroup of G. Similarly, we define the product of more than two Lie groups.
A Riemannian metric {⟨·, ·⟩g ; g ∈ G} on G is called left invariant if ⟨Dlg (X), Dlg (Y )⟩gh =
⟨X, Y ⟩h for any g, h ∈ G and X, Y ∈ Th G. A right invariant metric is defined by replacing
Dlg by Drg . If ⟨·, ·⟩ is an inner product on g, then it determines a unique left (resp. right)
invariant Riemannian metric ⟨·, ·⟩lg (resp. ⟨·, ·⟩rg ), defined by
⟨Dlg (X), Dlg (Y )⟩lg = ⟨X, Y ⟩
(resp. ⟨Drg (X), Drg (Y )⟩rg = ⟨X, Y ⟩)
for X, Y ∈ g and g ∈ G. Under this metric, lg (resp. rg ) is an isometry on G for any g ∈ G.
As an example of Lie group, let GL(n, R) be the group of all the n × n real invertible
matrices. Since such a matrix represents a linear automorphism on Rn , GL(n, R) is called
the general linear group on Rn . With the matrix multiplication and inverse, and with the
natural identification of GL(n, R) with an open subset of the n2 -dimensional Euclidean space
2
Rn , GL(n, R) is a Lie group of dimension n2 . The identity element e is the n × n identity
matrix In . The Lie algebra of GL(n, R) is the space gl(n, R) of all the n × n real matrices.
∑
k
The exponential map is given by the usual matrix exponentiation eX = I + ∞
k=1 X /k! and
the Lie bracket is given by [X, Y ] = XY − Y X for X, Y ∈ gl(n, R). Note that GL(n, R)
has two connected components and its identity component GL(n, R)+ is the group of all
g ∈ GL(n, R) with determinant det(g) > 0. We have
∀X ∈ gl(n, R),
det(eX ) = exp[Tr(X)],
∑
where Tr(X) = i Xii is the trace of X.
Similarly, the group GL(n, C) of n × n complex invertible matrices is a Lie group, called
the complex general linear group on Cn . Its Lie algebra gl(n, C) is the space of all the n × n
complex matrices with Lie bracket [X, Y ] = XY − Y X.
283
Let V be a d-dimensional linear space and let GL(V ) denote the set of the linear bijections: V → V . By choosing a basis of V , GL(V ) may be identified with the general linear
group GL(d, R), Therefore, GL(V ) is a Lie group and its Lie alebra is the space gl(V ) of all
the linear endomorphsims on V with Lie bracket [X, Y ] = XY − Y X.
A.2
Action of Lie groups and homogeneous spaces
A left action of a group G on a set S is a map F : G → S satisfying F (gh, x) = F (g, F (h, x))
and F (e, x) = x for g, h ∈ G and x ∈ S. When G is a Lie group, M is a manifold and
F is smooth, we say G acts smoothly on M on the left by F and we often omit the word
“smoothly”. A right action of G on M is defined similarly with F (gh, x) = F (h, F (g, x)). For
simplicity, we may write gx for the left action F (g, x) and xg for the right action. Because
left actions are encountered more often, therefore, an action in this book will mean a left
action unless explicitly stated otherwise.
The subset Gx = {gx; g ∈ G} of M is called an orbit of G on M . It is clear that if y ∈ Gx,
then Gx = Gy. The action of G on M will be called effective if F (g, ·) = idM =⇒ g = e,
and it will be called transitive if any orbit of G is equal to M . If G acts effectively on M , it
will be called a Lie transformation group on M . By Helgason [33, II.Proposition 4.3], if G is
transitive on a connected manifold M , then its identity component is also transitive on M .
Any X ∈ g induces a vector field X ∗ on M given by X ∗ f (x) = (d/dt)f (etX x) |t=0 for
any f ∈ C 1 (M ) and x ∈ M . We have (see in [33, II.Theorem 3.4]), [X, Y ]∗ = −[X ∗ , Y ∗ ] for
X, Y ∈ g. The Lie algebra g∗ = {X ∗ ; X ∈ g} is of finite dimensional. A vector field Y on M
is called complete if any solution of the equation (d/dt)x(t) = Y (x(t)) extends to all time t.
It is clear that X ∗ is complete. In general, if Γ is a collection of complete vector fields on
M such that Lie(Γ) is finite dimensional, then there is a Lie group G acting effectively on
M with Lie algebra g such that Lie(Γ) = g∗ , see Palais [71, IV.Theorem III]. Note that any
vector field on a compact manifold is complete.
Let K be a closed subgroup of G. The set of left cosets gK for g ∈ G is denoted by
G/K and is called a homogeneous space of G. It is equipped with the quotient topology,
the smallest topology under which the natural projection G → G/K given by g 7→ gK is
continuous. By [33, II.Theorem 4.2], there is a unique manifold structure on G/K under
which the natural action of G on G/K, defined by g ′ K 7→ gg ′ K, is smooth. Similarly, we
may consider the right coset space K\G = {Kg; g ∈ G} on which G acts naturally via the
right action Kg ′ 7→ Kg ′ g.
Suppose a Lie group G acts on a manifold M . Fix p ∈ M . Let K = {g ∈ G; gp = p}.
Then K is a closed subgroup of G, called the isotropy subgroup of G at p. By Theorem 3.2
and Proposition 4.3 in [33, Chapter II] if the action of G on M is transitive, then the map:
284
gK →
7 gp is a diffeomorphism from G/K onto M , therefore, M may be identified with G/K.
Let G be a Lie group and K be a closed subgroup. If N is a Lie subgroup of G, then for
any x = gK ∈ G/K, the orbit N x may be naturally identified with N/(N ∩ K) via the map
nx 7→ n(N ∩ K) and, hence, it is equipped with the manifold structure of a homogeneous
space. By [33, II.Proposition 4.4], N x is in fact a sub-manifold of G/K. Moreover, if K is
compact and N is closed, then N x is a closed topological sub-manifold of G/K.
Let G be a Lie group with Lie algebra g and let K be a closed subgroup with Lie algebra
k. Via the natural action of G on G/K, any g ∈ G is a diffeomorphism: G/K → G/K. Let
π: G → G/K be the natural projection. Since k ∈ K fixes the point π(e) = eK in G/K,
Dk is a linear bijection: TeK (G/K) → TeK (G/K) and
Dk ◦ Dπ = Dπ ◦ Ad(k) : g → TeK (G/K).
The homogeneous space G/K is called reductive if there is a subspace p of g such that
g = k ⊕ p (direct sum) and p is Ad(K)-invariant in the sense that Ad(k)p ⊂ p for any
k ∈ K. Let ⟨·, ·⟩ be an inner product on p that is Ad(K)-invariant in the sense that
⟨Ad(k)X, Ad(k)Y ⟩ = ⟨X, Y ⟩ for k ∈ K and X, Y ∈ p. Then it induces a Riemannian metric
{⟨·, ·⟩x ; x ∈ G/K} on G/K, given by
∀g ∈ G and X, Y ∈ p,
⟨Dg ◦ Dπ(X), Dg ◦ Dπ(Y )⟩gK = ⟨X, Y ⟩.
By the Ad(K)-invariance of ⟨·, ·⟩, this metric is well defined and is G-invariant in the sense
that
∀g, u ∈ G and X, Y ∈ TuK (G/K), ⟨Dg(X), Dg(Y )⟩guK = ⟨X, Y ⟩uK .
A.3
Stochastic processes
Let (Ω, F, P ) be a probability space. A random variable z taking values in a measurable
space (S, S) by definition is a measurable map z: Ω → S. Its distribution is the probability
measure zP = P ◦ z −1 on (S, S).
The expectation E(X) of a real-valued random variable X is just the integral P (X) =
∫
∫
XdP whenever it is defined. We may write E[X; B] = B XdP for B ∈ F. When
E(|X|) < ∞ or when X ≥ 0 P -almost surely, the conditional expectation of X given a
σ-algebra G ⊂ F, denoted by E[X | G], is defined to be the P -almost surely unique G∫
∫
measurable random variable Y such that A Y dP = A XdP for any A ∈ G.
A stochastic process or simply a process xt , t ∈ R+ = [0, ∞), in a state space S is a
family of S-valued random variables indexed by t, and may be regarded as a measurable
map x: Ω → S R+ , where S R+ is equipped with the product σ-algebra S R+ . The probability
measure xP on (S R+ , S R+ ) is called the distribution of the process, which is determined
285
by the collection of the joint distributions of xt1 , xt2 , . . . , xtn over finitely many time points
t1 < t2 < · · · < tn , called the finite dimensional distributions of the process. The process xt
may also be written as xt (ω) or x(t, ω) to indicate its dependence on ω ∈ Ω.
We normally identify two random variables that are equal P -almost surely and two
processes that are equal as functions of time t almost surely. A version of a process xt is
another process yt such that xt = yt almost surely for each t ∈ R+ . We may also consider a
process xt with a different time index set, for example, a ≤ t ≤ b or t = 0, 1, 2, 3, . . ..
A process xt in a topological space S, with S being the Borel σ-algebra B(S) on S, will
be called continuous, right continuous, etc. if for P -almost all ω, the path t 7→ xt (ω) is so.
A process is called rcll if almost all its paths are right continuous with left limits.
For a collection C of sets or functions, let σ(C) denote the smallest σ-algebra under which
all the elements in C are measurable, called the σ-algebra generated by C. A filtration on
(Ω, F, P ) is a family of σ-algebras Ft on Ω, t ∈ R+ , such that Fs ⊂ Ft ⊂ F for 0 ≤ s ≤
∪
t < ∞. We will write F∞ for σ( t≥0 Ft ). A probability space (Ω, F, P ) equipped with a
filtration {Ft } is called a filtered probability space and may be denoted by (Ω, F, {Ft }, P ).
Let N be the collection of all null sets in F∞ , that is, the subsets of sets in F∞ with zero
∩
P -measure. The filtration is called right continuous if Ft+ = u>t Fu is equal to Ft for all
t ≥ 0. It is called complete if N ⊂ Ft for all t ≥ 0. The completion of a filtration {Ft } is
the complete filtration {F̄t }, where F̄t = σ{Ft ∪ N } for 0 ≤ t < ∞. Note that A ∈ F̄t if
and only if there is B ∈ Ft such that (A − B) ∪ (B − A) ∈ N . We have {F̄t+ } is equal to
the completion of {Ft+ }.
