COUPLING PROPERTY AND GRADIENT ESTIMATES OF LÉVY
PROCESSES VIA THE SYMBOL
RENÉ L. SCHILLING
PAWEŠ SZTONYK
JIAN WANG
We derive explicitly the coupling property for the transition semigroup of a Lévy process and gradient estimates for the associated semigroup of
transition operators. This is based on the asymptotic behaviour of the symbol or
the characteristic exponent near zero and innity, respectively. Our results can be
applied to a large class of Lévy processes, including stable Lévy processes, layered
stable processes, tempered stable processes and relativistic stable processes.
Abstract.
Keywords: Coupling; gradient estimates; Lévy process; symbol.
MSC 2010: 60G51; 60G52; 60J25; 60J75.
1. Introduction and Main Results
Let X be a pure jump Lévy process on R with the symbol (or characteristic
exponent)
Z
d
t
1 − eiξ·z + iξ · z1B(0,1) (z) ν(dz),
Φ(ξ) =
where ν is the Lévy measure, i.e. a σ-nite measure on R \ {0} such that R (1 ∧
|z| )ν(dz) < ∞. There are many papers studying regularity properties of Lévy
processes in terms of the symbol Φ. For example, recently [14, Theorem 1] points
out the relations between the classic Hartman-Wintner condition (see [9] or (1.1)
below) and some smoothness properties of the transition density for Lévy processes.
In particular, the condition that the symbol Φ(ξ) of the Lévy process X satises
Re Φ(ξ)
=∞
(1.1)
lim inf
log(1 + |ξ|)
is equivalent to the statement that for all t > 0 the random variables X have a
transition density p (y) such that ∇p ∈ L (R ) ∩ C (R ), where C (R ) denotes
the set of all continuous functions which vanish at innity. The main purpose of
this paper is to derive an explicit coupling property and gradient estimates of Lévy
processes directly from the corresponding symbol Φ.
Let (X ) be a Markov process on R with transition probability function
{P (x, ·)}
. An R -valued process (X , X ) is called a coupling of the
Markov process (X ) , if both (X ) and (X ) are Markov processes which
z6=0
d
z6=0
2
t
|ξ|→∞
t
t
t
1
0
t
2d
t>0,x∈Rd
t t>0
R.L. Schilling:
P. Sztonyk:
J. Wang:
∞
d
∞
d
d
t t>0
t
d
0
t t>0
00
t t>0
00
t t>0
TU Dresden, Institut für Mathematische Stochastik, 01062 Dresden, Germany.
[email protected] .
Institute of Mathematics and Computer Science, Wrocªaw University of Technology,
Wybrze»e Wyspia«skiego 27, 50-370 Wrocªaw, Poland. [email protected] .
School of Mathematics and Computer Science, Fujian Normal University, 350007
Fuzhou, P.R. China
TU Dresden, Institut für Mathematische Stochastik, 01062 Dresden,
Germany. [email protected] .
and
1
2
RENÉ L. SCHILLING
PAWEŠ SZTONYK
JIAN WANG
have the same transition functions P (x, ·) but possibly dierent initial distributions. In this case, (X ) and (X ) are called the marginal processes of the
coupling process; the coupling time is dened by T := inf{t > 0 : X = X }. The
coupling (X , X ) is said to be successful if T is a.s. nite. If for any two initial
distributions µ and µ , there exists a successful coupling with marginal processes
starting from µ and µ , respectively, we say that X has the coupling property (or
admits successful couplings ). According to [15] and the proof of [23, Theorem 4.1],
the coupling property is equivalent to the statement that
lim kP (x, ·) − P (y, ·)k = 0
for any x, y ∈ R ,
where P (x, ·) is the transition function of the Markov process (X ) . By kµk
we denote the total variation norm of the signed measure µ. We know from [23,
Theorem 4.1] that every Lévy process whose transition functions have densities for
large enough t > 0 has the coupling property. In this case, the transition probability
function satises
(1.2) kP (x, ·) − P (y, ·)k 6 C(1 +√|xt − y|) ∧ 2 for t > 0 and x, y ∈ R .
It is clear that for any x, y ∈ R and t > 0, kP (x, ·) − P (y, ·)k 6 2, and
kP (x, ·) − P (y, ·)k is decreasing with respect to t. This shows that it is enough
to estimate kP (x, ·) − P (y, ·)k for large values of t. We will call√any estimate for
kP (x, ·) − P (y, ·)k an estimate of the coupling time. The rate 1/ t in (1.2) is not
optimal for general Lévy processes which admit successful couplings. For example,
for rotationally invariant α-stable Lévy processes we can prove, see [1, Example 2.3],
that
1
kP (x, ·) − P (y, ·)k as t → ∞,
t
where for any two non-negative functions g and h, the notation g h means that
there are two positive constants c and c such that c g 6 h 6 c g.
Let P (x, ·) and P be the transition function and the semigroup of the Lévy
process X , respectively. We begin with coupling time estimates of Lévy processes
which satisfy the following Hartman-Wintner condition
Re Φ(ξ)
(1.3)
lim inf
> 0;
log(1 + |ξ|)
this condition actually ensures that the transition function of the Lévy process X
is, for suciently large t > 0, absolutely continuous, see e.g. [9] or [14].
Suppose that (1.3) holds and
t
00
t t>0
0
t t>0
0
t
0
t
00
t t>0
1
00
t
2
1
2
t
t
t→∞
d
Var
t
t
Var
t t>0
t
d
Var
t
d
t
Var
t
Var
t
t
t
t
t
t
Var
Var
t
Var
t
1
t
1/α
2
1
2
t
t
|ξ|→∞
t
Theorem 1.1.
as |ξ| → 0,
where f : [0, ∞) → R is a strictly increasing function which is dierentiable near
zero and which satises
Re Φ(ξ) f (|ξ|)
lim inf f (r)| log r| < ∞
and
r→0
lim sup f −1 (2s)/f −1 (s) < ∞.
s→0
Then the corresponding Lévy process Xt has the coupling property, and for any x, y ∈
Rd ,
kPt (x, ·) − Pt (y, ·)kVar = O f −1 (1/t)
as t → ∞.
