MARKOV TRANSITION FUNCTIONS AND
SEMIGROUPS OF MEASURES
TIMOTHY LANT† AND HORST R. THIEME¦
Abstract. The application of operator semigroups to Markov
processes is extended to Markov transition functions which do not
have the Feller property. Markov transition functions are characterized as solutions of forward and backward equations which
involve the generators of integrated semigroups and are shown to
induce integral semigroups on spaces of measures.
1. Introduction
Continuous semigroups of bounded linear operators have played an
important role as functional analytic tools in the theory of Markov
processes [15, 16, 26]. The link is provided by (Markov) transition
functions K(t, x, D). Here x is a point in a state space Ω and D an
element of a σ-algebra B of subsets of Ω; t ≥ 0 is interpreted as time
and K(t, x, D) as the probability of the state being in the set D at
time t provided x was the state at time 0. There are at least two ways
of associating operator families with K (Section 2). The first type of
family operates on the Banach space of (signed) measures on B with
bounded variation, M(Ω), and is defined by
Z
(1.1)
[S(t)µ](D) =
µ(dx)K(t, x, D),
µ ∈ M(Ω),
Ω
[16, X.8], the second is the formally dual family on the Banach space
of bounded measurable functions on Ω, BM(Ω), with supremum norm,
Z
(1.2)
[S̃(t)f ](x) =
K(t, x, dy)f (y),
f ∈ BM(Ω),
Ω
Date: May 10, 2006.
1991 Mathematics Subject Classification. 47D06, 47D62, 60J35.
Key words and phrases. (Markov) transition functions, stochastic continuity,
semigroups (C0 -, integrated, integral), forward and backward equations, Feller
property, spaces of measures.
† partially supported by NSF grants DMS-0314529 and SES-0345945.
¦ partially supported by NSF grants DMS-9706787 and DMS-0314529.
1
2
T. Lant, H.R. Thieme
[19, Sec.1.9]. The space of measures has an important interpretation
in another application of Markov transition functions, structured population models, as a positive measure µ represents the population distribution with respect to the structure induced by the individual state
space Ω [9]. In this context, S(t)µ represents the structural distribution
of the population at time t. Both operator families, S and S̃, are oneparameter semigroups. The semigroup property, S(t)S(r) = S(t + r)
for all t, r ≥ 0, is equivalent to the Chapman-Kolmogorov equation
Z
(1.3)
K(t + r, x, D) =
K(r, x, dy)K(t, y, D)
∀t, r ≥ 0.
Ω
Among the many types of operator semigroups [21], C0 -semigroups [6,
7, 14, 19, 25, 31] are the most studied. Unfortunately, neither S(t)µ nor
S̃(t)f are continuous in t in general. However, meaningful conditions
concerning K can be found for S̃(t)f to be continuous in t if f is
a continuous function on a suitable Hausdorff topological space, f ∈
C(Ω). Let Ω be a compact metric space for the sake of exposition (we
will also look at the cases where Ω is a normal space or a locally compact
space). The continuity of S̃(t)f in t for f ∈ C(Ω) can most easily be
exploited in a semigroup setting if S̃(t) leaves C(Ω) invariant such that
the restrictions of S̃(t) to C(Ω) form a C0 -semigroup. Historically,
this property, called the Feller property [8], has flatly been postulated.
The associated transition function K is then called a Feller function
[26, p.56]. Feller functions are measure-theoretically characterized as
follows (Sections 4 and A, cf. [31]):
(•) If D is an open set which is the countable union of compact
sets (an open Kσ set) and t ≥ 0, then K(t, x, D) is a lower
semi-continuous function of x ∈ Ω. If D is a compact set which
is the countable intersection of open sets (a compact Gδ set),
then K(t, x, D) is an upper semi-continuous function of x ∈ Ω.
For K to induce a C0 -semigroup on C(Ω) it is necessary and sufficient
to have the respective semi-continuity properties (•) to hold in (t, x) ∈
R+ × Ω (Sections 4 and A). If they hold and Ω is a compact metric
space, S is a dual semigroup on M(Ω) [5, 6, 28, 32] as M(Ω) can be
identified with the dual space of C(Ω).
The theory of operator semigroups has been so attractive to stochastic processes, because C0 -semigroups have an infinitesimal generator
which allows the transition functions to be characterized as solutions
of two different types of operator differential equations, known as forward and backward equations. It is one of the purposes of this paper to
get rid of the restriction (•) and still preserve these characterizations of
Transition functions and semigroups
3
transition functions. Other approaches can be found in [15, 17]. We use
a more recent development in semigroup theory, integrated semigroups
[1]. If we define
Z t
(1.4)
[T (t)µ](D) =
[S(r)µ](D)dr,
0
the operator family {T (t)} is a (once) integrated semigroup (Section
6). In turn, S can be characterized in terms of T by the relation
T (t)S(r) = T (t + r) − T (r).
S is uniquely determined by T provided T is non-denegerate, i.e., if
µ 6= 0 then T (t)µ 6= 0 for at least one t > 0. S is called the integral semigroup associated with T (Section 6.1). Non-degeneracy holds
under mild restrictions on K (Section 7) which follow from stochastic
continuity (Section 3) if Ω is normal or σ-compact. A construction on
BM(Ω) analogous to (1.4), which uses S̃ instead of S, typically leads to
degenerate integrated semigroups and is less fruitful. A non-degenerate
integrated semigroup is associated with a generator A, the generalization of the infinitesimal generator of a C0 -semigroup. In our case, the
generator A is a Hille-Yosida operator in M(Ω). The transition function K is uniquely determined as the solution of the integral equation
Z t
(1.5)
K(t, x, ·) = δx + A
K(s, x, ·)ds,
t ≥ 0, x ∈ Ω.
0
Here δx is the Dirac measure concentrated at x ∈ Ω. This equation can
be formally rewritten as a differential equation which corresponds to
the (Kolmogorov) forward equation (cf. [16, Sec.X.3 (3.5)], [15, Chap.4
(9.58)]). In turn, A is uniquely determined by the transition function
as its resolvent can be expressed by the Laplace transform of K,
Z
Z ∞
−1
dte−λt K(t, x, ·),
µ ∈ M(Ω).
(1.6) (λ − A) µ = µ(dx)
Ω
0
There is a subspace BM(Ω)◦ of BM(Ω) which is total for M(Ω) and
invariant under the semigroup S̃ such that the restrictions of S̃(t) to
BM(Ω)◦ form a C0 -semigroup S̃◦ . The infinitesimal generator of S̃◦ ,
Ã◦ , is dual
R to A and has the following property: If f ∈ D(Ã◦ ) and
v(t)(x) = Ω K(t, x, dy)f (y), then v is the unique solution of
(1.7)
v 0 = Ã◦ v,
v(0) = f,
which corresponds to the (Kolmogorov) backward equation (cf. [16,
Sec.X.3 (3.3), Sec.X.10], [24, Thm.7.11]). K is uniquely determined by
this operator differential equation. For compact Ω, BM(Ω)◦ contains
4
T. Lant, H.R. Thieme
C(Ω) if the transition function K satisfies a sufficiently strong type of
stochastic continuity (Section 5).
While the riddance of (•) is an obvious goal for theoretical reasons,
it is also important for the application of transition functions to deterministic models of structured populations, where this restriction is
in the way of a satisfactory perturbation theory which would allow
death and birth to be incorporated in addition to individual growth
[9, 20, 22, 23].
2. Transition functions and operator semigroups
In the tradition of [21, Def.8.3.5], a (one-parameter) semigroup on
a vector space X is a family of linear transformations S(t), t > 0,
satisfying
(2.1)
S(t + r) = S(t)S(r) ∀t, r > 0.
2.1. B0 -semigroups. All semigroups S we are going to consider here
operate on a Banach space X and will satisfy the extra condition
(2.2)
lim sup kS(t)xk < ∞ ∀x ∈ X.
t&0
We call these semigroups B0 -semigroups. It follows from the uniform
boundedness principle and from (2.1) that any B0 -semigroup is exponentially bounded, i.e., there exist M ≥ 1, ω ∈ R such that
kS(t)k ≤ M eωt
∀t > 0.
2.2. C0 -semigroups. A semigroup S is called a C0 -semigroup if
(2.3)
kS(t)x − xk → 0,
t & 0, x ∈ X.
It is then convenient to extend S(t) to [0, ∞) by
(2.4)
S(0)x = x,
x ∈ X.
With this extension, (2.1) holds for all t, r ≥ 0, and S(t) is strongly
continuous in t ≥ 0. For C0 -semigroups the infinitesimal generator A
is defined by
(2.5)
Ax = lim (1/t)(S(t)x − x),
t&0
x ∈ D(A),
with D(A) consisting of all elements x ∈ X for which this limit exists.
Any C0 -semigroup is a B0 -semigroup and is exponentially bounded.
If S is a B0 -semigroup on a Banach space X, one introduces the space
(2.6)
X◦ = {x ∈ X; kS(t)x − xk → 0, t & 0}.
X◦ is a closed subspace of X that is invariant under S(t) for all t ≥ 0.
The restriction of S to X◦ , S◦ , is a C0 -semigroup on X◦ .
Transition functions and semigroups
5
2.3. Transition functions. Let Ω and Ω̃ be measurable spaces with
respective σ-algebras B and B̃.
Definition 2.1 ([3, 10.3.1]). A function K̃ : Ω̃ × B → R is called a
measure kernel if
(a) K̃(x, ·) is a non-negative measure on B for all x ∈ Ω̃.
(b) K̃(·, D) is a B̃-measurable function for all D ∈ B.
A measure kernel K̃ is called bounded if supx∈Ω̃ K̃(x, Ω) < ∞.
Definition 2.2. A function K : R+ × Ω × B → R is called a transition
function if
(a) K(t, x, ·) is a non-negative measure on B for all t ≥ 0, x ∈ Ω.
(b) K(t, ·, D) is a B-measurable function for all t ≥ 0, D ∈ B.
(In other words, K(t, ·) is a measure kernel.)
(c) K(0, x, D) = 1 if x ∈ D and K(0, x, D) = 0 if x ∈ Ω \ D.
(d) There exist δ, c > 0 such that K(t, x, Ω) ≤ c for all t ∈ [0, δ],
x ∈ Ω.
A transition function K is called a Markov transition function if it
satisfies the Chapman-Kolmogorov equations (1.3).
A transition function is called a transition kernel if K(·, D) is BR+ ×B
measurable where BR+ is the σ-algebra of Borel sets on R+ and BR+ ×B
is the product σ-algebra, in other words if K is a measure kernel.