A filtration {Ft } is said to be generated by a collection of processes xλt , λ ∈ Λ, and
random variables z α , α ∈ A, if Ft = σ{z α and xλu ; α ∈ A, λ ∈ Λ and 0 ≤ u ≤ t}. The
filtration generated by a single process is called the natural filtration of that process. Its
completion will be called the completed natural filtration of the process.
A process xt in a linear space is said to have independent increments if for s < t, xt − xs
is independent of σ{xu ; 0 ≤ u ≤ t}. This may also be defined for a group-valued process if
−1
the increment xt − xs over time interval [s, t] is replaced by x−1
s xt or xt xs . Let {Ft } be the
completed natural filtration of a rcll process xt with independent increments. Then {Ft } is
right continuous (see Theorem A.1 in Appendix A.7).
A stopping time τ under a filtration {Ft }, or an {Ft }-stopping time, is a random variable
taking values in [0, ∞] such that the set [τ ≤ t] belongs to Ft for any t ∈ R+ . For a stopping
time τ ,
Fτ = {B ∈ F∞ ; B ∩ [τ ≤ u] ∈ Fu for any u ∈ R+ }
is a σ-algebra, which is just Ft for τ = t. If τ ≤ σ are two stopping times, then Fτ ⊂ Fσ .
If the filtration Ft is right continuous, then [τ ≤ t] in the definitions of stopping time τ and
its σ-algebra Fτ may be replaced by [τ < t].
286
A process xt is said to be adapted to a filtration {Ft } if xt is Ft -measurable for each
t ∈ R+ . It is always adapted to its natural filtration {Ft0 } given by Ft0 = σ{xs ; 0 ≤ s ≤ t}.
It will be called adapted to another process yt if it is adapted to the natural filtration of yt .
Let xt be a process in a topological space S. It is called progressive, under a given
filtration Ft , if ∀t ≥ 0, the map [0, t] × Ω → S given by (s, ω) 7→ xs (ω) is measurable
under B([0, t]) × Ft . A progressive process is clearly adapted. It is easy to see that an
adapted process is progressive if it is either right or left continuous. It is well known that if
xt is progressive, then for any stopping time τ , xτ restricted to [τ < ∞] is Fτ -measurable.
Moreover, for the natural filtration Ft0 of a progressive process xt , the hitting time τ of a
Borel subset B of S, defined by τ = inf{t > 0; xt ∈ B}, is a stopping time under the
0
}.
completion of {Ft+
A kernel from (S, S) to another measurable space (T, T ) is a map K: S × T → R+ such
that K(x, ·) is a measure on T for each fixed x ∈ S and x 7→ K(x, B) is S-measurable for
each fixed B ∈ T . It is called a probability kernel if K(x, ·) is a probability measure for all
∫
x ∈ S. For a measurable function f on T , we may write Kf (x) = f (y)K(x, dy) whenever
the integral exists, and thus may regard K as an operator on a function space.
A family of probability kernels Pt , t ∈ R+ , from (S, S) to itself is called a transition
semigroup on S if Ps+t = Pt Ps and P0 (x, ·) = δx (unit mass at point x). A process xt taking
values in S will be called a Markov process with transition semigroup Pt if the following
Markov property holds:
∀t > s and ∀f ∈ S+ ,
E[f (xt ) | Fs0 ] = Pt−s f (xs ),
(A.2)
P -almost surely. The above Markov property may be strengthened by replacing the natural
filtration Ft0 by a larger filtration Ft , then xt is called a Markov process with respect to {Ft }
or an {Ft }-Markov process. The distribution of a Markov process is determined completely
by the transition semigroup Pt and the initial distribution ν, the distribution of x0 . In fact,
its finite dimensional distributions are given by
P [f1 (xt1 )f2 (xt2 ) · · · fk (xtk )]
∫
=
∫
ν(dx)
Pt1 (x, dz1 )f1 (z1 )
∫
Pt2 −t1 (z1 , dz2 )f2 (z2 ) · · ·
∫
Ptk −tk−1 (zk−1 , dzk )fk (zk )
for t1 < t2 < · · · < tk and f1 , f2 , . . . , fk ∈ S+ .
The Markov process will often be assumed to be rcll when S is a topological space with
S = B(S). In this case we may take Ω to be the canonical path space, that is, the space of
all the rcll maps ω: R+ → S with F being the σ-algebra generated by all the maps ω 7→ ω(t)
for t ∈ R+ . Then the process xt is the coordinate process xt (ω) = ω(t). Its distribution on Ω
will be denoted as Pν , where ν is the initial distribution, that is, the distribution of x0 . For
x ∈ S and ν = δx , Pν is simply written as Px . Note that x 7→ Px is a probability kernel from
287
S to Ω. When xt is continuous, we may use the space of continuous paths as the canonical
path space.
The state space S of a Markov process xt is often taken to be a second countable locally
compact Hausdorff space. We will allow a Markov process xt to have a possibly finite life
time ζ, a random variable taking values in [0, ∞]. To accommodate this, the state space
S is enlarged to include an additional point ∆, called the point at infinity, so that xt ∈ S
for t < ζ and xt = ∆ for t ≥ ζ. We will take ∆ to be the point at infinity of the onepoint compactification of a noncompact S, or an isolated point for a compact S, so that
S∆ = S ∪ {∆} is always compact. The canonical path space Ω is now taken to be all rcll
paths ω(·) in S∆ such that ω(t) ∈ S for t < ζω , for some ζω ∈ [0, ∞], and ω(t) = ∆ for
t ≥ ζω . Any function f on S is automatically extended to S∆ by setting f (∆) = 0. The
Markov property (A.2) is assumed to hold under this convention. In this case, the transition
semigroup Pt is a family of sub-probability kernels (that is, Pt 1 ≤ 1 instead of Pt 1 = 1). It
may be extended to be a probability kernel on S∆ by setting Pt (x, {∆}) = 1 − Pt (x, S), and
then xt becomes a Markov process in S∆ with an infinite life time.
Let xt be a rcll Markov process, regarded as the coordinate process on the canonical path
0
space Ω. For any probability measure µ on S, let {Ftµ } be the completion of {Ft+
} under Pµ .
µ
µ
Then Ft is right continuous, and so the hitting time of a Borel set by xt is an {Ft }-stopping
time. Let {Ft } be the filtration defined by Ft = ∩µ Ftµ , where the intersection ∩µ is taken
over all probability measures µ on S, which may be called the augmented filtration of the
Markov process xt with respect to all initial distributions. Then {Ft } is a right continuous
filtration, xt is adapted to it, and the hitting time of a Borel set is an {Ft }-stopping time.
The Markov process xt is said to have the strong Markov property if for any f ∈ S+ ,
u ∈ R+ and {Ft }-stopping time τ ,
E[f (xu+τ )1[τ <∞] | Fτ ] = Exτ [f (xu )] on [τ < ∞].
(A.3)
This strong Markov property can be written in the following equivalent but more useful
0
form: For any ξ ∈ (F∞
)+ ,
E[(ξ ◦ θτ )1[τ <∞] | Fτ ] = Exτ (ξ)
on [τ < ∞],
where θt is the time shift operator: Ω → Ω defined by (θt ω)(s) = ω(t+s), and so xt+s = xs ◦θt
for any s, t ∈ R+ .
An important class of Markov processes are Feller processes. Let S be a second countable
locally compact Hausdorff space. The space C0 (S) of the continuous functions on S vanishing
at ∞ is a Banach space under the norm ∥f ∥ = supx∈S |f (x)|. A transition semigroup Pt
on S (of sub-probability kernels) is called a Feller transition semigroup if C0 (S) is invariant
under Pt (that is, Pt f ∈ C0 (S) for f ∈ C0 (S) and t ∈ R+ ) and Pt f → f in C0 (S) as
288
t → 0 (which may be relaxed to the apparently weaker but in this case equivalent point-wise
convergence). Given such a transition semigroup and a probability measure ν on S, there is
a rcll Markov process xt in S, unique in distribution, with Pt as transition semigroup and ν
as the initial distribution. Such a process is called a Feller process. A Feller process has the
strong Markov property and its completed natural filtration is right continuous. Moreover,
any Markov process with a Feller transition semigroup has a rcll version, and so is a Feller
process up to a version.
The generator of the Feller process xt , or its transition semigroup Pt , is a linear operator
on C0 (S) defined by
Pt f − f
∀f ∈ D(L), Lf = lim
,
t→0
t
where the domain D(L) is the space of f ∈ C0 (S) for which the above limit exists in the
Banach space C0 (S). The generator L determines the transition semigroup Pt completely
and is a closed operator with D(L) dense in C0 (S). The reader is referred to a standard text
such as [46, chapter 19] for more details.
We may also consider an inhomogeneous Markov process. A family of probability kernels
Ps,t , s, t ∈ R+ with s ≤ t, from (S, S) to itself is called a two-parameter transition semigroup
on S if for r ≤ s ≤ t, Pr,t = Pr,s Ps,t and Pt,t (x, ·) = δx . A process xt taking values in S will
be called an inhomogeneous Markov process with (two-parameter) transition semigroup Ps,t
if the following Markov property holds:
∀t > s and ∀f ∈ S+ ,
E[f (xt ) | Fs0 ] = Ps,t f (xs ),
P -almost surely. If the above holds for a larger filtration {Ft }, a possibly stronger property,
then the inhomogeneous Markov process xt is said to be associated to {Ft }. The finite
dimensional distributions of an inhomogenous Markov process xt are given by
P [f1 (xt1 )f2 (xt2 ) · · · fk (xtk )]
∫
=
∫
ν(dx)
P0,t1 (x, dz1 )f1 (z1 )
∫
Pt1 ,t2 (z1 , dz2 )f2 (z2 ) · · ·
∫
Ptk−1 ,tk (zk−1 , dzk )fk (zk )
for t1 < t2 < · · · < tk and f1 , f2 , . . . , fk ∈ S+ , where ν is the initial distribution. We may
also allow an inhomogeneous Markov process xt to have a possibly finite life time, and then
its transition semigroup Ps,t consists of sub-probability kernels.