COUPLING PROPERTY AND GRADIENT ESTIMATES OF LÉVY PROCESSES
3
It can be seen from the above remark on rotationally invariant α-stable Lévy
processes that the estimate in Theorem 1.1 is sharp.
Now we turn to explicit gradient estimates for the semigroup of a Lévy process.
For a function u ∈ B (R ) we dene
b
d
∇u(x) := lim sup |u(y) − u(x)| ,
x ∈ Rd .
|y − x|
y→x
∇u(x)
x
Pt
Bb (Rd )
φ
(0, ∞)
If u is dierentiable at , then
is just the norm of the gradient of u at x. We
are interested in sub-Markov semigroups on
which satisfy that for some
positive function on
k∇Pt uk∞ 6 kuk∞ φ(t),
t > 0, u ∈ Bb (Rd ).
Similar uniform gradient estimates for Markov semigroups have attracted a lot of
attention in analysis and probability, e.g. see [17] and references therein. Because of
the Markov property of the semigroup P , φ(t) is deceasing with respect to t. Thus,
it is enough to obtain sharp estimates for φ(t) both as t → 0 and t → ∞. For Lévy
processes, we have
Assume that (1.1) holds. If there is a strictly increasing function f
t
Theorem 1.2.
which is dierentiable near innity and which satises
lim sup f −1 (2s)/f −1 (s) < ∞,
s→∞
and
as |ξ| → ∞,
then there exists a constant c > 0 such that for t > 0 small enough,
Re Φ(ξ) f (|ξ|)
(1.4)
k∇Pt uk∞ 6 ckuk∞ f −1 (1/t) ,
u ∈ Bb (Rd ).
Similarly, let f be a strictly increasing function which is dierentiable near zero
and which satises
lim inf f (r)| log r| < ∞,
r→0
lim sup f −1 (2s)/f −1 (s) < ∞
s→0
and
as |ξ| → 0.
Then there exists a constant c > 0 such that
holds for t > 0 large enough.
Re Φ(ξ) f (|ξ|)
(1.4)
We will see in Remark 3.3 below that Theorem 1.2 is also sharp for rotationally
invariant α-stable Lévy processes. Roughly speaking, Theorems 1.1 and 1.2 show
that the gradient estimate (1.4) for a Lévy process for small t 1 depends on the
asymptotic behaviour of the symbol Φ near innity, while (1.4) for large t 1 relies
on the asymptotic behaviour of the symbol Φ near zero. This situation is familiar
from estimates of the coupling time of Lévy processes. More details can be found
in the examples given below.
In order to illustrate the power of Theorems 1.1 and 1.2 we present two examples.
Let X be a subordinate Brownian motion with symbol f (|ξ| ), where
f (λ) = λ (log(1 + λ)) , α ∈ (0, 2) and β ∈ (−α, 2 − α). To see that f is indeed
Example 1.3.
α/2
2
t
β/2
4
RENÉ L. SCHILLING
PAWEŠ SZTONYK
JIAN WANG
a Bernstein function we observe that λ, log(1 + λ) and λ/ log(1 + λ) are complete
Bernstein functions and that for α, β > 0
λ · log(1 + λ)
is a complete Bernstein function if α2 + β2 6 1,
while for −α 6 β 6 0 6 α
α β
λ
is
a
complete
Bernstein
function
if
+ 6 1.
λ
·
log(1 + λ)
2
2
This follows easily from [22, (Proof of) Proposition 7.10], see also [24, Examples
5.15, 5.16].
For all x, y ∈ R ,
as t → ∞,
kP (x, ·) − P (y, ·)k = O t
and there exists c > 0 such that for all u ∈ B (R ),
 h
i
 c t
log(1 + t )
kuk
for small t 1;
k∇P uk 6

ct
kuk
for large t 1.
Let µ be a nite nonnegative measure on the unit sphere S and
assume that µ is nondegenerate in the sense that its support is not contained in any
proper linear subspace of R . Let α ∈ (0, 2), β ∈ (0, ∞] and assume that the Lévy
measure ν satises that for some constant r > 0 and any A ∈ B(R ),
β/2
α/2
β/2
(α−β)/2
d
t
t
−1/(α+β)
Var
d
b
t
−β/2 1/α
−1
−1
∞
∞
−1/(α+β)
∞
Example 1.4.
d
d
0
Z
r0
Z
1A (sθ)s
ν(A) >
0
−1−α
Z
∞
Z
ds µ(dθ) +
r0
S
Then, for all x, y ∈ R ,
1A (sθ)s−1−β ds µ(dθ).
S
d
kPt (x, ·) − Pt (y, ·)kVar = O t−1/(β∧2)
as
t → ∞,
and there exists a constant c > 0 such that for all u ∈ B (R ),

c kuk t
for small t 1;
k∇P uk 6
ckuk t
for large t 1.
b
∞
t
∞
∞
d
−1/α
−1/(β∧2)
The remaining part of this paper is organized as follows. In Section 2 we rst
present estimates for the derivatives of the density for innitely divisible distributions in terms of the corresponding Lévy measure; this part is of some interest on its
own. Then we use these estimates to investigate derivatives of the density for Lévy
processes, whose Lévy measures have (modied) bounded support. In Section 3 we
give the proofs of all the theorems and examples stated in Section 1, by using the
results of Section 2. Some remarks and examples are also included here to illustrate
the optimality and the eciency of Theorems 1.1 and 1.2.
COUPLING PROPERTY AND GRADIENT ESTIMATES OF LÉVY PROCESSES
5
2. Derivatives of Densities for Infinitely Divisible Distributions
Let π be an innitely
divisible distribution. It is well known that its characteristic
R
function πb(ξ) := e π(dy)
is of the form exp(−Φ(ξ)), where
Z
iξ·y
Rd
Φ(ξ) =
iξ·y
1−e
+ iξ · y1B(0,1) (y) ν(dy),
y6=0
and ν is a Lévy measure on R \ {0} such that R (1 ∧ |y| )ν(dy) < ∞. In this
section we rst aim to study estimates for derivatives of the density of π. As usual, we
denote for every n ∈ N by C (R ) the set of all n-times continuously dierentiable
functions on R which are, together with all their derivatives, bounded; for n = 0
we use the convention that C (R ) = C (R ) denotes the set of continuous and
bounded functions on R .
d
2
y6=0
d
0
b
d
Proposition 2.1.