Remark 2.1. Sometimes the term ‘transition function’ is used such that
the Chapman-Kolmogorov equations are included [15]. We follow the
use in [26, Sec.3.2] and [18, Sec.2.1] though it may not be clear whether
they use ‘Markov’ to highlight the Chapman-Kolmogorov equations or
the assumption that K(t, x, Ω) ≤ 1 (or = 1) which we do not make (cf.
[3, 10.3.1]). We use ‘Markov’ in order to emphasize the connection of
the Chapman-Kolmogorov equations to Markov processes [2, Sec.2.2]
[15, Ch.4 (1.9)].
If K is a transition function, (1.1) and (1.2) define families of bounded
linear operators on the space of measures on B of bounded variation,
M(Ω), and the space of bounded measurable functions, BM(Ω), respectively. The Chapman-Kolmogorov equations are equivalent to the semigroup property of these operator families. By property (d), kS(t)k ≤ c,
kS̃(t)k ≤ c for all t ∈ [0, δ] and both S and S̃ are B0 -semigroups and
so exponentially bounded.
Lemma 2.3. There exist ω ∈ R, M ≥ 1 such that kS(t)k ≤ M eωt ,
kS̃(t)k ≤ M eωt , K(t, x, Ω) ≤ M eωt for all t ≥ 0, x ∈ Ω.
6
T. Lant, H.R. Thieme
Through
Z
hµ, f i =
f dµ,
f ∈ BM(Ω), µ ∈ M(Ω),
Ω
BM(Ω) can be identified with a subspace of M(Ω)∗ , the topological
dual of M(Ω). Actually BM(Ω) is a sequentially weakly∗ closed subspace which norms M(Ω), i.e. sup{|hf, µi|; kf k ≤ 1} is an equivalent
norm on M(Ω). We see that the semigroup of dual operators S ∗ (t)
leaves BM(Ω) invariant and the restrictions of S ∗ (t) to BM(Ω) coincide with S̃(t).
2.4. A first shot at the forward and backward equations. Since
S and S̃ are B0 -semigroups on M(Ω) and BM(Ω) respectively, we
can use the construction (2.6) and obtain C0 -semigroups S◦ and S̃◦ on
M(Ω)◦ and BM(Ω)◦ respectively with infinitesimal generators A◦ and
Ã◦ . These operators are in duality, as
hA◦ µ, f i = hµ, Ã◦ f i
∀µ ∈ D(A◦ ), f ∈ D(Ã◦ ).
The equations
(2.7)
u0 = A◦ u
and
v 0 = Ã◦ v
correspond to the (Kolmogorov) forward and backward equations. The
backwards equation v 0 = Ã◦ v uniquely determines the transition function K if BM(Ω)◦ is a total subspace of M(Ω)∗ , i.e. it separates
measures: if µ ∈ M(Ω) is not the 0 measure, then there exists some
f ∈BM(Ω)◦ such that hµ, f i 6= 0. One of the tasks of this paper consists in finding practical conditions for this to be the case. The forward
equation in the form of u0 = A◦ u is less useful because it seems difficult
to come up with reasonable conditions which make M(Ω)◦ separate
points in BM(Ω). One of the reasons that this is difficult lies in the
fact that an important class of Markov transition functions is of the
form K(t, x, dy) = κ(t, x, y)dy, with Ω being a measurable subset of
Rn . Then S(t) maps M(Ω) into the closed subspace of measures which
are Lebesgue absolutely continuous and which can be identified with
L1 (Ω). This implies that M(Ω)◦ is contained in L1 (Ω). But L1 (Ω) does
not separate the zero function from functions which are 0 Lebesgue almost everywhere. For the same class of Markov transition functions,
the semigroup S̃ is degenerate, i.e. if f is 0 almost everywhere (but not
everywhere), we still have that S̃(t)f is the zero-function in BM(Ω) for
all t > 0. This makes S̃ by itself not such a useful object of study and is
one of the motivations to consider integrated and integral semigroups
on M(Ω).
Transition functions and semigroups
7
2.5. Feller’s warning. Feller notes in his classic [16, X.8] that semigroups on M(Ω) seem to be a natural path to study Markov transition
functions but warns that general semigroups on M(Ω) may be too large
a class to work with as not all of the semigroups are linked to transition
functions. The following holds, however.
Proposition 2.4. There is a bijective correspondance between Markov
transition functions and those B0 -semigroups on M(Ω) the duals of
which leave BM(Ω) invariant.
Proof. One direction of the correspondence is clear from the previous
considerations. Now let S(·) be a B0 -semigroup on M(Ω) such that,
for t ≥ 0, S ∗ (t) maps BM(Ω) into itself. We define
K(t, x, D) = [S ∗ (t)χD ](x),
x ∈ Ω, D ∈ B.
Then K(t, x, D) is a measurable function of x ∈ Ω. Now
K(t, x, D) = hδx , S ∗ (t)χD i = hS(t)δx , χD i = [S(t)δx ](D),
which shows that K(t, x, ·) is a measure on B. Let µ ∈ M(Ω). Then
Z
∗
[S(t)µ](D) = hS(t)µ, χD i = hµ, S (t)χD i =
µ(dx)K(t, x, D).
Ω
The other properties of a Markov transition function are easily checked
for K.
¤
3. Stochastic continuity of transition functions
Little progress can be made in the interface of Markov transition
functions and one-parameter semigroups unless the state space Ω is a
topological Hausdorff space. Exceptions are transition functions associated with Markov jump processes [16, X.3] [15, 4.1] which induce
C0 -semigroups on M(Ω) when Ω is just a measurable space [30]. Recall
the operator semigroups S̃ on BM(Ω) and S on M(Ω), introduced in
(1.2) and (1.1), associated with a Markov transition function K.
Let Ω be a topological Hausdorff space. Cb (Ω) denotes the Banach
space of real-valued bounded continuous functions on Ω with supremum
norm. C0 (Ω) denotes the set of continuous real-valued functions that
vanish at infinity. The latter means that for every ² > 0 there is a
compact subset C of Ω such that |f (x)| < ² whenever x ∈ Ω \ C.
When Ω is a topological space, the following two σ-algebras are typically considered: the σ-algebra of Borel sets generated by the open
subsets in Ω and the σ-algebra of Baire sets which is the smallest σalgebra such that all continuous functions from Ω to R are measurable.
In a topological space, we exclusively consider the σ-algebra of Baire
8
T. Lant, H.R. Thieme
sets which is denoted by B. The reason is that we need all finite measures on B to be regular in an appropriate sense; conditions which make
every Borel measure regular seem to imply that the Borel and Baire
sets coincide.
Remark 3.1. Borel and Baire sets coincide if Ω is a metric space [3,
7.2.4], in particular if Ω is a locally compact space with countable base
[3, 7.6.2, 7.6.3] or if, more generally, Ω is perfectly normal.
Recall that a topological space is perfectly normal if it is normal and
every open set is an Fσ -set, i.e. a countable union of closed sets [4,
Sec.11 Exc.10]. In a normal space, the Baire-σ-algebra is generated by
the open Fσ -sets [3, 7.2.3]. So, if every open set is an Fσ -set, the Baire
and Borel σ-algebras coincide.
3.1. Locally compact state space. Let Ω be a locally compact space,
i.e. it is a Hausdorff space and every x ∈ Ω is contained in an open set
with compact closure.
Definition 3.1. A measure kernel K is called weakly stochastically
continuous, if for every open Kσ -set U ⊆ Ω with compact closure
lim K(t, x, U ) = 1 whenever x ∈ U.
t→0
Definition 3.2. A function f : Ω̃ → R is upper semi-continuous at a
point x0 in the topological space Ω̃ if the set {x ∈ Ω̃; f (x) < f (x0 ) + ²}
is open for every ² > 0. f is called lower semi-continuous at x0 if −f
is upper semi-continuous at x0 .
Theorem 3.3. The following are equivalent for a locally compact space
Ω and a Markov transition function K:
(a) K is weakly stochastically continuous.
(b) If U is an open Kσ set and x ∈ Ω, then K(·, x, U ) is lower
semi-continuous at t = 0. If D is a compact Gδ set and x ∈ Ω,
then K(·, x, D) is upper semi-continuous at t = 0.
(c) For each x ∈ Ω, f ∈ C0 (Ω), [S̃(t)f ](x) is a right-continuous
function of t ≥ 0.
Proof. (a) =⇒ (b): The first part of (b) is an obvious consequence of
(a). Let D be a compact Gδ -set. Let x ∈ Ω. There exists a function
f ∈ C0 (Ω) such that f = 1 on D and f (x) = 1. Set U = {f > 1/2}.
Then U is an open Kσ -set, x ∈ U , D ⊆ U , and Ū ⊆ {f ≥ 1/2} is
compact. By assumption,
K(t, x, D) ≤ K(t, x, U ) → 1,
t→0+.
Now let x ∈ Ω \ D. Since x ∈ U , x is an element of the open
S set U \ D.
We claim that U \D is a Kσ set. Since U is a Kσ -set, U = ∞
n=1 Cn for a
Transition functions and semigroups
9
countable family of compact sets Cn . Since D is a Gδ -set, D =
for a countable family of open sets Uk . Now
n
∞
\
[
U \ D =U \ ( Uk ) =
(U \ Uk )
k=1
=
∞ µ³ [
∞
[
n=1
k=1
k=1
´
C n \ Uk
¶
=
∞
∞ µ[
[
k=1
Tn
k=1
Uk
¶
(Cn \ Uk ) .
n=1
So U \ D is the union of the countable family {Cn \ Uk ; k, n ∈ N} of
compact sets Cn \ Uk . By assumption, since x ∈ U \ D, as t → 0+,
K(D, t, x) = K(U, t, x) − K(U \ D, t, x) → 1 − 1 = 0 = K(D, 0, x).
So K(D, t, x) is upper semi-continuous at t = 0.
(b) =⇒ (c): (b) can be reformulated as follows:
If U is an open Kσ -set, [S̃(t)χU ](x) is upper semi-continuous at t = 0
for every x ∈ Ω. Further, if C is a compact Gδ set, [S̃(t)χC ](x) is lower
semi-continuous at t = 0 for every x ∈ Ω. The same proof as in
Theorem A.2 shows that [S̃(t)f ](x) is right continuous at t = 0. Since
S̃ is a semigroup,
Z
[S̃(t + h)f ](x) = [S̃(t)S̃(h)f ](x) = [S̃(h)f ](y)K(t, x, dy).