A.4
Stochastic integrals
By [46, Theorem 13.4], if Bt is a continuous process in Rd with independent increments, then
there are continuous functions b(t) in Rd and A(t) in the space of d × d nonnegative definite
symmetric matrices such that Bt − Bs has the normal distribution of mean b(t) − b(s) and
289
covariance A(t) − A(s) for s < t. If b(t) = 0 and Aij (t) = aij t for a constant matrix {aij },
then Bt is called a Brownian motion with covariance matrix {aij }. In this case, B0 = 0 and
Bt − Bs for s < t has the normal distribution of mean 0 and covariance {aij (t − s)}. If
aij = δij , then Bt is called a standard Brownian motion. More generally, if b(t) = 0, then Bt
will be called an inhomogeneous Brownian motion with covariance matrix function A(t). A
Brownian motion Bt , possibly inhomogeneous, is said to be associated to a filtration {Ft },
or an {Ft }-Brownian motion, if it is adapted to {Ft } and for s < t, Bt − Bs is independent
of Fs . A Brownian motion is always associated to its natural filtration.
Given a filtration {Ft }, a real-valued process Mt is called a martingale if it is adapted,
E[|Mt |] < ∞ for any t ∈ R+ and E[Mt | Fs ] = Ms for s < t. If {Ft } is right continuous, then
any martingale has a rcll version. From now on, a martingale will be always assumed to be
rcll. It is clear that an {Ft }-martingale is a {F̄t }-martingale, then by the right continuity of
Mt , it is also a {F̄t+ }-martingale. A martingale Mt is called an L2 -martingale if E[Mt2 ] < ∞
for t ∈ R+ . As a special case of Doob’s norm inequality (see for example [46, Proposition
7.16]), if Mt is an L2 -martingale, then
E[(Mt∗ )2 ] ≤ 4E(Mt2 ),
(A.4)
where Mt∗ = sup0≤s≤t |Ms |.
A process Ht stopped by a stopping time τ is the process Htτ = Ht∧τ , where t ∧ τ =
min(t, τ ). It is said to have a certain property locally or to be a local process of a certain
type if there are stopping times τn ↑ ∞ P -almost surely such that the stopped processes
(M − M0 )τt n = Htτn − M0 possess the property or are the processes of that type. For example,
a local martingale Mt is a process for which there are stopping times τn ↑ ∞ such that
Mtτn − M0 is a martingale.
A process may be regarded as a function on R+ × Ω. The predictable σ-algebra P on
R+ × Ω is the smallest σ-algebra under which all real-valued, adapted and left continuous
processes are measurable. A process is called predictable if it is measurable under P. Let Mt
and Nt be two local L2 -martingales. By the Doob-Meyer decomposition, there is a unique
predictable rcll process ⟨M, N ⟩t of finite variation such that ⟨M, N ⟩0 = 0 and Mt Nt −⟨M, N ⟩t
is a local martingale. The process ⟨M ⟩t = ⟨M, M ⟩t is increasing, and if ⟨M ⟩t = 0 for all
t > 0, then Mt = M0 almost surely for all t > 0. It is easy to show that a local L2 -martingale
Mt is an L2 -martingale if and only if E(M02 ) < ∞ and E[⟨M ⟩t ] < ∞ for any t ∈ R+ .
An inhomogeneous Brownian motion Bt with covariance matrix function A(t) is a continuous L2 -martingale with ⟨B i , B j ⟩t = Aij (t).
We now introduce stochastic integral. A process is called lcrl if its paths are left continuous with right limits. Given a complete and right continuous filtration {Ft }, we will
define Itô’s stochastic integral of a real valued adapted lcrl process Ht with respect to a local
290
∫
L2 -martingale Mt over time interval (0, t], denoted as (H · M )t or 0t Hs dMs . We first assume
Mt is an L2 -martingale and Ht is bounded. For any partition ∆: 0 = t0 < t1 < · · · < tk < · · ·
of R+ , where tk ↑ ∞ as k ↑ ∞, with mesh ∥∆∥ = supi≥1 |ti − ti−1 |, consider the following
special Riemann sum,
St∆ (H, M ) =
k
∑
Hti−1 (Mti − Mti−1 ) + Htk (Mt − Mtk ),
t ∈ [tk , tk+1 ).
(A.5)
i=1
We note that St∆ (H, M ) is an L2 -martingale. It is easy to show that as ∥∆∥ → 0, for any
t ∈ R+ , St∆ (H, M ) converge in L2 to a random variable, which is defined to be (H · M )t =
∫t
2
∆
0 Hs dMs . By Doob’s norm inequality (A.4), sup0≤s≤t |Ss (H, M ) − (H · M )s | → 0 in L for
any t ∈ R+ , so (H · M )t has a rcll version that is an L2 -martingale. The following basic
properties are easy to establish:
(i) (H · M )t is bilinear in H and M ;
∫
(ii) ⟨H · M, N ⟩t = 0t Hs d⟨M, N ⟩s for any L2 -martingale Nt ;
(iii) for any stopping time τ , (H · M )τt = (H · M τ )t = (H τ · M τ )t = (H1[0, τ ] · M )t ;
(iv) almost surely, (H · M )t − (H · M )t− = Ht (Mt − Mt− ) for any t > 0.
Now for any adapted lcrl process Ht , and a local L2 -martingale Mt , we may choose
stopping times τn ↑ ∞ such that Htτn − H0 are bounded and Mtτn − M0 are L2 -martingales,
and define
(H · M )t = H0 (Mt − M0 ) + [(H τn − H0 ) · (M τn − M0 )]t
for t ≤ τn . By (iii), this consistently defines (H · M )t for any t ∈ R+ . We have, (H · M )t
is a local L2 -martingale, and sup0≤s≤t |Ss∆ (H, M ) − (H · M )s | → 0 in probability P for any
t ∈ R+ . Properties (i) – (iv) still hold with N in (ii) being any local L2 -martingale. Moreover,
(H · M )t is the (almost sure) unique local L2 -martingale for which (ii) holds for any local
L2 -martingale Nt . It is then easy to show the following associativity property:
(v) for any two adapted lcrl processes Ht and Kt , and any local L2 -martingale Mt , [(HK) ·
M ]t = [H · (K · M )]t .
We note that in our definition of stochastic integrals, it is not necessary to assume {Ft }
is complete and right continuous as we can always replace {Ft } by {F̄t+ }, and because
sup0≤s≤t |Ss∆ (H, M ) − (H · M )s | → 0 in P , (H · M )t has a rcll version adapted to {Ft }.
A process Xt is called a semi-martingale if it can be written as Xt = Mt + At , where Mt is
a local martingale and At is an adapted rcll process of finite variation. By [46, Lemma 26.5],
one may assume Mt is a locally bounded martingale, and so may assume Mt is a local L2 ∫
martingale. For an adapted lcrl process Ht , the Itô stochastic integral (H · X)t = 0t Hs dXs is
∫
∫
defined to be 0t Hs dMs + 0t Hs dAs , where the second integral is a path-wise Lebesque-Stieljes
291
integral. Note that (H · X)t is the limit in P of St∆ (H, X) as ∥∆∥ → 0. The process (H · X)t
is also a semi-martingale.
Let Xt and Yt be two adapted rcll processes, and let
Q∆
t (X, Y ) =
∞
∑
[X(ti ∧ t) − X(ti−1 ∧ t)][Y (ti ∧ t) − Y (ti−1 ∧ t)],
i=1
where ∆: 0 = t0 < t1 < t2 < · · · < tk ↑ ∞ is a partition of R+ = [0, ∞). The quadratic
covariance process [X, Y ]t is defined to be an adapted rcll process At of finite variation such
that sup0≤t≤T |At − Q∆
t (X, Y )| converges to 0 in P for any T > 0 as ∥∆∥ → 0, provided that
such a process At exists. In fact, for the existence of [X, Y ]t , a slightly stronger convergence
property is required, see Protter [76, §V5] for more details. Note that in [76], [X, Y ] is
defined to be the present [X, Y ] plus X0 Y0 . The continuous part [X, Y ]ct of [X, Y ]t is defined
by
∑
[X, Y ]t = [X, Y ]ct +
∆s [X, Y ],
0<s≤t
where ∆t Z = Zt − Zt− and ∆0 Z = 0 for any rcll process Zt . Note that ∆t [X, Y ] =
(∆t X)(∆t Y ), hence, [X, Y ]t = [X, Y ]ct if Xt and Yt do not jump at the same time. Note also
that if either Xt or Yt has finite variation, then [X, Y ]ct = 0.
If Xt and Yt are semi-martingales, then [X, Y ]t is defined and we have the following
integration by parts formula.
∫
Xt Yt = X0 Y0 +
∫
t
0
Xs− dYs +
0
t
Ys− dXs + [X, Y ]t .
(A.6)
In this case, if H and K are adapted lcrl processes, then [H · X, K · Y ] = (HK) · [X, Y ].
If Mt and Nt are (local) L2 -martingales, then [M, N ]t is the unique adapted rcll process
At of finite variation such that A0 = 0, Mt Nt − At is a (local) martingale and ∆t A =
(∆t M )(∆t N ) for t > 0. Note that the quadratic covariation process [M, N ]t may not be
equal to the predictable quadratic covariation process ⟨M, N ⟩t defined earlier, and [M, N ]t −
⟨M, N ⟩t is a local martingale of finite variation. However, if Mt and Nt do not jump at the
same time, then [M, N ]t = ⟨M, N ⟩t .
The Stratonovich stochastic integral of an adapted rcll process Ht with respect to a
semi-martingale Xt is defined as
∫
t
0
Hs− ◦ dXs =
∫
t
0
1
Hs− dXs + [H, X]ct
2
provided [H, X] exists. If Ht and Xt do not jump at the same time, then
equal to the limit in P of the sum
k
∑
1
i=1
2
[H(ti−1 ) + H(ti )][X(ti ) − X(ti−1 )]
292
(A.7)
∫t
0
Hs− ◦ dXs is
as the mesh of the partition tends to zero. The reader is referred to [76, §V5] for more details
on Stratonovich integrals.