(2.5)
d
n
b
0
d
b
d
If for some n, m ∈ N0 ,
Z
e− Re Φ(ξ) (1 + |ξ|)n+m dξ < ∞,
and
(2.6)
Z
|y|2∨n ν(dy) < ∞,
|y|>1
then π has a density p ∈ Cbm+n (Rd ) such that for every β ∈ Nd0 with |β| 6 m,
|∂ β p(y)| 6 ψ(n, m, ν)(1 + |y|)−n ,
y ∈ Rd ,
where for n > 0
Z
2
2∨n
|y| + |y|
ψ(n, m, ν) = C(n, d) 1 +
n Z
ν(dy)
e− Re Φ(ξ) (1 + |ξ|)n+m dξ.
The existence of the density p ∈ C (R ) is a consequence of (2.5) and [19, Proposition 28.1] or [16, Proposition 0.2].
To prove the second assertion, we recall some necessary facts and notations. Given
a function f ∈ L (R ), its Fourier transform
is given by
Z
Proof of Proposition 2.1.
1
m+n
b
d
d
f (y)eiξ·y dy.
fb(ξ) =
Rd
For ξ ∈ R and a multiindex β = (β , β , · · · , β ) ∈ N , we set M (ξ) := ξ =
ξ ξ . . . ξ . If fb ∈ C (R ) and ∂ (M fb) ∈ L (R ) for N ∈ N and every γ ∈ N
such that |γ| 6 N , then, using the inverse Fourier transform and the integration by
parts formula, we obtain that for every δ ∈ N Z with |δ| 6 N
d
β1 β2
1 2
1
βd
d
N
d
γ
2
d
0
d
1
β
d
β
0
d
0
h
i
∂ δ Mβ fb (ξ) e−iy·ξ dξ.
y δ ∂ β f (y) = (2π)−d (−1)|β| (i)|β|−|δ|
This yields
(2.7)
|y ∂ f (y)| 6 (2π)
In particular, for every n ∈ N ,
(2.8)
|y | |∂ f (y)| 6 (2π)
δ
−d
β
Z h
i δ
b
∂ Mβ f (ξ) dξ.
0
k
n
β
Z n h
i ∂
−d
b
∂ξ n Mβ f (ξ) dξ.
k
β
d
0
6
RENÉ L. SCHILLING
PAWEŠ SZTONYK
JIAN WANG
For n = 0, the required assertion immediately follows from (2.7) if we use f = p
and δ = 0. If n > 0, then for every β ∈ N such that |β| = 1 we have
d
0
Z
β
∂ Φ(ξ) = −i
y β eiξ·y − 1B(0,1) (y) ν(dy).
By the Hölder inequality,
|∂ β Φ(ξ)|
Z
1/2 Z
2
6
|y| ν(dy)
2
1/2
(1 − cos(ξ · y)) ν(dy) + ν(B(0, 1) )
c
B(0,1)
(2.9)
Z
6
1/2 Z
2
2
|y| ν(dy)
|ξ|
1/2
|y| ν(dy) + ν(B(0, 1) )
2
c
B(0,1)
Z
|y|2 ν(dy) · |ξ|2 + 1
Z
6 (1 + |ξ|) |y|2 ν(dy).
6
1/2
On the other hand, for 1 < |β| 6 n, we have
|β|
β
Z
∂ Φ(ξ) = −(i)
y β eiξ·y ν(dy),
and so
(2.10)
β
|∂ Φ(ξ)| 6
Z
|y||β| ν(dy).
For symmetric Lévy measures ν similar estimates are due to Hoh [10], see also [12,
Theorem 3.7.13].
Let k ∈ {1, . . . , d} and M ∈ N with M 6 n. We use Faa di Bruno's formula, see
[6], to obtain
M X Y
M
X
M
∂
pb(ξ) = M ! exp(−Φ(ξ))
∂ξkM
j=1
u(M,j) l=1
∂ l (−Φ)
(ξ)
∂ξkl
λl
(λl !)(l!)λl
,
where
(
u(M, j) =
(λ1 , . . . , λM ) : λl ∈ N0 ,
M
X
l=1
λl = j,
M
X
l=1
)
lλl = M
.
COUPLING PROPERTY AND GRADIENT ESTIMATES OF LÉVY PROCESSES
This, (2.9) and (2.10) yield
7
M
λl
Z
M X Y
M
X
∂
M!
l∨2
(1 + |ξ|)
|y| ν(dy)
∂ξ M pb(ξ) 6 | exp(−Φ(ξ))|
(λl !)(l!)λl
Rd
k
j=1
l=1
u(M,j)
− Re Φ(ξ)
6e
M X
Z
6e
2∨n
|y| + |y|
(1 + |ξ|)
j X Y
M
ν(dy)
Rd
j=1
− Re Φ(ξ)
2
M
(1 + |ξ|)
u(M,j) l=1
M Z
X
2
2∨n
|y| + |y|
j X Y
M
ν(dy)
Rd
j=1
6 c1 (n) e− Re Φ(ξ) (1 + |ξ|)n 1 +
M!
(λl !)(l!)λl
u(M,j) l=1
n
Z
|y|2 + |y|2∨n ν(dy)
M!
(λl !)(l!)λl
.
We note that this inequality remains valid for M = 0.
For β ∈ N with |β| 6 m we can use the Leibnizrule to get
Rd
d
0
X
n
n j
∂
n ∂
∂ n−j
(M
p
b
)(ξ)
6
M
(ξ)
p
b
(ξ)
β
j
n−j
∂ξ n β
j
∂ξ
∂ξ
k
k
k
j=0
n n−j
X
n ∂
|β|
6 (1 + |ξ|)
n−j pb(ξ)
j ∂ξk
j=0
Z
− Re Φ(ξ)
n+m
6 c2 (n) e
(1 + |ξ|)
1+
By (2.8), we see
n
2∨n
|y| + |y|
n
ν(dy) .
Rd
β
2
Z
|yk | |∂ p(y)| 6 c2 (n) 1 +
Finally,
2
2∨n
|y| + |y|
n Z
ν(dy)
e− Re Φ(ξ) (1 + |ξ|)n+m dξ.