Ω
Since [S̃(h)f ](y) → f (y) as h & 0, pointwise in x ∈ Ω, the dominated
convergence theorem implies that [S̃(t+h)f ](x) → [S̃(t)f ](x) as h & 0,
pointwise in x ∈ Ω.
(c) =⇒ (a): Let x ∈ U and U open. Since Ω is locally compact,
there exists a function f ∈ C0 (Ω) such that f (x) = 1, 0 ≤ f ≤ χU .
Then
Z
K(t, x, U ) ≥
K(t, x, y)f (y)dy → f (x) = 1,
t → 0.
Ω
Since U has compact closure, there exist some f ∈ C0 (Ω) such that
f (y) = 1 for all y ∈ U . So, for x ∈ U ,
K(t, x, U ) ≤ [S̃(t)f ](x) → f (x) = 1,
t → 0.
¤
Ω is called σ-compact [12, XI.7] if it is locally compact and a Kσ set (countable at infinity [3, 7.4.4, 7.5]). Every σ-compact space is
paracompact [12, XI.7] and every paracompact space is normal [12,
VIII.2].
Corollary 3.4. Let Ω be σ-compact. Then, if the Markov transition
function K is weakly stochastically continuous, it is a transition kernel.
10
T. Lant, H.R. Thieme
Proof. Let f ∈ C0 (Ω). By Theorem 3.3, [S̃(t)f ](x) is right continuous
R in t ≥ 0 and Baire measurable in x ∈ Ω. By Proposition B.1,
f (y)K(t, x, dy) is jointly Baire-measurable in (t, x). Let U be an
Ω
open Kσ -set. Then there exists a sequence (fn ) in C0 (Ω) such
that fn %
R
χU pointwise on Ω as n → ∞. So K(t, x, U ) = limn→∞ Ω fn (y)K(t, x,
dy) is Baire-measurable in (t, x). It follows from our assumptions that
the σ-algebra of Baire sets is generated by the open Kσ -sets [3, 7.1.3,
7.4.3, 7.4.5]. This implies that K(t, x, U ) is Baire measurable in (t, x)
for all U ∈ B.
¤
A similar proof shows that C0 (Ω) separates point in M(Ω).
Proposition 3.5. Let Ω be a σ-compact space. Then C0 (Ω) is a total
subspace of M(Ω)∗ .
As we mentioned before, we have a greater interest in the semigroup
S than in the semigroup S̃ because the latter is degenerate for an important class of Markov transition functions. We therefore reformulate
Theorem 3.3 in terms of S using Lebesgue’s theorem of dominated
convergence.
Corollary 3.6. The following are equivalent for a locally compact space
Ω and a Markov transition function K:
(a) K is weakly stochastically continuous.
(b) For all f ∈ C0 (Ω) and µ ∈ M(Ω), hf, S(t)µi is a right-continuous function of t ≥ 0.
3.2. General topological Hausdorff spaces.
Definition 3.7. A transition function K is called stochastically continuous if for every x ∈ Ω and every open Baire-set U ⊆ Ω,
K(x, t, U ) → 1,
t → 0+,
whenever x ∈ U.
Theorem 3.8. (a) If K is stochastically continuous, then, for all f ∈
Cb (Ω) and µ ∈ M(Ω), hf, S(t)µi is a right continuous function of
t ≥ 0.
(b) If Ω is a completely regular space (e.g. a metric, normal, or
locally compact space), the stochastic continuity of K is equivalent to
the right continuity of hf, S(t)µi in t for all f ∈ Cb (Ω), µ ∈ M(Ω).
Proof. (a) This part can be proved similarly as for Theorem 3.3 and
Corollary 3.6.
(b) Let x ∈ Ω be fixed but arbitrary. Let U 3 x be an open subset
of Ω. Since Ω is completely regular, there exists a continuous function
f : Ω → [0, 1] such that f (x) = 1 and f (y) = 0 for y ∈ Ω \ U . Then
Transition functions and semigroups
11
f ∈ Cb (Ω) and f ≤ χU . Assume that hf, S(t)µi is continuous in t ≥ 0
for µ = δx , the Dirac measure concentrated at x. Then
Z
K(t, x, U ) ≥ f (y)K(t, x, dy) → 1,
t→0+.
Ω
Since χΩ ∈ Cb (Ω),
Z
0 ≤ K(t, x, U ) ≤
χΩ (y)K(t, x, dy) → χΩ (x) = 1,
t→0+.
Ω
We combine the two statements and K(t, x, U ) → 1 as t → 0+.
¤
Corollary 3.9. Let Ω be a normal Hausdorff space. Then Cb (Ω) is a
total subspace of M(Ω)∗ . If the Markov transition function K(t, x, B)
is stochastically continuous, it is a transition kernel.
Proof. Since Ω is normal, the σ-algebra of Baire sets is generated by
the set of closed Gδ sets (and also by the set of open Fσ sets) [3, 7.2.3].
By Urysohn’s characterization of normality, Cb (Ω) separates disjoint
closed sets. The proof proceeds now as the one of Corollary 3.4.
¤
4. Which transition functions induce C0 -semigroups on
C0 (Ω)?
Often semigroup theory has been applied to Markov transition functions by assuming the Feller property, namely that the appropriate
space of continuous functions is invariant under S̃ and a C0 -semigroup
is induced [16, X.10], [19, 9.11], [26, 3.2]. In this section, we clarify the
restrictions that the Feller property imposes on the transition function.
Theorem 4.1. Let Ω be a locally compact Hausdorff space. Then a
Markov transition function induces a C0 -semigroup on C0 (Ω) if and
only if the following are satisfied.
(i): For every compact Gδ -set C in Ω and every t > 0, K(t, ·, C)
is upper semi-continuous on Ω.
(ii): For every open Kσ -set U in Ω and every t > 0, K(t, ·, U ) is
lower semi-continuous on Ω.
(iii): For every compact subset C of Ω, every t > 0, and every ² >
0, there exists a compact subset C̃ of Ω such that K(t, x, C) < ²
for all x ∈ Ω \ C̃.
(iv): K is weakly stochastically continuous.
A similar statement can be found in [31, Thm 2.1]. We are not
able to prove the necessity-statement there (which has been left to
the reader) unless it is interpreted in the sense above. We need the
following abstract result [7, Prop.1.23][14, Ch.I, Thm.5.8].
12
T. Lant, H.R. Thieme
Theorem 4.2. Let {S(t); t ≥ 0} be a semigroup of bounded linear
operators on a Banach space X. Then S(·) is a C0 -semigroup if (and
only if ) it is weakly continuous at t = 0, i.e., hS(t)x, x∗ i → hx, x∗ i for
t & 0, x ∈ X, x∗ ∈ X ∗ .
Proof of Theorem 4.1. The necessity of (i), (ii), (iii) follows from Proposition A.2. (Sufficiency:) By Proposition A.2, S̃(t) maps C0 (Ω) into
itself for every t ≥ 0. Let S ¦ (t) be the restriction of S̃(t) to C0 (Ω).
By Theorem 3.3, for f ∈ C0 (Ω) and x ∈ Ω, [S ¦ (t)f ](x) = [S̃(t)f ](x)
is a continuous function of t at t = 0. It follows from the dominated
convergence theorem that
Z
Z
¦
[S (t)f ](x)µ(dx) →
f (x)µ(dx),
t → 0,
Ω
Ω
for every non-negative Borel measure µ on Ω and also for every signed
Borel measure of finite variation. Since C0 (Ω)∗ can be identified with
the Banach space of signed regular Borel measures of finite variation
[4, Thm.38.7], S ¦ (t)f is weakly continuous in t at t = 0. By Theorem
4.2, S ¦ (·) is a C0 -semigroup.
¤
Analogously one can characterize the Markov transition functions
which induce C0 -semigroups on Cb (Ω) for a normal space Ω by using
Theorem 3.8 and Proposition A.3.
Remark 4.1. By functional analytic magic (the Krein-Šmulian theorem that the closed convex hull of a weakly compact set is weakly
compact [13, V.6.Thm.3]), the weak stochastic continuity of K (together with the Feller property) implies the following locally uniform
time-continuity statements:
(i) Let C ⊂ U ⊂ Ω and C compact and U open. Then
lim inf inf K(t, x, U ) ≥ 1.
t→0
x∈C
(ii) Let C ⊂ Ω and C compact. Then
lim sup inf K(t, x, C) ≤ 1.
t→0
x∈C
If Ω is a locally compact metric space, C a compact set in Ω and
K(t, x, Ω) ≤ 1 for all t ≥ 0, x ∈ Ω, then it follows [26, Thm. 3.1] that
£
¤
lim sup sup 1 − K(t, x, B² (x)) = 0,
t&0
x∈C
for all ² > 0 where B² (x) is the open ball with center x and radius ².
As far as sufficient conditions are concerned, Theorem 4.1 shows that
this assumption in [26, Thm. 3.1] can be considerably relaxed (and the
separability of Ω be dropped).
Transition functions and semigroups
13
Proof. (i) Choose f ∈ C0 (Ω) with f = 1 on C, f = 0 on U , and
0 ≤ f (y) ≤ 1 for all y ∈ Ω. Since S̃ induces a C0 -semigroup on C0 (Ω),
¯Z
¯
¯
¯
sup¯ K(t, x, dy)f (y) − f (x)¯ → 0,
t → 0.
x∈Ω
Ω
Let ² > 0. Then there exists some δ > 0 such that
Z
f (x) + ² >
K(t, x, dy)f (y) > f (x) − ²
∀t ∈ [0, δ].
Ω
For x ∈ C,
Z
1 − ² = f (x) − ² <
K(t, x, dy)f (y) ≤ K(t, x, U ).
U
Further, for x ∈ C,
Z
1 + ² = f (x) + ² >
f (y)K(t, x, dy) = K(t, x, C).
C
¤
An obvious example of a Markov transition function which does not
have the Feller property is K(t, x, D) = e−tγ(x) δx (D) where γ is a nonnegative Baire measurable function which is not continuous. Then the
associated semigroup on BM(Ω), [S̃(t)f ](x) = f (x)e−tγ(x) , does not
map continuous functions f to continuous functions.
In the following example, with Ω = R+ , the semigroup S̃ preserves
continuity, but not the property of vanishing at infinity. We interpret
exponential growth, N (t) = N0 e(β−µ)t , in an age-structured population
model: β and µ are the per capita birth and mortality rates, N (t) the
population size at time t and N0 the initial
stratify
R ∞ population size. RWe
∞
the population along age: N (t) = 0 u(t, a)da, N0 = 0 u0 (a)da.