∫
The Stratonovich stochastic integral Vt = 0t Ys− ◦ dZs is always defined for two semimartingales Yt and Zt , and it is also a semi-martingale. If Xt is another semi-martingale,
∫
∫
then 0t Xs− ◦ dVs = 0t (Xs− Ys− ) ◦ dZs , which may be written concisely as Xt− ◦ (Yt− ◦ dZt ) =
(Xt− Yt− ) ◦ dZt .
Let Xt = (Xt1 , . . . , Xtd ) be a d-tuple of semi-martingales and f, g ∈ C 1 (Rd ). Then
[f (X), g(X)]t is defined and is given by
[f (X), g(X)]t =
d ∫
∑
t
i,j=1 0
∂i f (Xs )∂j g(Xs )d[X i , X j ]cs +
∑
∆s f (X) ∆s g(X),
(A.8)
0<s≤t
∫
where ∂i = (∂/∂xi ). In particular, the Stratonovich stochastic integral 0t f (Xs− ) ◦ dYs is
defined for any f ∈ C 1 (Rd ) and semi-martingale Yt . If f ∈ C 2 (Rd ), then f (Xt ) is a semimartingale and
∫
f (Xt ) = f (X0 ) +
= f (X0 ) +
+
∑
t
0
∑
Df (Xs− ) ◦ dXs +
d ∫
∑
[∆s f (X) − Df (Xs− )(∆s X)]
(A.9)
0<s≤t
t
i=1 0
∂i f (Xs− )dXsi +
d ∫ t
1 ∑
∂i ∂j f (Xs )d[X i , X j ]cs
2 i,j=1 0
[∆s f (X) − Df (Xs− )(∆s X)]
0<s≤t
∑
∑
where Df (Xt− ) ◦ dXt = i ∂i f (Xt− ) ◦ dXti and Df (Xt− )(∆t X) = i ∂i f (Xt− )∆t X i . This is
an extended version of Itô’s formula, see Theorem 20 in [76, §V5]. When Xt is continuous,
Itô’s formula looks like the usual rule of calculus:
∫
f (Xt ) = f (X0 ) +
A.5
0
t
Df (Xs ) ◦ dXs .
Stochastic differential equations
Let σij (x) and bi (x) be continuous functions on Rd for 1 ≤ i ≤ d and 1 ≤ j ≤ m, and let
Bt be an m-dimensional standard Brownian motion, associated to some filtration {Ft }. A
solution xt = (x1t , . . . , xdt ) of the following stochastic differential equation, or sde for short,
dxit
=
m
∑
σi (xt )dBtj + bi (xt )dt,
j=1
is a continuous semimartingale in Rd such that
xit = xi0 +
m ∫
∑
j=1 0
t
∫
σi (xs )dBsj +
293
t
0
bi (xs )ds.
(A.10)
It is well known that if σij and bi are uniformly Lipschitz continuous on Rd , then given
any initial value x0 (that is F0 -measurable), there is a unique solution xt , with an infinite
life. This can be proved rather easily by the usual successive approximation method, also
called Picard iteration, (see Ikeda and Watanabe [42, IV.3]). Moreover, xt is a Feller process
and its generator, restricted on Cc∞ (Rd ), is given by
Lf (x) =
d
d
∑
1 ∑
aij (x)∂i ∂j f (x) +
bi (x)∂i f (x),
2 i,j=1
i=1
(A.11)
∑
where aij (x) = m
p=1 σpi (x)σpj (x). It is then easy to show that the process xt possesses the
∫
following martingale property: For any f ∈ Cc∞ (Rd ), f (xt ) − 0t Lf (xs )ds is a martingale
under Ft .
When σij and bi are only locally Lipschitz continuous on Rd , that is, uniformly Lipschitz
continuous on any bounded open subset of Rd , then given any initial value x0 , the solution
xt of (A.10) still exits uniquely, and is a Markov process, but it may have a finite life time,
that is, it may go to infinity in finite time, and it may not be a Feller process. However, it
still possesses the above martingale property, see [42, IV.Theorem 6.1].
A rcll process xt taking values in a d-dimensional manifold M is called a semi-martingale
in M , under some filtration {Ft }, if ∀f ∈ Cc∞ (M ), f (xt ) is a semi-martingale under {Ft }.
In this case, if f ∈ C 1 (M ) and if Yt is a real-valued semi-martingale, then the Stratonovich
∫
integral 0t f (xs− ) ◦ dYs is defined. Moreover, if f ∈ C 2 (M ), then f (xt ) is a real-valued semimartingale. These claims follow easily from the results on Euclidean spaces, because by the
Whittney Embedding Theorem (see [10, II.Theorem 10.8]) and the Tubular Neighborhood
Theorem (see [39, 4.Theorem 5.1]), we may embed M into a Euclidean space Rn , and extend
any smooth function or vector field on M to be smooth function or vector field on Rn .
Let M be a d-dimensional manifold and let Y0 , Y1 , . . . , Ym be smooth vector fields on M .
Let Bt be an m-dimensional standard Brownian motion, associated to some filtration {Ft }.
A solution of the Stratonovich sde
dxt =
m
∑
Yi (xt ) ◦ dBti + Y0 (xt )dt
(A.12)
i=1
is a semimartingale xt in M (under {Ft }) such that ∀f ∈ Cc∞ (M ),
f (xt ) = f (x0 ) +
m ∫
∑
i=1 0
t
Yi f (xs ) ◦ dBsi +
∫
t
0
Y0 f (xs )ds.
If M is a compact manifold, then given x0 (that is F0 -measurable), there is a unique solution
xt with an infinite life time, because a compact manifold can be embedded in a Euclidean
space RN and Yi can be extended to be smooth vector fields on RN with compact supports.
The existence of a unique solution with an infinite life time holds also if M is a Lie group
294
and the Yi ’s are left (resp. right) invariant vector fields on M (see Kunita [51, Theorem
4.8.7]). In these cases, xt is a continuous Feller process in M , and its generator L restricted
to Cc∞ (M ) is given by
∀f ∈ Cc∞ (M ),
m
1∑
Yi Yi f + Y0 f.
2 i=1
Lf =
(A.13)
In general, the solution xt of the sde (A.12) exists uniquely for any initial value x0 (that
is F0 -measurable), but it may have a finite life time. It may not be a Feller process, but
is a continuous Markov process possessing the strong Markov property and the following
martingale property:
∀f ∈
Cc∞ (M ),
f (xt ) − f (x0 ) −
∫
0
t
Lf (xs )ds
is a martingale
(A.14)
under {Ft }, where L is given by (A.13).
We note that L is a differential operator on M which may be expressed in local coordinates
x1 , . . . , xn as
d
d
∑
1 ∑
L=
αij (x)∂i ∂j +
βi (x)∂i ,
(A.15)
2 i,j=1
i=1
where αij and βi are smooth in x with αij forming a nonnegative definite symmetric matrix.
Any differential operator L on M that has expression (A.15) in local coordinates will be
called a diffusion generator on M .
It can be shown that given any diffusion generator L on M , there is a family of continuous
processes xt in M , one for each initial point x0 , unique in distribution, such that they all
have the martingale property (A.14) under the natural filtration. Such a family of processes
is a Markov process with strong Markov property, and is called a diffusion process in M with
generator L, or an L-diffusion process. Indeed, the process xt may be obtained by solving
an sde in a collection of coordinate neighborhoods that cover M and then suitably piecing
these solutions together, and the uniqueness follows from the unique solution of the sde
(see Theorems 5.1 and 6.1, and Proposition 6.2 in [42, Chapter IV]). We note that it is not
necessary to have C ∞ -smoothness for αij and βi in (A.15) for the existence and uniqueness
of a diffusion process, but for simplicity, this is assumed in this work.
According to our definition, a continuous Feller process in M with generator L is an
L-diffusion process if Cc∞ (M ) ⊂ D(L) and L restricted to Cc∞ (M ) has the form (A.15).
In particular, a d-dimensional standard Brownian motion is a diffusion process in Rd with
∑
generator (1/2)∆, where ∆ = di=1 ∂i ∂i is the Laplace operator on Rd .
Let ϕ: M → M be a diffeomorphism. Recall the ϕ-invariance of Markov processes and
operators as defined in §1.1. We note that a L-diffusion process xt in M is ϕ-invariant if
and only if L is ϕ-invariant, see Proposition A.4 in Appendix A.7, and consequently, xt is
invariant under the action of a Lie group G on M if and only if L is G-invariant.
295
A.6
Poisson random measures
This section is devoted to stochastic integrals of Poisson random measures. The discussion
follows essentially the exposition in Ikeda-Watanabe [42, II.3] for integrals of homogeneous
Poisson random measures, but general Poisson random measures, which may be found in
[46, chapter 12], are treated here with essentially same proofs.
A Poisson distribution ν of mean c > 0 is a probability measure on the set Z+ of nonnegative integers defined by ν({k}) = e−c ck /k!. If c = 0 or c = ∞, then it is defined to be δ0 or
δ∞ . A Z+ -valued rcll process Nt with N0 = 0 and of independent and stationary increments
is called a Poisson process of rate c ≥ 0 if ∀t > 0, Nt has Poisson distribution of mean ct.
A random measure ξ on a measurable space (S, S) is a kernel from (Ω, F) to (S, S) and
the measure µ = E(ξ) on S is called its intensity measure. The ξ may also be regarded as
a random variable taking values in the space M(S) of measures on S, equipped with the
σ-algebra generated by all the projection maps: M(S) → (R+ ∪{∞}), ν 7→ ν(B), for B ∈ S.
An integer-valued random measure ξ on S, that is, a random measure ξ on S taking values
from Z̄+ = Z+ ∪ {∞}, is called a point process on S if its intensity measure is σ-finite. In
∑
this case, when S is a Borel space, ξ = κi=1 δσi for a Z̄+ -valued random variable κ and a
sequence of S-valued random variables σi , the latter are called the points of ξ.