Rd
(1 + |y|)n 6 2n−1 (1 + |y|n ) 6 2n−1 dn/2 1 +
d
X
!
|yk |n ,
and the required assertion follows with C(n, d) = 2 d (d + 1)c (n).
We will now study the derivatives of transition densities for Lévy processes with
(modied) bounded support. For this we need Proposition 2.1. Let Φ be the symbol
(i.e. the characteristic exponent) of a Lévy process and consider for every r > 0 the
semigroup of innitely divisible measures {P , t > 0} whose Fourier transform is of
the form Pb (ξ) = exp(−tΦ (ξ))Z, where
k=1
n−1 n/2
2
r
t
r
t
r
1 − eiξ·y + iξ · y ν(dy),
Φr (ξ) =
|y|6r
(ν is the Lévy measure of the symbol Φ). For ρ > 0 and t > 0, we dene
1
.
(2.11)
ϕ(ρ) = sup Re Φ(η)
and h(t) := ϕ (1/t)
Assume that (1.3) holds, and for some m ∈ N ,
Z
1
(2.12)
exp − t Re Φ(ξ) |ξ| dξ = O ϕ
as t → ∞.
t
−1
|η|6ρ
Proposition 2.2.
0
m
−1
m+d
8
RENÉ L. SCHILLING
PAWEŠ SZTONYK
JIAN WANG
Then there is a t0 = t0 (m, d) > 0 such that for any t > t0 , there exists a density
h(t)
h(t)
pt ∈ Cbm (Rd ) of Pt , and for every n ∈ N0 and β ∈ Nd0 with |β| 6 m − n,
h(t)
|∂yβ pt
d+|β| −n
(y)| 6 c(m, n, |β|, Φ) ϕ−1 (t−1 )
1 + ϕ−1 (t−1 )|y|
,
Proof. Step 1.
y ∈ Rd .
For ξ ∈ R ,
d
−t
(1 − cos(ξ · y)) ν(dy)
|y|<r
Z
= exp − t Re Φ(ξ) −
(1 − cos(ξ · y)) ν(dy)
|y|>r
6 exp − t Re Φ(ξ)
exp 2tν(B(0, r)c ) .
r Pbt (ξ) = exp
(2.13)
Z
By (1.3) and [19, Proposition 28.1], it follows that there exists t := t (d) > 0 such
that for all r > 0 and for any t > t , the measure P has a density p ∈ C (R ).
Step 2. For t > t , we dene g (y) = h(t) p (h(t)y). We consider the innitely
divisible distribution π (dy) = g (y)dy. Its Fourier transform is given by
1
1
1
d
t
t
h(t)
t
r
t
1
r
t
b
t
d
Z
h(t)
π
bt (ξ) = (h(t))
eiξ·y pt (h(t)y) dy
Z
h(t)
= eiξ·y/h(t) pt (y) dy
Z
iξ · y
iξ·y/h(t)
= exp −t
1−e
+
ν(dy)
h(t)
|y|6h(t)
Z
iξ·y
= exp −
1 − e + iξ · y λt (dy)
|y|61
= exp − Gt (ξ) ,
(2.14)
where λ is the Lévy measure of π , i.e. for any Borel set B ⊂ R
t
d
t
Z
λt (B) = t
|y|6h(t)
1B y/h(t) ν(dy).
\ {0}
,
d
COUPLING PROPERTY AND GRADIENT ESTIMATES OF LÉVY PROCESSES
9
For n > 2, we have
Z
Z
n
|y| λt (dy) = t
|y|6h(t)
Z
6t
|y|6h(t)
|y|
h(t)
n
|y|
h(t)
2
ν(dy)
ν(dy)
|y|/h(t)
Z
6 2t
2
2 ν(dy)
1 + |y|/h(t)
2
Z
|y|/h(t)
6 2t
2 ν(dy)
1 + |y|/h(t)
ZZ 1 − cos y/h(t) · ξ fd (ξ) dξ ν(dy)
= 2t
ZZ 1 − cos y · ξ/h(t) ν(dy) fd (ξ) dξ
= 2t
Z
ξ
= 2t Re Φ
fd (ξ) dξ,
h(t)
|y|6h(t)
where
1
fd (ξ) =
2
Z
∞
(2πρ)−d/2 e−|ξ|
2 /(2ρ)
e−ρ/2 dρ.
0
Obviously, f (ξ) possesses all moments,
see e.g. [20, (2.5) and (2.6)]. By using several
p
times the subadditivity of η 7→ Re Φ (η), we can easily nd, see e.g. the proof of
[21, Lemma 2.3],
d
Re Φ
So,
Z
Re Φ
ξ
h(t)
ξ
h(t)
6 2 1 + |ξ|2
sup
Re Φ(η) = 2 1 + |ξ|2
|η|61/h(t)
Z
fd (ξ) dξ 6 2
sup
Re Φ(η)
|η|61/h(t)
1
.
t
c0
1 + |ξ|2 fd (ξ) dξ =: .
t
According to the denition of h(t), we get that for any t > 0,
Z
(2.15)
|y| λ (dy) 6 2c .
n
Step 3.
and
t
0
It is easily seen from (2.14) that the characteristic exponent of π is G (ξ),
t
Re Gt (ξ) = t Re Φh(t) (h(t)−1 ξ) .
Thus,
Z
−1
Re Gt (ξ) = t Re Φ(h(t) ξ) −
1 − cos h(t) ξ · y ν(dy)
|y|>h(t)
> t Re Φ(h(t)−1 ξ) − 2t ν(B(0, h(t))c ).
−1
t
10
For any t > 0,
RENÉ L. SCHILLING
PAWEŠ SZTONYK
Z
c
ν(B(0, h(t)) ) 6 2
|y|>h(t)
JIAN WANG
(|y|/h(t))2
ν(dy)
1 + |y|2 /h(t)2
(|y|/h(t))2
ν(dy)
1 + (|y|/h(t))2
ZZ 1 − cos h(t)−1 ξ · y ν(dy)fd (ξ) dξ
=2
Z
ξ
fd (ξ) dξ
= 2 Re Φ
h(t)
6 2c0 sup Re Φ(η),
Z
62
|η|61/h(t)
where the last two lines follow from the same arguments as those leading to (2.15).