Then u satisfies the McKendrick equation
(4.1)
ut + ua = −µu,
u(t, 0) = βN (t),
u(0, a) = u0 (a),
where ut and ua are the partial derivatives with respect to t and a.
One readily checks that

u0 (a −Zt)e−µt ;
t < a,

∞
u(t, a) =
u0 (s)ds; t > a,
 βeβ(t−a) e−µt
0
is a solution of (4.1) for t 6= a if u0 is differentiable. Otherwise it
is a solution in an appropriately generalized sense. In this example,
the semigroup S leaves L1 (R+ ) invariant (identified with the subspace
of measures which are absolutely continuous relative to the Lebesgue
14
T. Lant, H.R. Thieme
measure) and S(t)u0 = u(t, ·). We use the formal duality hS̃(t)f, u0 i =
hf, S(t)u0 i to find
Z t
³
´
−µt
β(t−s)
(4.2)
[S̃(t)f ](a) = e
f (a + t) +
f (s)e
ds ,
0
for f ∈ BM(R+ ). For t > 0, S̃(t) maps continuous functions to continuous functions, but lim [S̃(t)f ](a) > 0 if f ∈ C0 (R+ ) is positive. To
a→∞
make this example rigorous, we observe that (4.2) defines a semigroup
S̃ on BM(R+ ) indeed and that the associated transition function is
Z
³
´
−µt
K(t, a, D) = e
δt+a (D) +
eβ(t−s) ds .
[0,t]∩D
Similar problems arise in body-size structured population models
which involve per capita birth rates β(x) that depend on body size
x. Constructing the associated C0 -semigroup [10] (strongly continuous
evolutionary system [11]) on C0 (R+ ) by perturbation requires β(x) → 0
as x → ∞, i.e. the birth rate must tend to 0 for large body sizes. This
assumption which may be unrealistic in certain applications can be
dropped if the Feller property does not need to be satisfied [20, 23].
5. More continuity results and a backward integral
equation
Motivated by the characterization of the Feller property in the previous section and its failure in the preceding examples, we want to work
without it and the restrictions it imposes on Markov transition kernels.
Recall the space BM(Ω)◦ of those bounded measurable functions f on
Ω for which S̃(t)f is a continuous function of t; the Markov transition
function K induces the C0 -semigroup S̃◦ on BM(Ω)◦ (Section 2.4 and
(2.6)). In the following we derive conditions for C0 (Ω) ⊆BM(Ω)◦ if Ω
is locally compact. This is of interest as the domain of infinitesimal
generator Ã◦ of S̃◦ involved in the backward equation (1.7) (see also
Section 2.4) is hard to characterize and one can alternatively consider
the integral version
Z t
(5.1)
v(t) = f + Ã◦
v(r)dr,
f ∈ BM(Ω)◦ ,
0
R
which is uniquely solved by v(t)(x) = Ω f (y)K(t, x, dy).
Throughout this section, Ω is a locally compact Hausdorff space and
K a Markov transition function. Further we assume that
Transition functions and semigroups
15
(¦) for every ² > 0, x ∈ Ω, there exist an open neighborhood U of
x and δ > 0 such that
K(t, z, Ω) ≤ 1 + ²
∀z ∈ U, t ∈ [0, δ].
In many stochastic applications, condition (¦) is trivially satisfied
as K(t, x, Ω) ≤ 1. The perturbation theory in [23] leads to Markov
transition kernels which only satisfy (¦), but not this more restrictive
inequality. In the next results, we investigate a stronger stochastic
continuity concept than the ones used in Section 3.
Theorem 5.1. C0 (Ω) ⊆ BM(Ω)◦ if and only if the following two statements hold:
(i) If U ⊂ Ω is an open Baire set, x ∈ U and ² > 0, then there
exist δ > 0 and an open set Ũ 3 x such that
K(t, z, U ) ≥ 1 − ²
whenever z ∈ Ũ , 0 ≤ t < δ.
(ii) If C ⊆ Ω is a compact Baire set and ² > 0, then there exist a
compact set C̃ ⊆ Ω and δ > 0 such that
K(t, x, C) ≤ ²
∀x ∈ Ω \ C̃, 0 ≤ t < δ.
Corollary 5.2. Make the additional assumptions (i), and (ii) of Theorem 5.1. Then C0 (Ω) ⊆ BM(Ω)0 and,
R t for every f ∈ C0 (Ω), the backward integral equation
R v(t) = f + Ã0 0 v(r)dr has the unique continuous solution v(t) = Ω K(t, ·, dy)f (y). If Ω is σ-compact, the transition
function K is uniquely determined by this fact.
The solution v does not necessarily take values in C0 (Ω). The first
part of the corollary follows directly from the previous theorem, the
properties of BM(Ω)0 , and standard semigroup theory. The uniqueness
of K follows from Proposition 3.5.
Proof of Theorem 5.1. ‘⇒’: Let U ⊆ Ω be open, x ∈ U and ² > 0.
Since Ω is locally compact, there exist an open set Ũ and a compact
set C such that x ∈ Ũ ⊆ C ⊆ U. Again, since Ω is locally compact,
there exist some f ∈ C0 (Ω) such that χC ≤ f ≤ χU . Then
K(t, z, U ) ≥ [S̃(t)f ](z) → f (z),
t & 0,
uniformly in z ∈ Ω. Since f (z) ≥ 1 for all z ∈ Ũ , (i) follows.
Let C ⊆ Ω be compact. Since Ω is locally compact, there exist
an open set U and a compact set C̃ such that C ⊆ U ⊆ C̃ ⊆ Ω.
Again, since Ω is locally compact, there exist f ∈ C0 (Ω) such that
χC ≤ f ≤ χU . Then
K(t, x, C) ≤ [S̃(t)f ](x) → f (x),
t & 0,
16
T. Lant, H.R. Thieme
uniformly in x ∈ Ω. Since f (x) = 0 for all x ∈ C̃, (ii) follows.
‘⇐’: By the semigroup property, it is sufficient to show that, for
f ∈ C0 (Ω), S̃(t)f is continuous at t = 0. Without loss of generality,
we can assume that f is a non-negative function. Suppose that S̃(t)f
is not continuous at t = 0. Then there exist η > 0, a sequence tn & 0
and a sequence (xn ) in Ω such that
¯
¯
¯
¯
∀n ∈ N.
(5.2)
¯[S̃(tn )f ](xn ) − f (xn )¯ > 2η
We can assume that η ≤ 1.
Claim: There exists a compact set C ⊆ Ω such that xn ∈ C for all
n ∈ N.
Suppose that the claim does not hold. By Definition 2.2 (d), there
exists some δ0 > 0, c > 0 such that
K(t, x, Ω) ≤ c
∀t ∈ [0, δ0 ], x ∈ Ω.
There exists some n0 ∈ N such that tn ≤ δ0 for all n ≥ n0 . Since
f ∈ C0 (Ω), there exists some compact set C1 ⊆ Ω such that
η
(5.3)
|f (x)| ≤
∀x ∈ Ω \ C1 .
8(1 + c)
For all x ∈ Ω, t ≥ 0,
Z
Z
|S̃(t)(x)| ≤
(5.4)
|f (y)|K(t, x, dy) +
C1
|f (y)|K(t, x, dy)
Ω\C1
η
≤kf kK(t, x, C1 ) + K(t, x, Ω).
8
By (ii), there exists a compact set C2 ⊆ Ω and some δ1 > 0 such that
η
(5.5)
K(t, x, C1 ) ≤
x ∈ Ω \ C2 , t ∈ [0, δ1 ].
8kf k + 1
Set δ = min{δ1 , δ0 }, C = C1 ∪ C2 ∪ {x1 , . . . xn0 }. Then C is compact.
Since (xn ) is not contained in any compact set, there exists some n ∈ N,
n ≥ n0 , such that xn ∈ Ω \ C. By (5.3), (5.4), and (5.5), for such an
xn we have
¯
¯ ¯
¯ ¯
¯
¯[S̃(tn )f ](xn ) − f (xn )¯ ≤ ¯[S̃(tn )f ](xn )¯+¯f (xn )¯ ≤ η,
a contradiction. This proves the claim.
After choosing subsequences, the real sequences (f (xn )) and ([S̃(tn )f ]
T
(xn )) converge. It follows from the claim that ∞
m=1 {xn ; n ≥ m} 3 x
for some x ∈ Ω. Since f is continuous and f (xn ) has a limit, f (xn ) →
f (x) as n → ∞. By (5.2), there exists some mη ∈ N such that
¯
¯
¯[S̃(tn )f ](xn ) − f (x)¯ ≥ η
(5.6)
∀n ≥ mη .
Transition functions and semigroups
17
In order to derive a contradiction, we let U = {y ∈ Ω; |f (y) − f (x)| <
η/8}. Then U is an open Baire set, x ∈ U , f (x) ≥ 0, and
Z
¯
¯
¯[S̃(tn )](xn )−f (x)¯ ≤
|f (y) − f (x)|K(tn , xn , dy)
U
Z
£
¤
+
|f (y)|K(tn , xn , dy) + |f (x)| K(tn , xn , Ω) − 1 + ,
Ω\U
where [r]+ = max{r, 0} is the positive part of a real number r. By the
definition of U ,
¯
¯
£
¤
¯[S̃(tn )](xn ) − f (x)¯ ≤ η K(tn , xn , U ) + kf k K(tn , xn , Ω) − 1
+
8
+ kf k(K(tn , xn , Ω) − K(tn , xn , U )).
Since every open set Ũ ⊆ Ω with x ∈ Ũ has non-empty intersection
with all sets {xn ; n > m}, m ∈ N, By (¦) and (i), there exists some
n > mη such that
η
η
K(tn , xn , Ω) < 1 +
≤ 2,
K(tn , xn , U ) > 1 −
.
8kf k + 1
8kf k + 1
For such an n > mη , we have
¯
¯
¯[S̃(tn )f ](xn ) − f (x)¯ < 3η ,
4
a contradiction to (5.6).
¤
For a locally compact metric space, K is called locally uniformly
stochastically continuous [26, 3.2] if for each δ > 0 and each compact
subset C of Ω,
K(t, x, Bδ (x)) → 1,
t → 0+, uniformly in x ∈ C.