Let N be a point process on R+ ×S with intensity µ such that almost surely, N ({t}×S) ≤
1 for all t ∈ R+ , and µt (·) = µ([0, t]×·) is σ-finite on S for any t ∈ R+ . The N is said to have
independent increments if the measure-valued process Nt = N ([0, t] × ·) have independent
increments. In this case, if µ({t} × S) = 0 for any t ∈ R+ , then by [46, Theorem 12.10],
for disjoint B1 , . . . , Bk ∈ B(R+ ) × S, N (B1 ), . . . , N (Bk ) are independent random variables
with Poisson distributions of means µ(B1 ), . . . , µ(Bk ) respectively. In this case, N is called
a Poisson random measure on R+ × S. If in addition, µ(dt dx) = dtν(dx) for some σ-finite
measure ν on G, then N is called a homogeneous Poisson random measure and ν is called
the characteristic measure of N . More generally, if N has independent increments, but the
condition µ({t}×S) = 0 may not hold, then N is called an extended Poisson random measure
on R+ × S. In this case, the time points t with µ({t} × S) > 0 are at most countable, and
µ({t} × S) ≤ 1 for any t ∈ R+ . Moreover, given any measure µ on R+ × S such that for
any t ∈ R+ , µ([0, t] × ·) is σ-finite and µ({t} × S) ≤ 1, there is an extended Poisson random
measure N on R+ × S, unique in distribution, with µ as intensity. Our definition of Poisson
random measures is essentially the same as in [45], but is less general than that in [46].
A Poisson random measure N on R+ × S is said to be associated to a filtration {Ft },
or is called an {Ft }-Poisson random measure, if the measure-valued process Nt is adapted
to {Ft }, and for s < t, Nt − Ns is independent of Fs . A Poisson random measure N is
clearly associated to its natural filtration FtN defined to be the natural filtration of Nt . This
definition also applies to an extended Poisson random measure.
296
A random variable τ ≥ 0 is said to have an exponential distribution of rate c > 0 if
P (τ > t) = e−ct for any t ∈ R+ . Then E(τ ) = 1/c. This definition may be applied to
c = 0 for τ = ∞, and to c = ∞ for τ = 0. A homogeneous Poisson random measure N on
R+ × S with a finite characteristic measure ν ̸= 0 can be constructed from a sequence of iid
exponential random variables τn of rate ν(S) and an independent sequence of iid S-valued
random variables σn with distribution ν/ν(S) by setting
N ([0, t] × B) = #{n > 0; Tn ≤ t and σn ∈ B}
∑
for t ∈ R+ and B ∈ S, where Tn = ni=1 τi . Note that Tn are stopping times of N (that is,
{FtN }-stopping times) and Tn ↑ ∞ almost surely. The points of N are (Tn , σn ), n ≥ 1.
Let F be the space of the functions g: R+ × S × Ω → R such that
(i) ∀t ∈ R+ , (x, ω) 7→ g(t, x, ω) is S × Ft -measurable; and
(ii) ∀(x, ω) ∈ S × Ω, t 7→ g(t, x, ω) is lcrl.
A simple monotone class argument shows that if N is a Poisson random measure on
R+ × S with intensity measure µ, then for f ∈ F ,
∫
t
∫
∫
E[
0
S
|f (u, x, ·)|N (du, dx)] = E[
t
∫
0
S
|f (s, x, ·)|µ(ds, dx)].
For simplicity, f (t, x, ω) may be written as f (t, x).
For a process xt taking values in a linear space and a fixed T ∈ R+ , the T -shifted process
(T )
(T )
xt is defined by xt = xT +t − xT . For a Poisson random measure N on R+ × S, let N (T )
be the T -shifted random measure defined by N (T ) ([0, t] × ·) = N ([T, T + t] × ·), which is
also a Poisson random measure. If N is homogeneous, then N (T ) is identical in distribution
to N .
Let {Ft } be a filtration and let N be an {Ft }-Poisson random measure on R+ × S with
intensity measure µ. The compensated random measure Ñ of N is defined by
Ñ (dt, dx) = N (dt, dx) − µ(dt, dx).
(A.16)
Then for B ∈ S with µt (B) < ∞ for t > 0, Ñt (B) = Ñ ([0, t] × B) is an L2 -martingale.
Define
∫ t∫
F α (N ) = {g ∈ F ; E[
|g(s, x)|α µ(ds, dx)] < ∞}
0
S
for α > 0. Then for g ∈ F (N ) ∩ F (N ),
1
∫
t
∫
2
∫
t
∫
g(s, x)Ñ (ds, dx) =
0
G
0
G
g(s, x)N (ds, dx) −
∫
t
∫
g(s, x)µ(ds, dx)
0
G
is an L2 -martingale satisfying
∫
t
∫
∫
E{[
0
g(s, x)Ñ (ds, dx)] } = E[
t
2
G
297
0
∫
g(s, x)2 µ(ds, dx)].
G
(A.17)
For g ∈ F 2 (N ), choose gn ∈ F 1 (N ) ∩ F 2 (N ) such that
∫
t
∫
E[
0
G
|gn (s, x) − g(s, x)|2 µ(ds, dx)] → 0
as n → ∞
and define the stochastic integral
∫
t
∫
g(s, x)Ñ (ds, dx)
0
G
∫ ∫
as the limit in L2 (P ) of 0t G gn (s, x)Ñ (ds, dx). By Doob’s norm inequality (A.4), it has
a rcll adapted version that is an L2 -martingale satisfying (A.17). Note that in general
∫t∫
2
0 G g(s, x)N (ds dx) may not be defined for g ∈ F (N ).
2
Let Floc
(N ) be the space of functions f ∈ F such that there are stopping times Tn ↑
∞ satisfying fn ∈ F 2 (N ), where fn (t, x) = f (t ∧ Tn , x). Then the stochastic integral
∫t∫
∫t∫
0 G f (u, x)Ñ (du dx) may be defined by setting it equal to 0 G fn (u, x)Ñ (du dx) for t ≤ Tn .
Then it is a local L2 -martingale.
∫ ∫
We will write (g · N )t = 0t G g(u, x)N (du, dx), and similarly for (g · Ñ )t and (g · µ)t .
Then
⟨(g · Ñ ), (h · Ñ )⟩t = ((gh) · µ)t
(A.18)
and
[(g · Ñ ), (h · Ñ )]t = ((gh) · N )t .
(A.19)
To prove these, let (g · Ñ )s,t = (g · Ñ )t −(g · Ñ )s for s < t. Then E[(g · Ñ )s,t | Fs ] = 0. Because
a Poisson random variable has the same mean and variance, E[(g · Ñ )2s,t | Fs ] = (g 2 · µ)s,t ,
E[(g · Ñ )2t | Fs ] = E[(g · Ñ )2s,t − 2(g · Ñ )s (g · Ñ )s,t + (g · Ñ )2s | Fs ] = (g 2 · µ)s,t + (g 2 · Ñ )s .
This shows (g · Ñ )2t − (g 2 · µ)t is a martingale, and hence proves (A.18) for h = g. The general
case follows from an interpolation. To prove (A.19), note that (g · Ñ )t (h · Ñ )t − ((gh) · N )t
is a martingale, and ∆t (g · Ñ )∆t (h · Ñ ) = ∆t ((gh) · N ) because N ({t} × S) = 0 or 1.
Given a filtration {Ft }, a family of rcll processes {xλt ; λ ∈ Λ}, adapted to {Ft } and
taking values on, possibly different, linear spaces, are called independent under {Ft } if for
λ (T )
all T ∈ R+ , the T -shifted processes xt
and FT are independent. This is equivalent to
saying that for any s < t, the random variables xλt − xλs and Fs are independent.
Let Bt be a Brownian motion and let N be a homogeneous Poisson random measure
(regarded as the measure-valued process Nt ), associated to the same filtration {Ft }. Then
by Theorem 6.3 in [42, chapter II] (see also (6.12) in its proof), they are automatically
independent under {Ft }. In this case, if Ht is an adapted left continuous process satisfying
∫
E( 0t Hs2 ds) < ∞ and g ∈ F 2 (N ), then
⟨(H · B), (g · Ñ )⟩t = [(H · B), (g · Ñ )]t = 0.
298
(A.20)
To prove (A.20), just check that (H · B)t (g · Ñ )t is a martingale by a direct computation.
We note that given a Brownian motion Bt and a homogeneous Poisson random measure
N on R+ × S, if they are independent, then they are independent under the filtration {Ft }
generated by {Bt } and N (regarded as a measure-valued process Nt ), and by possibly other
independent processes and random variables.
A.7
Some miscellaneous results
We collect some miscellaneous results here to be used in the main text, which may be well
known, but seem to be difficult to locate in the standard literature.
Right continuity of filtration: The independent increments and the completion of a
filtration mentioned in the following proposition are defined in Appendix A.3.
Theorem A.1 Let xt be a rcll process with independent increments and let {Ft } be its
natural filtration. Then the completion {F̄t } of {Ft } is right continuous.
Proof: For any two sets A and B, we will write A ≈ B if (A − B) ∪ (B − A) is a null
set in F∞ . Fix t ≥ 0. Because F̄t = {A ⊂ Ω; A ≈ B for some B ∈ Ft }, if A ∈ F̄t+ , then
A ∈ F̄u for any u > t, so A ≈ Au ∈ Fu . Let Bv = ∩u∈(t, v] Au for v > t. Then Bv ∈ Fv and
Bv ↑ B ∈ Ft+ as v ↓ t for some B ∈ Ft+ with A ≈ B. Therefore, to show A ∈ F̄t , we may
assume A ∈ Ft+ .
For v > u, let Gu,v be the σ-algebra generated by the increments of xt over intervals (a, b]
for a, b ∈ [u, v]. Then for u > t, Fu = σ{Ft ∪ Gt,u }. For v > u, let Ht,u,v = σ{Ft ∪ Gu,v }. Let
ξ ∈ (Fu )+ and η ∈ (Ht,u,v )+ , we want to show E(ξη | Ft ) = E(ξ | Ft )E(η | Ft ), that is, ξ and
η are conditionally independent given Ft . Using a monotone class type argument, the proof
may be reduced to the case when ξ = ξ1 ξ2 and η = η1 η2 with ξ1 , η1 ∈ (Ft )+ , ξ2 ∈ (Gt,u )+ and
η2 ∈ (Gu,v )+ . Then
E(ξη | Ft ) = ξ1 η1 E(ξ2 η2 | Ft ) = ξ1 η1 E(ξ2 )E(η2 )
(because ξ2 , η2 and Ft are independent)
= E(ξ1 ξ2 | Ft )E(η1 η2 | Ft ) = E(ξ | Ft )E(η | Ft ).