Hence, for any t > 0, we have
t ν(B(0, h(t))c ) 6 2c0 t
By (2.12), for m ∈ N and c
any t > t ,
Z
0
sup
Re Φ(η) = 2c0 .
|η|61/h(t)
1
, there exists t
>0
0
:= t0 (m, Φ, c1 ) > t1
such that for
0
Therefore,Z we obtain
exp − t Re Φ(ξ)
|ξ|m dξ 6 c1 h(t)−(m+d) .
Z
i m
exp − t Re Φ(ξ/h(t)) |ξ| dξ
Z
h
i
4c0
m+d
= e h(t)
exp − t Re Φ(ξ) |ξ|m dξ
exp − Re(Gt (ξ)) |ξ|m dξ 6 e4c0
(2.16)
h
= c1 e4c0 < ∞.
According to (2.15), (2.16) and Proposition 2.1, g ∈ C
t > t , and for every n ∈ N and β ∈ N with |β| 6 m − n we get
Step 4.
t
0
d
0
0
−n
|∂yβ gt (y)| 6 c(m, n, |β|, Φ) 1 + |y| ,
d
m
b (R )
for any
y ∈ Rd .
This nishes the proof since ∂ g (y) = h(t) ∂ p (h(t)y).
The following result is the counterpart of Proposition 2.2, which presents estimates
for the derivatives of the densities p for small time. Recall the denitions of ϕ
and h from (2.11).
Assume that (1.1) is satised, and for some m ∈ N ,
Z
h
i
(2.17)
exp − t Re Φ(ξ) |ξ| dξ = O ϕ 1/t
as t → 0.
β
y t
d+|β|
β h(t)
y t
h(t)
t
Proposition 2.3.
0
−1
m
m+d
Then there is some t0 > 0 such that for all t 6 t0 , there exists a density ph(t)
∈
t
h(t)
m
d
d
Cb (R ) of Pt . Moreover, for every n ∈ N0 and β ∈ N0 with |β| 6 m − n,
h(t)
|∂yβ pt (y)|
−1
−1
6 c(m, n, |β|, Φ, t0 ) ϕ (t )
d+|β| −n
1 + ϕ (t )|y|
,
−1
−1
y ∈ Rd .
COUPLING PROPERTY AND GRADIENT ESTIMATES OF LÉVY PROCESSES
11
The proof is similar to that of Proposition 2.2, and we only sketch some
key dierences. We continue to use the notations of the proof of Proposition 2.2.
According to (2.13) and (1.1), for all r > 0 and t > 0, the measure P is absolutely
continuous with respect to Lebesgue measure. Denote by p its density. Following
the argument of Proposition 2.2, we nd that (2.15) is still valid, and according
to (2.17), there exists some t > 0 such that (2.16) holds for all 0 < t 6 t . The
required assertion follows now from Proposition 2.1.
3. Proofs of the Main Theorems and Further Examples
We will now give the proofs for Theorem 1.1 and 1.2. For this we need to estimate
the coupling time of a general Lévy process. We will use the functions ϕ and h
dened in (2.11).
Assume that (1.3) holds, and
Z
h
i
(3.18)
exp − t Re Φ(ξ) |ξ| dξ = O ϕ 1/t
as t → ∞.
Proof.
r
t
r
t
0
0
Theorem 3.1.
2d+2
−1
d+2
Then, the Lévy process Xt has the coupling property, and for any x, y ∈ Rd ,1
(3.19)
Proof.
as t → ∞.
kPt (x, ·) − Pt (y, ·)kVar = O ϕ−1 1/t
Set
Z
1 − eiξ·y + iξ · y ν(dy)
Φr (ξ) =
|y|6r
and
Z
iξ·y
Ψr (ξ) := Φ(ξ) − Φr (ξ) =
1−e
Z
ν(dy) − iξ ·
|y|>r
y ν(dy).
1<|y|6r
Let Y and Z be two independent Lévy processes whose symbols are Φ (ξ) and
b (ξ), respectively. Denote by Q and Q (x, ·) the semigroup and the transition
Φ
function of Y . Similarly, R and R (x, ·) stand for the semigroup and the transition
function of Z . Then,
t
t
r
r
t
t
t
t
t
t
kPt (x, ·) − Pt (y, ·)kVar = sup Pt f (x) − Pt f (y)
kf k∞ 61
= sup Qt Rt f (x) − Qt Rt f (y)
(3.20)
kf k∞ 61
6 sup Qt g(x) − Qt g(y)
kgk∞ 61
= kQt (x, ·) − Qt (y, ·)kVar .
Now, we take r = h(t). Then, according to (3.18) and Proposition 2.2, there exists
t > 0 such that for any t > t , the kernel Q has a density q ∈ C (R ), and for
all y ∈ R ,
∇q (y) 6 c(d, Φ) h(t)
(3.21)
1 + h(t) |y|
.
0
0
t
t
d
t
−(d+1)
−1
d+2
b
d
−(d+1)
1We prove here actually that kP (x, ·) − P (y, ·)k
t
t
Var 6 c|x − y|/h(t) and in this form we use it
later in the proof of Theorem 3.2. Maybe it would be better to write the estimate explicitly here.
12
RENÉ L. SCHILLING
PAWEŠ SZTONYK
Thus, for any t > t ,
JIAN WANG
0
kQt (x, ·) − Qt (y, ·)kVar
(3.22)
Z
Z
= sup f (z) Qt (x, dz) − f (z) Qt (y, dz)
kf k∞ 61
Z
Z
= sup f (z)qt (z − x) dz − f (z)qt (z − y) dz kf k∞ 61
Z
= qt (z − x) − qt (z − y) dz.