Theorem 5.3. Consider the statements (i) and (ii) in Theorem 5.1
and the following statements:
(iii) K is locally uniformly stochastically continuous.
(iv) C0 (Ω) ⊆BM(Ω)◦ .
Then we have the following equivalences,
[(i) ∧ (ii)] ⇐⇒ [(ii) ∧ (iii)] ⇐⇒ (iv).
Proof. The equivalence [(i) ∧ (ii)] ⇐⇒ (iv) has been established in
Theorem 5.1.
(iii) =⇒ (i): Let U be an open set and x ∈ U . Then there exists
some η > 0 such that Bη (x) ⊆ U . Since Ω is locally compact, there
exists an open set Ũ such that x ∈ Ũ and the closure of Ũ is compact.
18
T. Lant, H.R. Thieme
So K(t, x, Bη (x)) → 1 as t → 0+ uniformly on Ũ . Let ² > 0. Then
there exists some δ > 0 such that
K(t, x, U ) ≥ K(t, x, Bη (x)) > 1 − ²
whenever t ∈ [0, δ], x ∈ Ũ .
(iv) =⇒ (iii) follows from the proof of [26, Thm.3.1].
¤
The equivalence (ii) ∧ (iii) ⇐⇒ (iv) is proved directly in [26,
Thm.3.1]. There it is assumed that K has the Feller property, i.e. S̃
leaves C0 (Ω) invariant, but this property is not really used. In fact if it
is satisfied, weak stochastic continuity instead of (iii) (or (i)) is sufficient
in combination with (ii) to make the restriction of S̃ to C0 (Ω) a C0 semigroup (see Theorem 4.1). For further continuity results we refer
to [22].
6. Integrated semigroups
Our quest for a forward equation and a backward equation each of
which uniquely characterizes the Markov transition function leads us to
(once) integrated semigroups, T , strongly continuous operator families
which satisfy
Z t+r
Z t
Z r
T (t)T (r) =
T (s)ds −
T (s)ds −
T (s)ds, t, r ≥ 0,
(6.1)
0
0
0
T (0) = 0.
Rt
These relations can be motivated by formally defining T (t) = 0 S(s)ds,
with a semigroup S. For an authoritative survey and bibliographic
notes concerning integrated semigroups we refer to [1, Chap.3].
One is mainly interested in non-degenerate integrated semigroups,
i.e., T (t)x = 0 for all t > 0 occurs only for x = 0. The generator A
of a non-degenerate integrated semigroup is given as follows [27]: if
x, y ∈ X,
Z t
(6.2) x ∈ D(A), y = Ax
⇐⇒
T (t)x−tx =
T (s)yds ∀t ≥ 0.
0
Notice that this definition makes sense and defines a closed operator
A, even if T is not an integrated semigroup. Actually one has the
following result:
Theorem 6.1. Let T (t), t ≥ 0, be a non-degenerate strongly continuous family of bounded linear operators on X and let the closed linear
Transition functions and semigroups
19
operator A Rbe defined by (6.2). Then T is an integrated semigroup if
t
and only if 0 T (s)ds ∈ D(A) for all t ≥ 0 and
Z t
A
T (s)xds = T (t)x − tx ∀t ≥ 0.
0
Proof. The “only if” part follows from [27, Lemma 3.4]. The “if” part
follows from the proof of [27, Thm.6.2].
¤
If T (t) is exponentially bounded, i.e., there exist M, ω > 0 such that
kT (t)k ≤ M eωt
∀t ≥ 0,
one has the following useful relation between the Laplace transform of
the integrated semigroup and the resolvent of the generator. It follows
by combining [1, Prop.3.2.4] and [27, Prop.3.10].
Theorem 6.2. Let T (t), t ≥ 0, be a strongly continuous exponentially
bounded family of bounded linear operators on X and A : D(A) →
X be a linear operator in X. Then T is a non-degenerate integrated
semigroup and A its generator if and only if there exists some ω > 0
such that any λ > ω is contained in the resolvent set of A and the
resolvent of A can be expressed in terms of Laplace transforms of T ,
Z ∞
−1
e−λt T (t)xdt.
(6.3)
(λ − A) = λT̂ (λ) := λ
0
Actually formula (6.3) can be used to define the generator A in the
case of exponentially bounded integrated semigroups [1, Def.3.2.1].
A particularly interesting family of (once) integrated semigroups are
those that are locally Lipschitz continuous (l.L.c.) [1, Sec.3.5].
Theorem 6.3. The following statements (i), (ii), and (iii) are equivalent for a closed linear operator A in a Banach space X:
(i) A is the generator of a non-degenerate integrated semigroup T
that is l.L.c.: for any b > 0, there exists some Λ > 0 such that
kT (t) − T (r)k ≤ Λ|t − r|,
0 ≤ r, t ≤ b.
(ii) A is the generator of a non-degenerate integrated semigroup T
and there exist constants M ≥ 1, ω ∈ R such that
Z t
kT (t) − T (r)k ≤ M
eωs ds, 0 ≤ r ≤ t < ∞.
r
(iii) A is a Hille-Yosida operator, i.e., there exist M ≥ 1, ω ∈ R
such that (ω, ∞) is contained in the resolvent set of A and
k(λ − A)−n k ≤ M (λ − ω)−n ,
λ > ω, n = 1, 2, . . .
20
T. Lant, H.R. Thieme
The constants M, ω in (ii), (iii) can be chosen to be identical.
• Moreover, if one (and then all) of (i), (ii), (iii) holds, D(A) coincides with those x ∈ X for which T (t)x is continuously differentiable.
The derivatives S◦ (t) = T 0 (t)x, t ≥ 0, x ∈ D(A), provide bounded linear
operators S◦ (t) from X◦ = D(A) into itself forming a C0 -semigroup on
X◦ which is generated by the part of A in X◦ , A◦ . Finally T (t) maps
X into X◦ and
(2.10)
T 0 (r)T (t) = T (t + r) − T (r),
r, t ≥ 0.
Proof: The statements follow from combining the results in [1, Chap.3].
Remark 6.1. X◦ = D(A) can be characterized in various ways:
(6.4)
X◦ ={x ∈ X; kλ(λ − A)−1 x − xk → 0,
λ → ∞}
={x ∈ X; k(1/h)T (h)x − xk → 0, h & 0}.
We define the closed subspace X ¯ (pronounced “X sun”) of the dual
space of X, X ∗ ,
(6.5)
X ¯ = {x∗ ∈ X ∗ ; kλ(λ − A)−1∗ x∗ − x∗ k → 0, λ → ∞}.
If we want to emphasize the dependence of X ¯ on the generator A, we
write XA¯ . The resolvent identity implies that, for λ ∈ ρ(A), (λ − A)−1∗
maps X ∗ into X ¯ and actually
X ¯ = (λ − A)−1∗ X ∗ .
Notice that X ¯ separates points in X and norms X◦ :
kx◦ k ≤ M sup{|hx◦ , x¯ i|; x¯ ∈ X ¯ , kx¯ k ≤ 1}.
Vice versa, X◦ norms X ¯ . The restriction of (λ − A)−1∗ to X ¯ forms
a family of pseudoresolvents that is actually the resolvent of a closed
linear operator A¯ in X ¯ . It is easy to show that A¯ is densely defined
in X ¯ and, of course, a Hille-Yosida operator, and thus the infinitesimal generator of a C0 -semigroup S ¯ on X ¯ . We have the following
relations:
X ¯ ={x∗ ∈ X ∗ ; k(1/h)T ∗ (h)x∗ − x∗ k → 0, h & 0}
Z t
∗
¯
(6.6)
T (t)x =
S ¯ (r)x¯ dr, t ≥ 0, x¯ ∈ X ¯ ,
0
hS◦ (t)x◦ , x¯ i =hx◦ , S ¯ (t)x¯ i,
t ≥ 0, x◦ ∈ X◦ , x¯ ∈ X ¯ .
Proposition 6.4. Let X̃ be a total subspace of X ∗ , i.e. X̃ separates
points in X: if x ∈ X, x 6= 0, then there exists some x∗ ∈ X̃ such that
hx, x∗ i 6= 0. Assume that X̃ is invariant under T ∗ (·), or equivalently
under (λ − A)−1∗ .
Transition functions and semigroups
21
Then X ® := X̃ ∩ X ¯ is a total subspace of X ∗ . Further the C0 semigroup S ¯ leaves X ® invariant and its restrictions form a C0 semigroup S ® on X ® . The generator of S ® is the part of A¯ in X ® , i.e.
the restriction of A¯ to D(A® ) = {x¯ ∈ D(A¯ ) ∩ X ® ; A¯ x¯ ∈ X ® }.
A acts like the dual operator of A® : iff x, y ∈ X, then
(6.7) x ∈ D(A), Ax = y ⇐⇒ hx, A® x¯ i = hy, x¯ i ∀x¯ ∈ D(A® ).
Proof. Let x ∈ X and hx, x∗ i = 0 for all x∗ ∈ X ® . Let y ∗ ∈ X̃.
Since X̃ is invariant under (λ − A)−1∗ , (λ − A)−1∗ y ∗ ∈ X ® . So 0 =
hx, (λ − A)−1∗ y ∗ i = h(λ − A)−1 x, y ∗ i. Since this holds for all y ∈ X̃
and X̃ is a total subspace of X ∗ , (λ − A)−1 x = 0 and x = 0. Since
(λ−A¯ )−1 is the restriction of (λ−A)−1∗ to X ¯ , it leaves X ® invariant
and so does the C0 -semigroup S ¯ which is generated by A¯ . (6.7)‘⇒’
follows from the construction of A® . Let x, y satisfy the right hand
side of (6.7). Let y ¯ ∈ X ® and set
x¯ = (λ − A)−1∗ y ¯ = (λ − A® )−1 y ¯
for some sufficiently large λ > 0. Then x¯ ∈ D(A® ) and
hy, (λ − A)−1∗ y ¯ i = hy, x¯ i = hx, A¯ x¯ i
=hx, A¯ (λ − A¯ )−1 y ¯ i = hx, −y ¯ + λ(λ − A¯ )−1 y ¯ i.
By duality,
h(λ − A)−1 y, y ¯ i = h−x + λ(λ − A)−1 x, y ¯ i.
Since y ¯ ∈ X ® has been arbitrary and X ® is a total subspace of X ∗ ,
(λ − A)−1 y = −x + λ(λ − A)−1 x. Thus x ∈ D(A) and Ax = y.
¤
6.1. Integral semigroups. We start with the following observation
concerning two operator families on a Banach space X [29, 2.4].