We may let ξ = 1A for A ∈ Ft+ . By the right continuity of xt , the σ-algebra generated by all
Ht,u,v as u ranges over (t, v) is equal to Fv . It follows that η can be taken to be 1A as well.
Then E(1A | Ft ) = E(1A | Ft )E(1A | Ft ), and hence E{1A [1−E(1A | Ft )]} = 0. This implies
that E(1A | Ft ) = 1 almost surely on A. Replacing A by A2 , we obtain E(1Ac | Ft ) = 1
almost surely on Ac . Because E(1A | Ft ) + E(1Ac | Ft ) = 1, it follows that E(1A | Ft ) = 1A
299
almost surely. Let B be the set on which E(1A | Ft ) = 1. Then B ∈ Ft and B ≈ A. This
shows A ∈ F̄t . 2
Section maps: Let G be a topological group equipped with a lcscH (locally compact
second countable Hausdorff) topology, let K be a closed subgroup, and let X = G/K be the
topological homogeneous space equipped with the quotient topology. We define a section
map S on X as a Borel map X → G such that π ◦ S = idX , where π: G → X = G/K
is the natural projection and idX is the identity map on X. Let e be the identity element
of G and o = eK in X. A continuous map ϕ from an open neighborhood U of o to G is
called a continuous local section map on X if π ◦ ϕ = idU . Some sufficient conditions for the
existence of a continuous local section map may be found in [47, 68], and see also [47] for
some non-existence examples.
Proposition A.2 Let ϕ: U → G be a continuous local section map. Then the map F :
(x, k) 7→ ϕ(x)k is a homeomorphism from U × K onto the open neighborhood π −1 (U ) of e.
Proof: It is easy to see that F : U × K → π −1 (U ) is continuous and is 1-1 onto. To show
F −1 is continuous, let gn = ϕ(xn )kn , xn ∈ U and kn ∈ K, and gn → g in π −1 (U ) as n → ∞.
Then xn = π(gn ) → x = π(g), and hence kn = ϕ(xn )−1 gn → ϕ(x)−1 g. This proves the
continuity of F −1 . 2
Proposition A.3 If there is a continuous local section map on X = G/K, then there is a
section map S on X. Moreover, for any x ∈ X, S may be chosen to be continuous near x.
Proof: Let ϕ: U → G be a continuous local section map. Because G is second countable,
there is a sequence of gn ∈ G with g1 = e such that gn π −1 (U ) cover G, and hence gn U cover
X. To define a section map S: X → G, first set S(x) = ϕ(x) for x ∈ U . For x ∈ (g2 U ) − U ,
define S(x) = g2 (ϕ(g2−1 x), and for x ∈ (g3 U ) − (g2 U ) ∩ U , define S(x) = g3 ϕ(g3−1 x). Continue
in this way, S can be defined on X. It is clearly Borel and satisfies π ◦ S = idX , and so is a
section map and is continuous near e. Now for any x = gK, let S ′ (y) = gS(g −1 y) for y ∈ X.
Then S ′ is another section map that is continuous near y. 2
Invariant diffusion processes: Let M be a manfold, and let ϕ: M → M be a diffeomorphism. Recall the ϕ-invariance of Markov processes and operators as defined in §1.1.
Proposition A.4 Let xt be a diffusion process in M with generator L as defined in Appendix A.5. Then xt is a ϕ-invariant Markov process if and only if L is ϕ-invariant.
Proof: We first assume xt is ϕ-invariant. Then the process ϕ(xt ) is equal in distribution to
∫
the same diffusion process starting at ϕ(x0 ). Thus, for f ∈ Cc∞ (X), f ◦ϕ(xt )− 0t Lf (ϕ(xs ))ds
300
∫
is a martingale. On the other hand, f ◦ ϕ(xt ) − 0t L(f ◦ ϕ)(xs )ds is also a martingale, this
∫
∫
implies that 0t Lf (ϕ(xs ))ds = 0t L(f ◦ ϕ)(xs )ds. Dividing by t and then letting t → 0 yields
Lf (ϕ(x0 )) = L(f ◦ ϕ)(x0 ) for any x0 ∈ X. This proves the ϕ-invariance of L. Next assume L
∫
∫
is ϕ-invariant. Then f ◦ ϕ(xt ) − 0t Lf (ϕ(xs ))ds = f ◦ ϕ(xt ) − 0t L(f ◦ ϕ)(xs )ds is a martingale.
Because the process having this martingale property is unique in distribution, it follows that
ϕ(xt ) is the same diffusion process starting at ϕ(x0 ). This proves the ϕ-invariance of xt . 2
Regular conditional kernels: A measurable space is called a Borel space if it is measurably isomorphic to a Borel subset of [0, 1] under a bijective map. It is well known that
a complete and separable metric space, equipped with its Borel σ-algebra, is a Borel space.
By the standard probability theory, if the underlying probability space (Ω, F, P ) is a Borel
space, then for any σ-algebra G ⊂ F , there is a regular conditional distribution Q(ω, B)
of P given G, which is a probability kernel from (Ω, G) to (Ω, F), such that for B ∈ F,
P (B | G) = Q(·, B) P -almost surely. We will need an extension of this result to a probability kernel Px in place of a single probability measure P as stated in the following proposition.
Theorem A.5 Let (Ω, F) and (H, H) be two measurable spaces with the former being a
Borel space, let {Px } be a probability kernel from (H, H) to (Ω, F), and let G be a σ-algebra
contained in F. If G is countably generated (that is, it is generated by a countable collection
of sets), then there is a probability kernel {Qx (ω, ·)} from (H × Ω, H × G) to (Ω, F) such that
for all x ∈ H and B ∈ F, Px (B | G) = Qx (·, B) Px -almost surely. The kernel {Qx (ω, ·)}
will be called the regular conditional distribution of the kernel {Px } given G. Moreover, it is
unique in the sense that if {Q′x } is another regular conditional distribution of {Px } given G,
then for all x ∈ H, Qx (ω, ·) = Q′x (ω, ·) for Px -almost all ω.
Proof We may assume Ω is a Borel subset of [0, 1] not containing 0 and regard Px as
a probability kernel from H to [0, 1] supported by Ω. Because G is countably generated,
there are σ-algebras Gn ↑ G ′ with G = σ(G ′ ) such that each Gn is generated by a finite
partition {Bn1 , Bn2 , . . . , Bnmn } of Ω. For x ∈ H and a rational r ∈ [0, 1], let fn (x, ω, r) =
Px ([0, r] ∩ Bni )/P (Bni ) for ω ∈ Bni if P (Bni ) > 0. For ω ∈ Bni with P (Bni ) = 0, put
fn (x, ω, r) = 0 for r < 1 and fn (x, ω, 1) = 1. Then for each rational r ∈ [0, 1], fn (x, ω, r)
is (H × Gn )-measurable in (x, ω) and for all x ∈ H, Px ([0, r] | Gn ) = fn (x, ·, r) Px -almost
surely. Moreover, fn (x, ω, r) is nondecreasing in r, fn (x, ω, 0) = 0 and fn (x, ω, 1) = 1.
Let f (x, ω, r) = lim supn→∞ fn (x, ω, r). Then f (x, ω, r) is (H × G)-measurable in (x, ω)
for each fixed r, and it is nondecreasing in r with f (x, ω, 0) = 0 and f (x, ω, 1) = 1. Because
for all (x, r), Px ([0, r] | Gn ) as a bounded martingale converges to Px ([0, r] | G) Px -almost
surely as n → ∞, it follows that Px ([0, r] | G) = f (x, ·, r) Px -almost surely.
Now for each real t ∈ [0, 1), let g(x, ω, t) = lim f (x, ω, r) as rational r ↓ t. The limit
exists because f (x, ω, r) is nondecreasing in r. Set g(x, ω, 1) = 1. Then g(x, ω, t) is (H × G)301
measurable in (x, ω), is right continuous and nondecreasing in t, and satisfies Px ([0, t] | G) =
g(x, ·, t) Px -almost surely. For each (x, ω), there is a probability measure Qx (ω, ·) on [0, 1]
such that for any t ∈ [0, 1], Qx (ω, [0, t]) = g(x, ω, t) which is H × G-measurable in (x, ω).
A simple monotone class argument shows that for any Borel subset B of [0, 1], Qx (ω, B)
is H × G-measurable and Px (B | G) = Qx (·, B) Px -almost surely. Because Qx (·, Ω) =
Px (Ω | G) = 1 Px -almost surely, we may modify Qx (ω, ·) on the (H × G)-measurable set
{(x, ω) ∈ H × Ω; Qx (ω, Ω) < 1} so that Qx (ω, Ω) = 1 for all (x, ω). This proves that Qx is
a regular conditional distribution of Px given G.
The uniqueness follows from Qx (·, B) = Q′x (·, B) Px -almost surely for any x ∈ X and
B ∈ F , and the fact that F is countably generated as the σ-algebra of a Borel space. 2
Separability of Cc∞ (X): Let X be a d-dimensional manifold. Let ξ1 , . . . , ξp be vector
fields on X which span the tangent space Tx X at every point x ∈ X. For f ∈ Cc∞ (X), let
∥f ∥0 = supx∈X |f (x)| be the sup norm, and for any integer k > 0, let
∥f ∥k = ∥f ∥0 +
p
∑
∑
∥ξj f ∥0 +
j=1
∥ξj1 · · · ξjk f ∥0 .
j1 ,...,jk
Theorem A.6 The space Cc∞ (X) is separable in the norm ∥f ∥k for any inter k > 0, that
is, there is a countable subset Γ of Cc∞ (X) such that for any f ∈ Cc∞ (X), there are fn ∈ Γ
with ∥fn − f ∥k → 0 as n → ∞.
Proof: We first prove the result for X = Rd . Let Rd be the one-point compactification
of Rd . Because Rd is compact, it is well known that C(Rd ) is separable under ∥f ∥0 -norm.