Let t > t . Assume that |x − y| > h(t). Then,
0
Z
qt (z − x) − qt (z − y) dz 6 2 6 2|x − y| .
h(t)
If |x − y| 6 h(t), then, by (3.21),
Z
qt (z − x) − qt (z − y) dz
Z
qt (z − x) − qt (z − y) dz +
=
|z−x|>2h(t)
qt (z − x) − qt (z − y) dz
|z−x|62h(t)

6 c(d, Φ)
Z
|x − y| 

h(t)d+1
Z
"

#
h(t)d+1
sup
∇qt (w) dz +
w∈B(z−x,|y−x|)
|z−x|>2h(t)

dz 
|z−x|62h(t)

|x − y| 
6 c(d, Φ)

h(t)d+1
Z

Z
−d−1
|z − x|

1+
dz + cd (2h(t))d 
2h(t)
|z−x|>2h(t)
|x − y|
= c(d, Φ)
h(t)d+1
C |x − y|
6
.
h(t)
Z |z − x|
1+
2h(t)
−d−1
dz + 2d cd c(d, Φ)
|x − y|
h(t)
Therefore, there exists C > 0 such that for all x, y ∈ R and t > t ,
Z
q (z − x) − q (z − y) dz 6 C |x − y| .
(3.23)
h(t)
The assertion follows now from (3.20), (3.22) and (3.23).
Next, we turn to the proof of Theorem 1.1.
Proof of Theorem 1.1. Under the conditions of Theorem 1.1 we can suppose that
for any ξ ∈ R ,
d
t
0
t
d
Re Φ(ξ) > F (|ξ|),
where F (r) is a strictly increasing and dierentiable function on (0, ∞) such that
(
c f (r)
if r ∈ (0, c );
F (r) =
c log(c + c r) if r ∈ [c , ∞),
1
3
2
4
5
2
COUPLING PROPERTY AND GRADIENT ESTIMATES OF LÉVY PROCESSES
for some constants c > 0, i = 1, 2, 3, 4, 5. Thus,
2
i
Z
13
exp − t Re Φ(ξ) |ξ|d+2 dξ 6
Z
exp − tF (|ξ|) |ξ|d+2 dξ
Z ∞
2d+1
e−tr F −1 (r)
dF −1 (r)
= cd
0
Z ∞
2(d+1)
cd
=
e−tr d F −1 (r)
.
2(d + 1) 0
Since lim inf f (r)| log r| < ∞ and lim sup
s→0
r→0
, we have
f −1 (2s)/f −1 (s) < ∞
lim inf F (r)| log r| < ∞
r→0
and
lim sup F −1 (2s)/F −1 (s) < ∞.
s→0
Note that the function F also satises
3
lim F (s)/ log s = c3 .
s→∞
Then, following the proof of [1, Theorem 2.1], we obtain that for t → ∞,
Z
∞
2(d+1) −1
2(d+1)
e−tr d F −1 (r)
F (1/t)
0
2(d+1)
= f −1 (1/t)
2(d+1)
ϕ−1 (1/t)
.
The desired assertion follows from Theorem 3.1.
The following result is the short-time analogue of Theorem 3.1 which gives, additionally, gradient estimates for general Lévy processes.
Assume that (1.1) holds and let φ and h be as in (2.11). If
Z
(3.24) exp −t Re Φ(ξ) |ξ| dξ = O ϕ (1/t)
as t → 0 t → ∞
Theorem 3.2.
−1
d+2
2d+2
then there exists a constant c > 0 such that for all u ∈ Bb (Rd )
(3.25)
k∇Pt uk∞ 6 ckuk∞ ϕ−1 (1/t) ,
for all t 1
t1 .
We will treat the short- and large-time cases separately.
Recall the notations used in the proof of Theorem 3.1: Q and R are the semigroups corresponding to Φ (ξ) and Ψ (ξ), respectively. According to (3.24) and
Proposition 2.3, for small enough t 1, and r = h(t), the measure Q has a density
q ∈ C (R ) such that for any y ∈ R ,
∇q (y) 6 c(d, Φ) ϕ (t )
(3.26)
1 + ϕ (t )|y|
.
Proof.
t
r
t
r
t
t
d+2
b
d
d
t
2I added c here.
d
3I replaced 1 by c here.
3
−1
−1
d+1
−1
−1
−(d+1)
14
RENÉ L. SCHILLING
PAWEŠ SZTONYK
Then, for all u ∈ B (R ),
JIAN WANG
d
b
sup k∇Qt uk∞ = sup sup |∇Qt u(x)|
kuk∞ 61
kuk∞ 61 x∈Rd
Z
= sup sup ∇ qt (z − x) · u(z) dz kuk∞ 61 x∈Rd
Z
= sup sup ∇qt (z − x) · u(z) dz (3.27)
x∈Rd kuk∞ 61
Z
= sup ∇qt (z − x) dz
x∈Rd
Z
= ∇qt (z) dz
6 c ϕ−1 (t−1 ),
where we used (3.26) and dominated convergence. This calculation shows
k∇Qt uk∞ 6 c ϕ−1 (t−1 ) kuk∞ .
Therefore,
(3.28) k∇P uk = k∇Q R uk 6 c ϕ (t ) kR uk 6 c ϕ (t ) kuk ,
which nishes the proof for small t 1.
If t 1 is suciently large, we can apply (3.19), to nd for any u ∈ B (R ) with
kuk = 1,
t
∞
t
∞
t
−1
−1
t
∞
−1
−1
∞
b
d
∞
|Pt u(x) − Pt u(y)|
|y − x|
y→x
supkwk∞ 61 |Pt w(x) − Pt w(y)|
6 lim sup
|y − x|
y→x
kPt (x, ·) − Pt (y, ·)kVar
6 lim sup
|y − x|
y→x
|∇Pt u(x)| 6 lim sup
6 C ϕ−1 (t−1 ).
This nishes the proof for large t 1.
Let X be a rotationally invariant α-stable Lévy process on R , and
p be its density function. By the scaling property, for any t > 0 and x ∈ R ,
p (x) = t
p (t
x). On the other hand, it is well known that, see e.g. [8],
Remark 3.3.
d
t
d
t
t
−d/α
1
−1/α
∇p1 (x) 6
c
.
1 + |x|d+α
Denote by P the semigroup of X . Then, according to the proof of (3.27), we have
t
t
sup k∇Pt uk∞ = t
kuk∞ 61
−1/α
Z
∇p1 (z) dz.
This implies that Theorem 1.2 is optimal.
We can now use Theorem 3.2 to prove Theorem 1.2.