Lemma 6.5. Let S(t), T (t), t ≥ 0, be two families of bounded linear
operators on X, T non-degenerate, T (0) = 0, such that
(6.8)
T (r)S(t) = T (r + t) − T (t),
t, r ≥ 0.
Then S is uniquely determined by T and is a semigroup satisfying
S(0)x = x for all x ∈ X. If T (t) is strongly continuous, then S is
non-degenerate.
Proof. S is uniquely determined by T because T is non-degenerate. By
(6.8),
T (r)S(t)S(u) = (T (r + t) − T (t))S(u)
=T (r + t + u) − T (u) − (T (t + u) − T (u)) = T (r + t + u) − T (t + u)
=T (r)S(t + u),
∀r, t, u ≥ 0.
22
T. Lant, H.R. Thieme
As T is non-degenerate, S(t)S(u) = S(t + u). Similarly we conclude
from T (r)S(0) = T (r) − T (0) = T (r) that S(0) = I. Now assume
that T (t), t ≥ 0, is strongly continuous, x ∈ X, and S(t)x = 0 for all
t > 0. By (6.8), T (r)x is constant on every interval [t, ∞), t > 0, and
hence constant on (0, ∞). By continuity and T (0) = 0 we have that
T (r)x = 0 for all r ≥ 0. Since T is non-degenerate, x = 0.
¤
Definition 6.6. Let S(t), t ≥ 0, be a family of bounded linear operators on X. S is called an integral semigroup if there exists a l.L.c.
integrated semigroup T (t), t ≥ 0, such that
T (r)S(t) = T (r + t) − T (t),
t, r ≥ 0.
S is called the integral semigroup associated with T . If A is the generator of T , S is called the integral semigroup generated by A.
A closed linear operator A is called the generator of an integral semigroup if A generates a l.L.c. integrated semigroup T and there exists
an integral semigroup S associated with T .
The integral semigroup S is uniquely determined by the generator
A because A uniquely determines T , and we will see later (Proposition
3.6) that S uniquely determines A and, equivalently, T . The definition
of an integral semigroup also makes sense if the associated integrated
semigroup is not l.L.c. The following theorem also holds in this more
general case.
Theorem 6.7. Let A be the generator of an integrated semigroup T .
Then the following statements are equivalent:
(i) A generates an integral semigroup.
(ii) T (t) maps X into D(A) for all t ≥ 0.
If one and then both statements hold we have the following relations
for the integral semigroup S generated by A:
S(t)x = x + AT (t)x,
x ∈ X, t ≥ 0.
Proof. (ii) ⇒ (i): Set S(t)x = x + AT (t)x. Since T (r) and A commute
[27, L.3.4], T (r)S(t)x = T (r)x + AT (r)T (t)x. By (6.1) and Theorem
6.1,
Z r
Z t
³Z t+r
´
T (r)S(t)x =T (r)x + A
T (u)xdu −
T (u)xdu −
T (u)xdu
0
0
0
= T (t + r) − T (t)x.
Hence (6.8) holds and S is the integral semigroup generated by A.
Transition functions and semigroups
23
(i) ⇒ (ii): We have to show that T (t) maps into D(A). We use the
definition in (6.2). By (6.8) and (6.1),
d
T (r)(S(t)x − x) = T (t + r)x − T (r)x − T (t)x = T (r)T (t)x − T (t)x.
dr
After integration, by (6.2), T (t)x ∈ D(A) and AT (t)x = S(t)x − x. ¤
We now restrict our discussion to the case that the integral semigroup
is associated with a l.L.c. integrated semigroup. We recall that X◦ =
D(A) coincides with the space of those x ∈ X such that T (t)x is
continuously differentiable in t ≥ 0 and that S◦ (t) = T 0 (t), t ≥ 0, form
a C0 -semigroup on X◦ which is generated by the part of A in X◦ , A◦ .
Lemma 6.8. Let x ∈ X. Then x ∈ X◦ if and only if S(t)x is continuous in t ≥ 0. Further S(t) extends S◦ (t) from X◦ to X and S◦ (t)T (r) =
T (r)S(t) for all r, t ≥ 0 and S◦ (t)(λ − A)−1 = (λ − A)−1 S(t).
Proof. Let x ∈ X and S(t)x be continuous in t ≥ 0. Then, by the
integrated semigroup property,
Z t
Z t
T (r)
S(u)xdu =
(T (r + u) − T (u))xdu = T (r)T (t)x.
0
0
Rt
As T is non-degenerate, 0 S(u)xdu = T (t)x. Hence T (t)x is continuously differentiable and dtd T (t)x = S(t)x. On the other hand, if T (t)x is
continuously differentiable, then by the integrated semigroup property,
d
T (r) T (t)x = T (r + t)x − T (t)x = T (r)S(t)x.
dt
As T is non-degenerate, dtd T (t)x = S(t)x and S(t)x is continuous in t.
The remaining statements follow from Theorem 6.3 and (6.3).
¤
Corollary 6.9. Let A be the generator of an integral semigroup S.
Then S ∗ (t) extends S ¯ (t) from X ¯ to X ∗ .
Proof. By Theorem 6.7, S is given by S(t)x = x + AT (t)x. Let x¯ ∈
D(A¯ ). Then
hx, S ∗ (t)x¯ i = hS(t)x, x¯ i = hx + AT (t)x, x¯ i
D Z t
E
¯
∗
¯ ¯
¯
=hx, x i + hx, T (t)A x i = hx, x i + x,
S ¯ (r)A¯ xdr, x¯
0
=hx, S ¯ (t)x¯ i.
¤
The example in [5, Sec.4] shows the following: If S ∗ (t)x∗ is continuous in t ≥ 0, then x∗ is not necessarily an element in X ¯ . However,
the following holds:
24
T. Lant, H.R. Thieme
Proposition 6.10. a) Let x∗ , y ∗ ∈ X ∗ . Then x∗ ∈ D(A¯ ) and A¯ x∗ =
y ∗ if and only if (1/h)(S ∗ (h)x∗ − x∗ ) → y ∗ as h → 0.
b) An integral semigroup uniquely determines its generator.
Proof. (a) ‘⇒’ is obvious since S ∗ extends S ¯ . Now let y ∗ = limh&0
(1/h) (S ∗ (h)x∗ − x∗ ). By Corollary 6.9,
(λ − A)−1∗ y = lim (1/h)(S ∗ (h) − I)(λ − A)−1∗ x∗
h&0
= lim (1/h)(S ¯ (h) − I)(λ − A)−1∗ x∗
h&0
=A¯ (λ − A)−1∗ x∗ = −x∗ + λ(λ − A)−1∗ x∗ .
This implies that x∗ ∈ X ¯ and y ∗ = limh&0 (1/h)(S ¯ (h)x∗ − x∗ ), hence
x∗ ∈ D(A¯ ), y ∗ = A¯ x∗ .
(b) By (a), A¯ is uniquely determined by S ∗ and thus by S. Since
A¯ uniquely determines A by (6.7), A is uniquely determined by S. ¤
Proposition 6.11. Let X̃ be a subspace of X ∗ which separates points in
X and Y a subset of X which separates points in X̃. Let {S(t); t ≥ 0}
and {T (t); t ≥ 0} be families of bounded linear operators such that S ∗ (t)
and T ∗ (t) map X̃ into itself for all t ≥ 0. Assume that T (t) is strongly
continuous in t ≥ 0 and hS(t)x, x∗ i is Borel measurable in t ≥ 0 for all
x ∈ X and x∗ ∈ X̃ and
Z t
∗
hS(r)x, x∗ idr
∀x ∈ X, x∗ ∈ X̃.
(6.9)
hT (t)x, x i =
0
Finally let A be a Hille-Yosida operator in X such that (λ−A)−1∗ maps
X̃ into itself for all λ > ω and S(t)x = x+AT (t)x for all t ≥ 0, x ∈ Y .
Then T is an integrated semigroup and A its generator and S the
associated integral semigroup, further
Z ∞
−1
∗
h(λ − A) x, x i =
e−λt hS(t)x, x∗ idt,
λ > ω, x ∈ X, x∗ ∈ X̃.
0
Proof. Recall Proposition 6.4. Let x¯ ∈ D(A® ). By (6.7),
hx, S ∗ (t)x¯ i = hx, x¯ i + hx, T ∗ (t)A® x¯ i
for all x ∈ Y . Since S ∗ (t)x¯ ∈ X ® and T ∗ (t)A® x¯ ∈ X ® and Y
separates point in X̃ ⊇ X ® ,
S ∗ (t)x¯ = x¯ + T ∗ (t)A® x¯ .
For all x ∈ X, by duality,
(6.10)
hS(t)x, x¯ i = hx, x¯ i + hT (t)x, A® x¯ i.
Transition functions and semigroups
25
By Proposition 6.4, T (t)x ∈ D(A) and
(6.11)
S(t)x = x + AT (t)x.
We integrate (6.10) in time and use (6.9),
DZ t
E
¯
¯
hT (t)x, x i = thx, x i +
T (r)xdr, A® x¯ .
0
Again by Proposition 6.4, (6.7),
Z
t
T (t)x = tx + A
T (r)xdr.
0
Since A is a Hille-Yosida operator, it generates an integrated semigroup T̃ (Theorem
R t 6.3) and u(t) = T̃ (t)x is the unique solution of
u(t) = tx + A 0 u(r)dr [27, Thm.6.1]. This implies T = T̃ and T is
an integrated semigroup and A its generator. By (6.11) and Theorem
6.7, S is the integral semigroup associated with T . The relation between the resolvent of A and the Laplace transform of S follows from
Theorem 6.2 and the assumed relation between T and S.
¤
7. Markov transition functions and integral semigroups
For a Markov transition kernel K we can define the operator families
Z t ³Z
´
(7.1) [T (t)µ](D) =
µ(dx)K(s, x, D) ds, µ ∈ M(Ω), D ∈ B.
0
Ω
Since K is jointly measurable, [S̃(t)f ](x) in (1.2) is jointly measurable in (t, x) and we can change the order of integration ad libitum.
This implies that T (·) is an integrated semigroup and S the associated
integral semigroup,
Z t
hf, S(r)µidr
∀f ∈ BM(Ω), µ ∈ M(Ω).
(7.2)
hf, T (t)µi =
0
Further the dual integrated semigroup T ∗ leaves BM(Ω) invariant and
Z t ³Z
Z t
´
∗
[T (t)f ](x) =
K(t, x, dy)f (y) ds =
[S̃(s)f ](x)ds,
(7.3)
0
Ω
0
t ≥ 0, x ∈ Ω, f ∈ BM(Ω).