Then as a subspace of C(Rd ), Cc (Rd ) is separable, and hence so is Cc (Rd )n for any integer
n > 0 under the norm
∥(f1 , . . . , fn )∥(n) =
n
∑
∥fi ∥0 .
i=1
For any k > 0, let n be the number of operators of the form ξj1 ξj2 · · · ξji , where i = 1, 2, . . . , k
and j1 , j2 , . . . , ji = 1, 2, . . . , d. Then Cc∞ (Rd ) under the norm ∥f ∥k is isometrically embedded in Cc (Rd )n+1 via the map f 7→ (f0 , f1 , . . . , fn ), where f0 = f and (f1 , . . . , fn ) is an
arrangement of ξj1 ξj2 · · · ξji f . It follows that Cc∞ (Rd ) is separable under the norm ∥f ∥k .
A general manifold X is covered by countably many open subsets Uα , each of which is
diffeomorphic to Rd . Then for each α, there is a countable subset Γα of Cc∞ (Uα ) ⊂ Cc∞ (X)
such that any f ∈ Cc∞ (Uα ) can be approximated in norm ∥ · ∥k by functions in Γα . Let {ϕα }
be a partition of unity subordinated to the open cover {Uα }. Let f ∈ Cc∞ (X). Then all
∑
ϕα = 0 on the support of f , a compact set, except finitely many, and hence f = α ϕα f is
actually a finite sum. Because ϕα f ∈ Cc∞ (Uα ), there are fnα ∈ Γα such that ∥ϕα f − fnα ∥k → 0
∑
as n → ∞. Let fn = α fnα (a finite sum over α with ϕα f ̸= 0). Then ∥f − fn ∥k → 0 as
302
n → ∞. The theorem is proved by setting Γ to be the set of functions which are finite sums
of f ∈ Γα . 2
303
304
Bibliography
[1] Applebaum, D. and Kunita, H., “Lévy flows on manifolds and Lévy processes on Lie
groups”, J. Math. Kyoto Univ. 33, pp 1105-1125 (1993).
[2] Applebaum, D., “Compound Poisson processes and Lévy processes in groups and symmetric spaces”, J. Theo. Probab. 13, pp 383-425 (2000).
[3] Applebaum, D., “Lévy processes and stochastic calculus”, second ed., Cambridge Univ.
Press (2009).
[4] Applebaum, D. and Dooley, T., “A generalised Gangolli-Levy-Khintchine formula for
infinitely divisible measures and Levy processes on semisimple Lie groups and symmetric
spaces , to appear in Annales Institut Henri Poincare (Prob.Stat.) (2013).
[5] Applebaum, D., “Probability on compact Lie groups”, Springer (2014).
[6] Azencott, R. el al. “Géodésiques et diffusions en temps petit”, Asterisque 84-85 (1981).
[7] Babullot, M., “Comportement asymptotique due mouvement Brownien sur une variété
homogène à courbure négative ou nulle”, Ann. Inst. H. Poincaré (prob et Stat) 27, pp
61-90 (1991).
[8] Bertoin, J., “Lévy processes”, Cambrige Univ. Press (1996).
[9] Born, E., “An explicit Lévy -Hinc̆in formula for convolution semigroups on locally compact groups”, J. Theor. Prob. 2, 325-342 (1989).
[10] Bredon, G.E., “Topology and geometry”, Springer 1997.
[11] Bourbaki, N., “Lie groups and Lie algebras”, Chap. 1-3, Springer-Verlag (1989).
[12] Bröcker, T. and Dieck, T. “Representations of compact Lie groups”, Springer-Verlag
(1985).
[13] Cheng, S.Y, Li, P. and Yau S-T, “On the upper estimate of the heat kernel of a complete
riemannian manifold”, Amer. J. Math. 103, 1021-1063 (1981).
305
[14] Chatelin, F. “Eigenvalues of matrices”, Wiley (1993).
[15] Dani, S.G. and McCrudden, M., “Embeddability of infinitely divisible distributions on
linear Lie groups”, Invent. Math. 110, 237C261(1992).
[16] Dani, S.G., and McCrudden, M., “Convolution roots and embedding of probability
measures on Lie groups”, Adv. Math. 209, 198-211 (2007).
[17] Diaconis, P. “Group representations in probability and statistics”, IMS, Hayward, CA.
(1988).
[18] Dynkin, E.B., “Nonnegative eigenfunctions of the Laplace-Betrami operators and Brownian motion in certain symmetric spaces”, Dokl. Akad. Nauk SSSR 141, pp 1433-1426
(1961).
[19] Eells, J. and Elworthy, K.D., “Stochastic dynamical systems”, Control Theory and
Topics in Functional Analysis, III, Intern. Atomic Energy Agency, Vienna, pp 179-185
(1976).
[20] Elworthy, K.D., “Stochastic differential equations on manidfolds”, Cambridge Univ.
Press (1982).
[21] Elworthy, K.D., “Geometric aspects of diffusions on manifolds”, in Ecole d’Eté de Probabilités de Saint-Flour XVII, July 1987, ed. P.L. Hennequin, Lecture Notes Math. 1362,
pp 276-425 (1988).
[22] Ethier, S.N. and Kurtz, T.G., “Markov processes, characterization and convergence”,
John Wiley (1986).
[23] Émery, M., “Stochastic calculus in manifolds”, Universitext. Springer-Verlag (1989).
[24] Evans, S.N., Hening, A. and Wayman, E., “A skew-product decomposition counterexample”, to appear in Eletronic J. of Probab (2015).
[25] Feinsilver, P., “Processes with independent increments on a Lie group”, Trans. Amer.
Math. Soc. 242, 73-121 (1978).
[26] Furstenberg, H. and Kesten, H., “Products of random matrices”, Ann. Math. Statist.
31, pp 457-469 (1960).
[27] Furstenberg, H., “Noncommuting random products”, Trans. Am. Math. Soc. 108, pp
377-428 (1963).
306
[28] Galmarino, A.R., “Representation of an isotropic diffusion as a skew product”, Z.
Wahrscheinlichkeitstheor. Verw. Geb. 1, 359C378 (1963).
[29] Gangolli, R. “Isotropic infinitely divisible measures on symmetric spaces”, Acta Math.
111, pp 213-246 (1964).
[30] Gilbarg, D. and Trudinger, N.S., “Elliptic partial differential equations of second order”,
Springer-Verlag (1977).
[31] Guivarc’h, Y. and Raugi, A., “Frontière de Furstenberg, propriétés de contraction et
convergence”, Z. Wahr. Gebiete 68, pp 187-242 (1985).
[32] Hazod, W. and Siebert, ., “Stable probability measures on Euclidean spaces and on
locally compact groups. Structural properties and limit theorems”. Mathematics and
its Applications 531. Kluwer Academic Publishers, Dordrecht (2001).
[33] Helgason, S., “Differential geometry, Lie groups, and symmetric spaces”, Academic
Press (1978).
[34] Helgason, S., “Groups and geometric analysis”, Academic Press (1984).
[35] Heyer, H., “Probability measures on locally compact groups”, Springer-Verlag (1977).
[36] Heyer, H., “Convolution semigroups of probability measures on Gelfand pairs”, Expo.
Math 1, 3-45 (1983)
[37] Heyer, H. and Pap, G., “Convolution hemigroups of bounded variation on a Lie projective group,” J. London Math. Soc. 59, 369-84 (1999).
[38] Heyer, H. and Pap, G., “Martingale characterizations of increment processes in a local
compact group”, in Infinite Dimensional Analysis, Quantum Probability and Related
Topics 6, 563-595 (2003).
[39] Hirsch, M.W., “Differential topology”, Springer-Verlag (1976).
[40] Hsu, E. “Stochastic analysis on manifolds”, Am. Math. Soc. (2002).
[41] Hunt, G.A., “Semigroups of measures on Lie groups”, Trans. Am. Math. Soc. 81, pp
264-293 (1956).
[42] Ikeda, N. and Watanabe, S., “Stochastic differential equations and diffusion processes”,
2nd ed., North-Holland-Kodansha (1989).
307
[43] Itô, K. and Mckean, H.P. Jr., “Diffusion processes and their sample paths”, second
printing, corrected, Springer-Verlag 1974.
[44] Itô, K., “Stochastic processes”, Lecture Notes Series No. 16, Aarhus University 1969.
[45] Jacod, J. and Shiryaev, A.N., “Limit theorems for stochastic processes”, second ed.,
Springer (2003).
[46] Kallenberg, O., “Foundations of modern probability, second ed.”, Springer-Verlag
(2002).
[47] Karube, T., “On the local cross sections in locally compact groups”, J. Math. Soc.
Japan 10, pp 343-347 (1958).
[48] Kesten, H., “Hitting probabilities of single points for processes with stationary independent increments”, Memoirs of Amer. Math. Soc. 93 (1969).
[49] Klyachko, A.A., “Random walks on symmetric spaces and inequalities for matrix spectra”, Linear Algebra Appl. 319, pp 37C59 (2000).
[50] Kobayashi, S. and Nomizu, K., “Foundations of differential geometry”, vols I and II,
Interscience Publishers (1963 and 1969).
[51] Kunita, H., “Stochastic flows and stochastic differential equations”, Cambrige Univ.
Press (1990).
[52] Kunita, H. “Stochastic differential equations and stochastic flows of diffeomorphisms”,
École d’Été de Probabilites de Saint-Flour XII-1982, Lecture Notes in Math. 1097, 143303, Springer (1984).
[53] Lang, S., “Fundamentals of differential geometry”, Springer 1999
[54] Liao, M., “Lévy processes in semi-simple Lie groups and stability of stochastic flows”,
Trans. Am. Math. Soc. 350, pp 501-522 (1998).
[55] Liao, M., “Lévy processes in Lie groups”, Cambridge Univ. Press (2004).
[56] Liao, M., “Lévy processes and Fourier analysis on compact Lie groups, Annals of Probability 32, pp 1553-1573 (2004).
[57] Liao, M., “A decomposition of Markov processes via group actions”, J. Theoret. Probab.
22, pp 164-185 (2009).
308
[58] Liao, M. and Wang, L., “Lévy -Khinchin formula and existence of densities for convolution semigroups on symmetric spaces”, Potential Analysis 27, 133-150 (2007).