COUPLING PROPERTY AND GRADIENT ESTIMATES OF LÉVY PROCESSES
15
The second assertion easily follows from Theorem 1.1 and the
proof of Theorem 3.2. It is therefore enough to consider the rst conclusion. Under
the conditions assumed in Theorem 1.2, we know that for any ξ ∈ R ,
Proof of Theorem 1.2.
d
Re Φ(ξ) > F (|ξ|),
where F (r) is an increasing function
on (0, ∞) such that
(
0
if r ∈ (0, c ];
F (r) =
c f (r) if r ∈ (c , ∞),
for some constants c > 0, i = 1, 2, and f is strictly increasing and dierentiable on
(c , ∞). Therefore,
Z
Z
1
2
1
i
1
exp − t Re Φ(ξ) |ξ|d+2 dξ 6
exp − tF (|ξ|) |ξ|d+2 dξ.
Since lim sup f (2s)/f (s) < ∞, we can choose c > 2 and s > 0 such that
we have f (2s) 6 cf (s) for all s > s . For any k > 1 the monotonicity of f
gives
f (2 s) 6 c f (s) = 2 f (s),
whereZwe use α = log c. Then, for suciently small t 1,
−1
−1
s→∞
−1
0
−1
−1
0
−1
k −1
k
kα −1
2
exp − tF (|ξ|) |ξ|d+2 dξ
Z
Z ∞
d+2
=
|ξ| dξ + cd
e−c2 tf (r) r2d+1 dr
|ξ|<c1
c1
Z ∞
2d+2
6 C 1 + cd
e−c2 s ds f −1 (s/t)
)
(0Z
∞ Z 2n
1
X
2d+2
+
e−c2 s ds f −1 (s/t)
6 C 1 + cd
0
n=1
2n−1
∞
X
−1
2d+2
2d+2
6 C1 + cd f (1/t)
+ cd
exp − c2 2n−1 f −1 (2n /t)
n=1
6 C 1 + cd
1+
∞
X
!
n−1
exp − c2 2
nα(2d+2)
2
−1
2d+2
f (1/t)
n=1
2d+2
6 C1 + C2 f −1 (1/t)
.
Because of (1.1) and Re Φ(ξ) f (|ξ|) as |ξ| → ∞, we nd f
Thus,
Z
Z
−1
(1/t) → ∞
as t → 0.
exp − t Re Φ(ξ) |ξ|d+2 dξ 6
exp − tF (|ξ|) |ξ|d+2 dξ
2d+2
6 C3 f −1 (1/t)
.
Now the assertion follows from Theorem 3.2.
If we assume in the statement of Theorem 1.2 that f (s) = s `(s)
for some α > 0 and some positive function ` which is slowly varying at ∞that
is, lim `(λs)/`(s) = 1 for every λ > 0, then standard Abelian and Tauberian
Remark 3.4.
s→∞
−1
α
16
RENÉ L. SCHILLING
PAWEŠ SZTONYK
JIAN WANG
arguments (see, e.g. [2, Theorems 1.7.1 and 1.7.1 ,] or [7, Chapter XIII.5, Theorems
1 and 3]), weZ can obtain that
exp − tF (|ξ|) |ξ| dξ f (1/t)
as t → 0.
Let us nally turn to the examples from Section 1.
Proof of Example 1.3. The symbol of the subordinate Brownian motion here satises
Re Φ(ξ) |ξ|
as |ξ| → 0,
and Re Φ(ξ) |ξ| log(1 + |ξ|) as |ξ| → ∞.
For r > 0, set f (r) = r (log(1 + r)) and g(r) = r(log(1 + r)) . Then, for
r → ∞, we have
0
2d+2
−1
d+2
α+β
β/2
α
α
−β/2 1/α
β/2
−β/2 h
1/α iβ/2
f g(r) = r log(1 + r)
log 1 + r(log(1 + r))−β/2
β/2
−β/2 2 log r − β log log r
r (log r)
α
β/2
2 log r − β log log r
=r
α log r
r.
This shows that f (r) g(r) for r → ∞, and now Theorems 1.1 and 1.2 apply. Proof of Example 1.4. Let Y and Z be Lévy processes whose Lévy measures are
given by
Z Z
Z Z
−1
t
t
∞
r0
1A (sθ)s−1−β ds µ(dθ)
1A (sθ)s−1−α ds µ(dθ) +
ν Y (A) :=
and
ν (dz) := ν(dz) − ν (dz) > 0,
respectively. After some elementary calculations, we see that the symbol Φ of
Y satises Re Φ (ξ) |ξ| as |ξ| → ∞ and Re Φ (ξ) |ξ|
as |ξ| → 0. Let
P (x, ·) and P denote the transition function and the semigroup of Y . According
to Theorems 1.1 and 1.2, we can prove the claim rst for Y (if we replace in these
Theorems Φ, P and P (x, ·) by the corresponding objects Φ , P and P (x, ·).
To come back to the original process resp. semigroup we can now use (3.20) and
(3.28).
Example 1.4 applies to a large number of interesting and important Lévy processes, whose Lévy measures isZof the
following polar coordinates form:
Z
0
r0
S
Z
S
Y
Y
t
Y
t
Y
α
Y
β∧2
Y
t
t
t
Y
t
Y
t
Y
t
∞
ν(A) =
1A (sθ)Q(θ, s) ds µ(dθ);
is a nonnegative function on S × (0, ∞). For instance, Example 1.4 is applicable for the following processes:
(1) Stable Lévy processes ([25]):
0
S
Q(θ, s)
where α ∈ (0, 2).
Q(θ, s) s−α−1 ,
COUPLING PROPERTY AND GRADIENT ESTIMATES OF LÉVY PROCESSES
17
(2) Layered stable processes ([11]):
Q(θ, s) s−α−1 1(0,1] (s) + s−β−1 1[1,∞) (s),
where α ∈ (0, 2) and β ∈ (0, ∞).
(3) Tempered stable processes ([18]):
Q(θ, s) s−α−1 e−cs ,
where α ∈ (0, 2) and c > 0.
(4) Relativistic stable processes ([5, 3]):
Q(θ, s) s−α−1 (1 + s)
where α ∈ (0, 2).
(5) Lamperti stable processes ([4]):
Q(θ, s) = s−α−1
d+α−1
2
e−s ,
exp(sf (θ)) s1+α
,
(es − 1)1+α
where α ∈ (0, 2) and f : S → R such that sup
(6) Truncated stable processes ([13]):
θ∈S
f (θ) < 1 + α
.