See [22] for details. We mention that there are one-parameter semigroups S on M(Ω) (not induced by a Markov transition kernel) for
which (1.4) does not lead to an integrated semigroup T [5, Sec.1].
26
T. Lant, H.R. Thieme
7.1. Non-degeneracy. A transition function K is called non-degenerate if a measureR µ ∈ M(Ω) is necessarily the zero measure whenever,
for any D ∈ B, Ω K(t, x, D)µ(dx) = 0 for almost all t > 0.
Obviously a Markov transition kernel K is non-degenerate if and only
if the associated integrated semigroup is non-degenerate. We derive
topological conditions for K to be non-degenerate.
Theorem 7.1. Let Ω be a topological space and B the Baire σ-algebra.
Then a Markov transition function K is a non-degenerate Markov transition kernel if one of the following conditions hold:
(a) Ω is a normal space and K is stochastically continuous
(b) Ω is σ-compact and K is weakly stochastically continuous.
Proof. (a) The joint measurability has already been proved in Corollary
3.9. Let µ ∈ M(B) and T (t)µ = 0 for all t ≥ 0. Since hf, S(t)µi is
right-continuous in t ≥ 0, it follows from (7.2) that hf, µi = 0 for all
f ∈ Cb (Ω) and µ = 0 because Cb (Ω) is a total subspace of M(Ω) by
Corollary 3.9.
(b) The proof is similar to the one for (a) except that we use Proposition 3.4 and Proposition 3.5.
¤
7.2. Forward and backward equations. Let us now assume that
Ω is a measurable space with σ-algebra B and K is a non-degenerate
Markov transition kernel. Then the generator, A, of the associated
non-degenerate integrated semigroup, T , is defined and a Hille-Yosida
operator, and the resolvents of A and the Laplace transform of T are
related by Theorem 6.2. By (7.1), this yields the following relation.
Theorem 7.2. Let K be a non-degenerate Markov transition kernel.
Then there exists a Hille-Yosida operator A in M(Ω) such that
Z
Z ∞
−1
[(λ − A) µ](D) =
µ(dx)
dte−λt K(t, x, D)
Ω
0
for all λ > ω, D ∈ B, µ ∈ M(Ω). Through this formula, K and A
determine each other in a unique way.
Obviously A is uniquely determined by this equation. In turn, the integrated semigroup T is uniquely determined by A through its Laplace
transform, and T uniquely determines its associated integral semigroup
S (Lemma 6.5 and Definition 6.6) which uniquely determines K by
(1.1).
By Theorem 6.7, T (t) maps into the domain of A and AT (t)µ + µ =
S(t)µ. Choosing µ = δx for x ∈ Ω yields the forward equation (1.5).
Transition functions and semigroups
27
Theorem 7.3. Let K be a non-degenerate transition kernel and A the
Hille-Yosida operator associated with K in Theorem 7.2. Then
Z t
K(t, x, ·) = δx + A
dsK(s, x, ·)
0
for all t ≥ 0, x ∈ Ω. Further K is uniquely determined by this equation.
Even the following holds: if K is a transition kernel which satisfies
this equation and
sup
k(t, x, Ω) < ∞
for all σ > 0,
0≤t≤σ,x∈Ω
then
K is a Markov transition kernel, the operator family S(t)µ =
R
µ(dx)K(t,
x, ·) is the integral semigroup generated by A, and the
Ω
Laplace transform of K is related to the resolvent of A as in Theorem
7.2.
Proof. To see the last statement, we apply Proposition 6.11 with X =
M(Ω) and X̃R= BM(Ω), Y = {δx ; x ∈ Ω}, S(t) as just defined and
t
[T (t)µ](D) = 0 [S(r)µ](D)dr.
¤
X̃ =BM(Ω) can be identified with a closed subspace of X ∗ which
is invariant under T ∗ . By Proposition 6.4, X̃ ∩ X ¯ is a total closed
subspace of X ∗ . By (2.6) and (7.3), X̃ ∩ X ¯ = X̃◦ and the restriction of S ¯ to X̃ ∩ X ¯ coincides with S̃◦ , the restriction of S̃ to X̃◦ .
Let
R Ã◦ be the generator of S̃◦ . Then, for f ∈ D(Ã◦ ) and v(t)(x) =
K(t, x, dy)f (y), x ∈ Ω, v is the unique solution of the backward
Ω
equation (1.7), v 0 = Ã◦ v, v(0) = f . The Markov transition function K
is uniquely determined by the backward equation. Indeed,
let K̃ be anR
other Markov transition function such that ṽ(t)(x) = Ω K̃(t, x, dy)f (y)
also
solves the backward
equation for all f ∈ D(Ã◦ ). By uniqueness,
R
R
K(t, x, dy)f (y) = Ω K̃(t, x, dy)f (y) for all f ∈ D(Ã◦ ). Since D(Ã◦ )
Ω
is dense in X̃◦ , this holds for all f ∈ X̃◦ . Since X̃◦ = X̃ ∩ X ¯ is a total
subspace of M(Ω)∗ , K(t, x, ·) = K̃(t, x, ·).
Appendix A. A characterization of the Feller property
Proposition
A.1. Every f ∈ Cb (Ω) is the uniform limit of
Pnfunctions
Pn
g = k=1 γk χCk and the uniform limit of functions h = k=1 γk χUk
where 0 < γj < ∞ and
(i) all Ck are closed Gδ -sets,
(ii) all Uk are open Fσ -sets.
If f ∈ C0 (Ω), one can arrange that
(iii) all Ck are compact Gδ -sets,
28
T. Lant, H.R. Thieme
(iv) all Uk are open Kσ -sets with compact closure.
Proof. We can assume that f ∈ Cb (Ω) is non-negative. Then f can be
uniformly approximated by functions
n
X
g(x) =
αk χ{αk−1 ≤f <αk } (x) + αn χ{αn ≤f } (x)
k=1
where 0 < α0 < · · · < αn are appropriately chosen numbers. g can be
rewritten as
n
X
¡
¢
g=
αk χ{αk−1 ≤f } − χ{αk ≤f } + αn χ{αn ≤f }
k=1
=
n−1
X
αk+1 χ{αk ≤f } −
k=0
n−1
X
αk χ{αk ≤f }
k=1
n−1
X
(αk+1 − αk )χ{αk ≤f } .
=α1 χ{α0 ≤f } +
k=1
The sets Ck = {αk ≤ f } are closed and, if f ∈ CT
0 (Ω), compact because
1
αk > 0. Moreover Ck is a Gδ -set because Ck = ∞
m=1 {αk − m < f }.
Alternatively f is the uniform limit of functions h of the form
h(x) =
n
X
αk χ{αk−1 <f ≤αk } (x) + αn χ{αn <f } (x),
k=1
0 < α0 < αk < αk+1 for k ∈ N. Similarly as before, we see that
h is the linear combination (with positive scalars) of characteristic
functions χUk where Uk = {αk < f }. Uk is open and has compact
closure contained in the compact set {αk ≤ f } if f ∈ C0 (Ω) because S
αk > 0. It is also an Fσ -set (Kσ -set if f ∈ C0 (Ω)), because
1
Uk = ∞
¤
m=1 {αk + m ≤ f }.
Let Ω, Ω̃ be Hausdorff topological spaces and B the σ-algebra of Baire
sets in Ω.
Let K̃ : Ω̃ × B → R+ be a bounded measure kernel. We consider the
map S̃ :BM(Ω) → BM(Ω̃) defined by
Z
(A.1)
[S̃(f )](x) =
K̃(x, dy)f (y),
x ∈ Ω̃.
Ω
Proposition A.2. Let K̃ be a bounded measure kernel. S maps C0 (Ω)
into C0 (Ω̃) (in particular, K̃ has the Feller property [26, p.56]) if the
following three conditions hold:
Transition functions and semigroups
29
(i): For every compact Gδ Baire set C in Ω, K̃(·, C) is upper
semi-continuous on Ω̃.
(ii): For every open Kσ Baire set U in Ω with compact closure,
K̃(·, U ) is lower semi-continuous on Ω̃.
(iii): For every compact subset C of Ω and every ² > 0, there
exists a compact subset C̃ of Ω̃ such that K̃(x, C) < ² for all
x ∈ Ω̃ \ C̃.
Condition (iii) is also necessary. Condition (i) and (ii) are necessary
as well if Ω is locally compact.
Proof. We first notice that upper and lower semi-continuity are preserved under uniform limits of functions. By Proposition A.1 (iii), f
is the uniform limit of certain functions g which are positive linear
combinations of χCk with compact Gδ -sets Ck . By (i)
Z
n
X
γk K̃(x, Ck )
g(y)K̃(x, dy) =
Ω
k=1
is a linear combination (with positive scalars) of upper semi-continuous
functions and so upper semi-continuous
itself. Since f is the uniform
R
limit of functions g of this form, Ω f (y)K̃(x, dy) is the uniform limit of
upper semi-continuous functions andR so upper semi-continuous itself.
By (ii) and Proposition A.1 (iv), Ω h(y)K̃(x, dy) is the linear combination (with positive scalars) of lower semi-continuous functions and
so lower semi-continuous
itself. Since f is the uniform limit of funcR
tions of the form h, Ω f (y)K̃(·, dy) is the uniform limit of lower semicontinuous functions
and so lower semi-continuous itself. We have
R
shown that Ω f (y)K̃(·, dy) is both lower and upper semi-continuous.
Hence it is continuous.
Now let ² > 0. Then there exists some compact subset C of Ω such
that f (x) < 2² for all x ∈ Ω \ C. By (iii), there exists a compact subset
²
C̃ of Ω̃ such that K̃(x, C) < 2(1+kf
for all x ∈ Ω̃ \ C̃. So, for all
k)
x ∈ Ω̃ \ C̃,
Z
Z
Z
f (y)K̃(x, dy) = f (y)K̃(x, dy) +
Ω
C
≤kf kK̃(x, C) +
f (y)K̃(x, dy)
Ω\C
²
< ².
2
Now let f ∈ C0 (Ω) be arbitrary. Then f = f+ − f− with non-negative
f+ , f− ∈ C0 (Ω). Then S(f ) = S(f+ ) − S(f− ) ∈ C0 (Ω̃) by our previous
consideration.