[59] Liao, M. and Wang, L., “Isotropic self-similar Markov processes”, Stochastic Process.
Appl. 121, 2064-2071 (2011).
[60] Liao, M. and Wang, L., “Limiting Properties of Levy Processes in Symmetric Spaces of
Noncompact Type, Stochastics and Dynamics 12, pp 1-16 (2012).
[61] Liao, M., “Inhomogeneous Lévy Processes in Lie Groups and Homogeneous Spaces”, J.
Theoret. Probab. 27, 315C357 (2014).
[62] Liao, M., “Fixed jumps of additive processes”, Stat. and Probab. Letters 83, 820-823
(2013).
[63] Liao, M., “Convolution of probability measures on Lie groups and homogeneous spaces,
Potential Analysis 43, 707-715 (2015).
[64] Li, X., “Stochastic Homogenization on Homogeneous Spaces”, preprint (http://
www2.warwick.ac.uk/fac/sci/maths/people/staff/xue mei li/HomogenisationHomogeneous-spaces.pdf) (2016).
[65] Mainardi, F., Luchko, Y., Pagnini, G., “The fundamental solution of the space-time
fractional diffusion equation”, Fract. Calc. Appl. Anal. 4, pp 153C192 (2001).
[66] Major, P. and Shlosman, S.B., “A local limit theorem for the convolution of probability
measures on a compact connected group”, Z.Wahr. verw. Gebiete 50, pp 137 - 148
(1979).
[67] Malliavin, M.P. and Malliavin, P., “Factorizations et lois limites de la diffusion horizontale audessus d’un espace Riemannien symmetrique”, Lecture Notes Math. 404, pp
164-271 (1974).
[68] Nagami, K., “Cross sections in locally compact groups”, J. Math. Soc. Japan 15, pp
301-303 (1963).
[69] Norris, J.R., Rogers, L.C.G. and Williams, D., “Brownian motion of ellipsoids”, Trans.
Am. Math. Soc. 294, pp 757-765 (1986).
[70] Orihara, A., “On random ellipsoids”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 17, pp
73-85 (1970).
309
[71] Palais, R.S., “A global formulation of the Lie theory of transformation groups”, Mem.
Am. Math. Soc. 22 (1957).
[72] Pap, G., “General solution of the functional central limit problems on a Lie group”, in
Infin. Dim. Anal. Quantum Probab. Rel. Top. 7, 43-87 (2004).
[73] Pauwels, E.J., Rogers, L.C.G., “Skew-product decompositions of Brownian motions”,
Contemp. Math. 73, Am. Math. Soc., pp. 237C262, (1988).
[74] Picard, J. and Sanova, C., “Smoothness of law of manifold-valued Markov processes
with jumps”, Bernoulli 19(5A), pp 1880C1919 (2013).
[75] Parthasarathy, K.R., “Probability measures on metric spaces”, Academic Press (1967).
[76] Protter, P., “Stochastic integration and differential equations, a new approach”,
Springer-Verlag (1990).
[77] Ramaswami, S., “Semigroups of measures on Lie groups”, J. Indiana Math. Soc. 38, pp
175-189 (1974).
[78] Raugi, A., “Fonctions harmoniques et théorèmes limites pour les marches gléatoires sur
les groupes”, Bull. Soc. Math. France, mémoire 54 (1997).
[79] Revuz, D. and Yor, M., “Continuous martingales and Brownian motion”, 3rd ed.,
Springer-Verlag (1999).
[80] Rosenthal, J.S. “Random rotations: characters and random walks on SO(N)”, Ann.
Probab. 22, pp 398-423 (1994).
[81] Saloff-Coste, L., “Analysis on compact Lie groups of large dimension and on connected
compact groups”, Colloq. Math. 118, pp 183-199 (2010).
[82] Sato, K., “Lévy processes and infinitely devisible distributions”, translated from 1990
Japanese original and revised by the author, Cambridge Studies Adv. Math. 68, Cambridge Univ. Press (1999).
[83] Siebert, E. “Fourier analysis and limit theorems for convolution semigroups on a locally
compact groups”, Adv. Math. 39, pp 111-154 (1981).
[84] Stroock, D.W. and Varadhan, S.R.S., “Limit theorems for random walks on Lie groups”,
Sankhyā: Indian J. of Stat. Ser. A, 35, pp 277-294 (1973).
310
[85] Taylor, J.C., “The Iwasawa decomposition and the limiting behavior of Brownian motion
on a symmetric space of non-compact type”, in Geometry of random motion, ed. by R.
Durrett and M.A. Pinsky, Contemp. Math. 73, Am. Math. Soc., pp 303-332 (1988).
[86] Taylor, J.C., “Brownian motion on a symmetric space of non-compact type: asymptotic
behavior in polar coordinates”, Can. J. Math. 43, pp 1065-1085 (1991).
[87] Tutubalin, V.N., “Limit theorems for a product of random matrices”, Theory Probab.
Appl. 10, pp 25-27 (1965).
[88] Valery V.Volchkov and Vitaly V.Volchkov, “Harmonic Analysis of mean periodic functions on symmetric spaces and the Heisenberg group”, Springer-Verlag (2009).
[89] Virtser, A.D., “Central limit theorem for semi-simple Lie groups”, Theory Probab. Appl.
15, pp 667-687 (1970).
[90] Warner, F.W., “Foundations of differential manifolds and Lie groups”, Springer-Verlag
(1971).
311
Index
Symbols:
C(X), Cb (X), Cc (X), C0 (X), 7
C k , Cck , 40
Cb2 (G), 51
GL(n, C), 283
GL(n, R), 283
GL(n, R)+ , 283
Id , 55
L2 (X), L2K (X), 142
M (n), 131
O(1, n), 141
O(d), 28
SL(n, R), 139
SO(1, n)+ , 141
SO(n), 131
SU (n), 98
SU (n), su(n), 119
U (n), 98
U δ , 98
Ad, 71, 282
Ad(G)-invariant matrix, 114
Ad(K)-invariant matrix, 73
∥f ∥(m) , 127
ad, 72
≍, 109
N, 85
Q, 30
R+ , 7
χδ , 99
Ĝ, 98
Ĝ+, 98
ĜK , ĜK+ , 142
B(X), Bb (X), B+ (X), 6
gl(n, R), 283
sl(n, R), 139
ψδ , 99
supp, 86
Tr, 55
∨, 62
∧, 51
ξ l , ξ r , 40
cg , l g , r g , 6
o(1, n), 141
o(n), 139
su(n), 98
u(n), 98
xet , 9
subjects:
additive process, 157
adjoint action, 71
admissible quadruple, 179
angular process, 28
Borel space, 301
Brownian motion, 290
associated to a filtration, 290
Riemannian, 89
standard, 290
canonical path space, 287
Cartan involution, 134
center of a Lie algebra or a Lie group, 283
312
character, 99
general linear group, 283
complex, 283
characteristic measure of a Poisson random
generator
measure, 296
of a convolution semigroup, 128
characteristics of Lévy processes, 76
of a Feller process, 289
characteristics of a Lévy process, 41
group action, 6, 284
components of a drift
on G/K, 252
Harr measure, 14
on a Lie group G, 175
homogeneous space, 13, 284
conjugation map, 282
irreducible, 78
conservative, 9
isotropic, 130
convolution
hyperbolic space, 141
of functions, 135
ideal, 283
of measures, 11, 14
iid, 52
convolution semigroup, 12, 14
inhomogeneous Lévy process
two parameter, 22, 25
in group, 21
coordinate functions
in homogeneous space, 24
on G/K, 75
intensity measure, 296
on a Lie group G, 40
invariant
core of an operator, 43
left,right,bi-,conjugate, 7
covariance matrix, 41
invariant vector field, 281
covariance matrix function, 170, 248
isotropy subgroup, 13, 284
diffuse measure, 23
jump counting measure, 45, 165
diffusion part, 43
jump intensity measure, 165, 247
hypoelliptic, 103
nondegenerate, 103
kernel, 287
diffusion process, 46, 295
killed process, 10
drift, 169, 248
Killing form, 137
extended, 170
Lévy measure, 41, 76
Lévy measure function, 169, 248
Euclidean motion group, 131
extended, 169
exponential map of a Lie group, 282
Lévy process
Feller process, 288
associated to a filtration, 11
filtration, 286
in group, 9
natural, 286
in homogeneous space, 17
finite dimensional distribution, 286
right, 12
fixed jump, 158
Lévy triple, 170, 248
313
extended, 170, 248
Lévy-Khinchin formula, 129
lcrl, 290
lcscH, 7
Lie algebra, 281
semisimple, 118
simple, 118
unimodular, 91
Lie bracket, 281
Lie group, 281
semisimple, 118
simple, 118
life time of a process, 288
linear action, 6
local mean, 169, 247
local section, 13
Lorentz group, 141
quaternion, 121
radial process, 28
Radon measure, 14
random measure, 296
rcll, 9
regular conditional distribution, 301
representation, 98, 136
equivalent, 98, 136
faithful, 98
irreducible, 98, 136
spherical, 137
unitary, 98, 136
root, 150
root space, 150
partition of unity, 25
Peter-Weyl Theorem, 99
Poisson distribution, 296
Poisson process, 296
Poisson random measure, 296
associated to a filtration, 296
extended, 296
homogeneous, 296
positive definite function, 133
section map, 14
semi-martingale, 291
sie, 58
Skorohod metric, 31
special linear group, 139
special orthogonal group, 131
special unitary group, 119
spherical function, 126
spherical transform, 127
stochastic differential equation (sde), 293
stochastic integral
Stratonovich, 292
with respect to a Poisson random measure, 298
stochastically continuous, 12
stopping time, 286
symmetric pair, 134
symmetric spaces, 134
compact, noncompact and Euclidean types, 138
quadratic covariation, 292
predictable, 292
topological group, 6
transition semigroup, 287
Markov process, 7, 287
inhomogeneous, 7, 289
invariant, 7
left invariant, 9, 21
martingale, 290
measure function, 164
orthognal group, 28
314
Feller, 289
two parameter, 289
unimodular, 14
unitary group, 98
version, 286
315