Q(θ, s) s−α−1 1(0,1] (s),
where α ∈ (0, 2).
Motivated by Example 1.4, we can present a short proof of (one part of) F.-Y.
Wang's result on explicit gradient estimates for the semigroup of a general Lévy
processes.
(F.-Y. Wang [26, Theorem 1.1]) Let X be a Lévy process on R with
Theorem 3.5
.
d
t
Lévy measure ν . Assume that there exists some r ∈ (0, ∞] such that
ν(dz) > |z|−d f (|z|−2 )1{|z|6r} dz,
where f is Bernstein function such that
lim inf
r→∞
f (r)
= ∞ and
log r
lim sup
s→∞
f −1 (2s)
∈ (0, ∞).
f −1 (s)
Then there exists a constant c > 0 such that for any t > 0,
k∇Pt uk∞ 6 c kuk∞ f
−1
1
t∧1
,
u ∈ Bb (Rd ).
According to the proofs of Example 1.4 and [26, Theorem 1.1], we see that
X can be decomposed into two independent Lévy processes Y and Z , such that
the symbol Φ (ξ) of Y satises Φ (ξ) f (|ξ| ) as |ξ| → ∞. Now we can apply
Theorem 1.2 and the claim follows.
Financial support through DFG (grant Schi 419/5-1) and
DAAD (PPP Kroatien) (for René L. Schilling), MNiSW, Poland (grant N N201
397137) (for Paweª Sztonyk) and the Alexander-von-Humboldt Foundation and the
Natural Science Foundation of Fujian (No. 2010J05002) (for Jian Wang) is gratefully
acknowledged.
Proof.
t
t
Y
Acknowledgement.
t
Y
2
t
18
RENÉ L. SCHILLING
PAWEŠ SZTONYK
JIAN WANG
References
[1] Böttcher, B., Schilling, R.L. and Wang, J.: Constructions of coupling processes for Lévy
processes,
, TU Dresden 2010. See also arXiv:1009.5511.
[2] Bingham, N.H., Goldie, C.M. and Teugels, J.L.:
, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge 2001.
[3] Byczkowski, T., Maªecki, J. and Ryznar, M.: Bessel potentials, hitting distribution and Green
functions,
361 (2009), 48714900.
[4] Caballero, M.E., Pardo, J.C. and Pérez, J.L.: On the Lamperti stable processes,
. 30 (2010), 128.
[5] Carmona, R., Masters, W.C. and Simon, B.: Relativistic Schrödinger operators: asymptotic
behavior of the eigenfunctions,
91 (1990), 117142.
[6] Constantine, G. M. and Savits, T.H.: A multivariate Faa di Bruno formula with applications,
348 (1996), 503520.
[7] Feller, W.:
, vol. 2 (2nd ed.), Wiley,
New York 1971.
[8] Gªowacki, P.: Stable semigroups of measures as a commutative approximate identities on
non-graded homogeneous groups,
83 (1986), 557582.
[9] Hartman, P. and Wintner, A.: On the innitesimal generators of integral convolutions,
64 (1942), 273298.
[10] Hoh, W.: A symbolic calculus for pseudo dierential operators generating Feller semigroups,
35 (1998), 798820.
[11] Houdré, C. and Kawai, R.: On layered stable processes,
13 (2007), 261287.
[12] Jacob, N.:
, Imperial College Press, London 2001.
[13] Kim, P. and Song R.M.: Boundary behavior of harmonic functions for truncated stable processes,
21 (2008), 287321.
[14] Knopova, V. and Schilling, R.L.: A note on the existence of transition probability densities
for Lévy processes,
, TU Dresden 2010. See also arXiv 1003.1419v2.
[15] Lindvall, T.:
, Wiley, New York 1992.
[16] Picard, J.: On the existence of small densities for jump processes,
105 (1996), 481511.
[17] Priola, E. and Wang, F.Y.: Gradient estimates for diusion semigroups with singular coecients,
236 (2006), 244264.
[18] Rosinski, J.: Tempering stable processes,
117 (2007), 677707.
[19] Sato, K.:
, Studies Adv. Math. vol. 69,
Cambridge University Press, Cambridge 1999.
[20] Schilling, R.L.: Growth and Hölder conditions for the sample paths of Feller processes,
112 (1998), 565611.
[21] Schilling, R.L.: Conservativeness and extensions of Feller semigroups,
2 (1998),
239256.
[22] Schilling, R.L., Song, R. and Vondra£ek, Z.:
,
Studies in Math. vol. 37, de Gruyter, Berlin 2010.
[23] Schilling, R.L. and Wang, J.: On the coupling property of Lévy processes, to appear in
See also arXiv 1006.5288.
[24] Song, R. and Vondra£ek, Z.: Potential theory of subordinate Browinan motion, in: Bogdan,
K. et al.:
, Lect. Notes Math. vol.
1980, Springer, Berlin 2009, 87176.
[25] Sztonyk, P.: Regularity of harmonic functions for anisotropic fractional Laplacians,
283 (2010), 289311.
[26] Wang, F.Y.: Gradient estimate for Ornstein-Uhlenbeck jump processes, to appear in
, 2010. See also arXiv 1005.5023v3.
Preprint
Regular Variation
Trans. Am. Math. Soc.
Math. Stat
J. Funct. Anal.
Trans. Am. Math. Soc.
An Introduction to Probability Theory and its Applications
Probab.
Invent. Math.
Am.
J. Math.
Osaka J. Math.
Bernoulli
Pseudo Dierential Operators and Markov Processes. Vol. 1: Fourier Analysis and
Semigroups
J. Theor. Probab.
Preprint
Lectures on the Coupling Method
Probab. Theor. Rel. Fields
J. Funct. Anal.
Stoch. Proc. Appl.
Lévy processes and Innitely Divisible Distributions
Theor. Rel. Fields
Positivity
Bernstein FunctionsTheory and Applications
Ann.
Inst. Henri Poincaré: Probab. Stat.
Potential Analysis of Stable Processes and its Extensions
Nach.
Proc. Appl.
Probab.
Math.
Stoch.