30
T. Lant, H.R. Thieme
To see the necessity of (iii), let C be a compact subset of Ω. Then
there exists a continuous nonnegative function
f with compact support
R
in Ω such that f ≥ χC . By assumption Ω f (y)K̃(·, dy) ∈ C0 (Ω̃). Let
² > 0. Then there exists a compact subset C̃ of Ω̃ such that
Z
K̃(x, C̃) ≤
f (y)K̃(x, dy) < ²
∀x ∈ Ω̃ \ C̃.
Ω
We now assume that Ω is locally compact. To show the necessity of
(i), let C be a compact Gδ set in Ω. Then χC is the pointwise limit of
a monotone decreasing sequence of continuous function fn on Ω with
compact support. By the theorem of dominated convergence,
Z
K̃(x, C) = lim
fn (y)K̃(x, dy).
n→∞
Ω
By assumption, K̃(C, ·) is the pointwise limit of a monotone decreasing sequence of continuous functions. Hence K̃(C, ·) is upper semicontinuous [12, III.10.4].
To show the necessity of (ii), let U be an open Kσ -set. Then χU
is the pointwise limit of a monotone increasing sequence of continuous
functions fn on Ω with compact support. A similar argument as before
shows that K̃(·, U ) is lower semi-continuous.
¤
A similar proof provides the following result.
Proposition A.3. Let Ω be a Hausdorff topological space. Then S
maps Cb (Ω) into Cb (Ω̃) if the following three conditions hold:
(i): For every closed Gδ Baire set C in Ω, K̃(·, C) is upper semicontinuous on Ω̃.
(ii): For every open Fσ Baire set U in Ω, K̃(·, U ) is lower semicontinuous on Ω̃.
The conditions (i) and (ii) are also necessary if Ω is normal.
The necessity proof is based on Urysohn’s characterization of normality [12, Sec. VII.4].
Appendix B. Joint measurability
Let (Θ, Σ) be a measurable space and Ω be a normal topological
space with the σ-algebra B of Baire sets. Let [0, τ ) be equipped with
the standard topology and the associated σ-algebra Bτ of Borel sets.
Proposition B.1. Let Ω be normal and f : [0, τ ) × Θ → Ω such
that f (t, θ) is a measurable function of x ∈ Θ for every t ∈ (0, τ ) and
f (t, θ) : [0, τ ) → Ω is a right continuous function of t ≥ 0 for all θ ∈ Θ.
Transition functions and semigroups
31
Then f : ([0, τ ) × Θ, Bτ × Σ) → (Ω, B) is measurable.
Proof. Since the Baire-σ-algebra on the normal space Ω is generated
by open Fσ sets [3, 7.2.3], we can assume that D is S
open and the union
of an increasing sequence of closed set C̃n , D = ∞
n=1 C̃n , and show
that f −1 (D) ⊆ Bτ × Σ. By Urysohn’s lemma, there exist continuous
functions gn : Ω → [0, 1] such that gn (x) = 0 for x ∈ Ω \ D and
gn (x) = 1 for x ∈ C̃n . Define
g(x) =
∞
X
2−n gn (x),
x ∈ Ω.
n=1
Then g is continuous and D = {x ∈ Ω; g(x) > 0}. Set Un = {x ∈
1
Ω; g(x) > n+1
} and Cn = {x ∈ Ω; g(x) ≥ n1 }. Then
D=
[
n∈N
Cn =
[
Un ,
Cn ⊆ Un ⊆ Cn+1 .
n∈N
Claim 1:
The following two statements are equivalent for t ∈ [0, τ ), ω̃ ∈ Θ:
(a) f (t, ω̃) ∈ D.
(b) There exists some n ∈ N such that for all k ∈ N there exists
some q ∈ (0, τ ) ∩ Q ∩ [t, t + 1/k) such that f (q, ω̃) ∈ Cn .
‘(a)⇒(b)’: Choose n ∈ N such that f (t, ω̃) ∈ Cn . Since Cn ⊆ Un and
Un is open and f (·, ω̃) is right continuous, for all k ∈ N, there exists
some q ∈ (0, τ ) ∩ Q ∩ [t, t + 1/k), such that
f (q, ω̃) ∈ Un ⊆ Cn+1 .
This implies (b).
‘(b)⇒(a)’: Choose n ∈ N such that for all k ∈ N there exists q ∈
(0, τ ) ∩ Q ∩ [t, t + 1/k) with f (q, ω̃) ∈ Cn . We can choose a sequence
qk & t in (0, τ ) ∩ Q such that f (qk , ω̃) ∈ Cn . Since f (·, ω̃) is rightcontinuous and Cn is closed, f (t, ω̃) ∈ Cn ⊆ D.
Claim 1 can easily be rewritten in the following way.
Claim 2: The following two statements are equivalent for ω̃ ∈ Θ:
(a) t ∈ [0, τ ), f (t, ω̃) ∈ D.
(b) (∃n ∈ N)(∀k ∈ N)(∃q ∈ (0, τ ) ∩ Q): t ∈ (q − 1/k, q] ∩ [0, τ ) and
f (q, ω̃) ∈ Cn .
Claim 2 can be rewritten set-theoretically as
32
T. Lant, H.R. Thieme
f
−1
¶
1 i
(D) =
q − , q ∩ [0, τ )
k
n∈N k∈N q∈Q∪(0,τ )
n
o
× ω̃ ∈ Ω̃; f (q, ω̃) ∈ Cn .
[ \
[
µ³
Observe that the sets on the left of × are in Bτ . Since f (q, ·) is measurable for fixed q, and Cn ∈ B, the sets on the right of × are elements
of Σ. Hence f −1 (D) ∈ Bτ × Σ, because countable intersections and
unions of measurable sets are again measurable.
¤
Acknowledgement. The authors thank Doug Blount and Mats Gyllenberg for useful comments.
References
[1] W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-valued
Laplace Transforms and Cauchy Problems, Birkhäuser, Basel 2001
[2] L. Arnold, Stochastic Differential Equations: Theory and Applications, John
Wiley, New York 1974
[3] H. Bauer, Probability Theory and Elements of Measure Theory, Academic
Press, London 1981
[4] C.D. Aliprantis, Owen Burkinshaw, Principles of Real Analysis, third
edition, Academic Press, San Diego 1998
[5] Ph. Clément, O. Diekmann, M. Gyllenberg, H.J.A.M. Heijmans,
H.R. Thieme, A Hille-Yosida theorem for a class of weakly continuous semigroups, Semigroup Forum 38 (1989), 157-178
[6] Ph. Clément, H.J.A.M. Heijmans, S. Angenent, C.J. van Duijn, B. de
Pagter, One-Parameter Semigroups, North-Holland, Amsterdam 1987
[7] E.B. Davies, One-Parameter Semigroups, Academic Press, London 1980
[8] M. Demuth, J.A. van Casteren, Stochastic Spectral Theory for Selfadjoint
Feller Operators, Birkhäuser, Basel 2000
[9] O. Diekmann, M. Gyllenberg, J.A.J. Metz, H.R. Thieme, On the formulation and analysis of deterministic structured population models. I. Linear
theory, J. Math. Biol. 43 (1998), 349-388
[10] O. Diekmann, M. Gyllenberg, H.R. Thieme (1993), Perturbing semigroups by solving Stieltjes renewal equations, Diff. Integ. Eqn. 6 (1993), 155181
[11] O. Diekmann, M. Gyllenberg, H.R. Thieme (1995), Perturbing evolutionary systems by step responses and cumulative outputs. Diff. Integ. Eqn. 8,
1205-1244
[12] J. Dugundji, Topology, Allyn and Bacon, Boston 1966, Prentice-Hall of India,
New Delhi 1975
[13] N. Dunford, J.T. Schwartz, Linear Operators Part I: General Theory,
John Wiley, New York 1988
Transition functions and semigroups
33
[14] K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution
Equations, Springer, New York 2000
[15] S.N. Ethier, T.G. Kurtz, Markov Processes. Characterization and Convergence, Wiley, New York 1986
[16] W. Feller, An Introduction to Probability Theory and its Applications, Vol
II, Wiley 1965
[17] W. Feller, Semi-groups of transformations in general weak topologies, Annals Math. 57 (1953), 287-308
[18] A. Friedman, Stochastic Differential Equations and Applications, Vol. 1, Academic Press, New York 1975
[19] J.A. Goldstein, Semigroups of Linear Operators and Application, Oxford
University Press, New York 1985
[20] M. Gyllenberg, T. Lant, H.R. Thieme, Perturbing dual evolutionary
systems by cumulative outputs, Differential Integral Equations 19 (2006), 401436
[21] E. Hille, R.S. Phillips, Functional Analysis and Semi-Groups, AMS, Providence 1957
[22] T. Lant, Transition Kernels, Integral Semigroups on Spaces of Measures, and
Perturbation by Cumulative Outputs, Dissertation, Arizona State University,
Tempe, December 2004
[23] T. Lant, H.R. Thieme, Perturbation of transition functions and a FeynmanKac formula for the incorporation of mortality, Positivity, to appear
[24] B. Øksendal, Stochastic Differential Equations, Springer, Berlin Heidelberg
1985
[25] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York Berlin Heidelberg 1982
[26] K. Taira, Semigroups, Boundary Value Problems, and Markov Processes,
Springer, Berlin Heidelberg 2004
[27] H.R. Thieme, Integrated semigroups and integral solutions to abstract
Cauchy problems, J. Math. Anal. Appl. 152 (1990), 416-447.
[28] H.R. Thieme, Positive perturbations of dual and integrated semigroups, Adv.
Math. Sci. Appl. 6, 445-507, 1996
[29] H.R. Thieme, Balanced exponential growth for perturbed operator semigroups, Adv. Math. Sci. Appl. 10 (2000), 775-819
[30] H.R. Thieme, J. Voigt, Stochastic semigroups: their construction by perturbation and approximation, Proc. Positivity IV - Theory and Applications,
(M.R. Weber, J. Voigt, eds.), 135-146, Technical University of Dresden, Dresden 2006
[31] J.A. van Casteren, Generators of strongly continuous semigroups, Pitman,
Boston 1985
[32] J. van Neerven, The Adjoint of a Semigroup of Linear Operators, Lecture
Notes in Mathematics 1529, Springer, Berlin Heidelberg 1992
34
T. Lant, H.R. Thieme
†
present address: Decision Center for a Desert City, Arizona
State University, PO Box 878209, Tempe, AZ 85287-8209, U.S.A.
E-mail address: [email protected]
†¦
Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804, U.S.A.
E-mail address: [email protected]