Linear Evolution Equations with Cylindrical L\'evy Noise: Gradient Estimates and Exponential Ergodicity

0.26 MB·English

by  Jian Wang

LINEAR EVOLUTION EQUATIONS WITH CYLINDRICAL LÉVY
NOISE: GRADIENT ESTIMATES AND EXPONENTIAL
ERGODICITY
JIAN WANG
Abstract. Explicitcouplingpropertyandgradientestimatesareinvestigatedfor
5
1 the linear evolution equations on Hilbert spaces driven by an additive cylindrical
0 Lévy process. The results are efficiently applied to establish the exponential er-
2 godicityfortheassociatedtransitionsemigroups. Inparticular,ourresultsextend
n recentdevelopmentsonrelatedtopicforcylindricalsymmetricα-stable processes.
a
Keywords: CylindricalLévyprocesses;linearevolutionequations;couplingprop-
J
erty; gradient estimates; exponential ergodicity
5
2 MSC 2010: 60H15; 60J75; 60G51;35R60.

]
R
P 1. Introduction and Main Results
.
h Let H be a real separable Hilbert space with the norm , and (A,D(A)) be a
t k·k
a linear possibly unbounded operator which generates a C -semigroup (i.e. a strongly
0
m continuous one-parameter semigroup of linear operators) (T ) on H. Consider the
t t>0
[
following linear evolution equation:

1
v (1.1) dX = AX dt+dZ , t > 0, X = x H,
t t t 0
1 ∈
3 where Z = (Z ) is an infinite dimensional Lévy process which may take values in
t t>0
1 a Hilbert space U usually greater than H. The Markov process X = (Xx) deter-
6 t t>0
0 mined by (1.1) (if exists) is called a Lévy driven Ornstein-Uhlenbeck process, e.g.
1. see [1, 2, 9, 21, 28] and the references therein. The associated transition semigroup
0 acting on B (H), the class of all bounded measurable functions on H, is given by
b
5
1 P f(x) := Ef(Xx),
: t t
v
i which is called a generalized Mehler semigroup in the sense of [6, 12, 17].
X
Throughout this paper we suppose that the infinite dimensional Lévy process
r
a Z = (Z ) in (1.1) is defined by the orthogonal expression
t t>0
∞
(1.2) Z = Zne , t > 0,
t t n
n=1
X
where (e ) is an orthonormal basis of H and (Zn) are independent real valued
n t t>0,n>1
purejumpLévyprocessesdefinedonafixedstochasticbasis. Wecall(Z ) givenby
t t>0
(1.2) an additive cylindrical Lévy process, see [3] for the recent study on cylindrical
Lévy processes. We also need the following assumption on the operator (A,D(A)):
J. Wang: School of Mathematics and Computer Science, Fujian Normal University, 350007,
Fuzhou, P.R. China. [email protected].
1
2 JIAN WANG
Hypothesis (H). The operator (A,D(A)) is a self-adjoint operator such that for
the fixed basis (e ) in (1.2) it verifies that (e ) D(A), Ae = γ e with γ > 0
n n n n n n
⊂ −
for any n > 1, and lim γ = .
n
n→∞ ∞
That is, we require that the basis (e ) from the representation (1.2) are eigen-
n
vectors of A. Then, we can identify H with l2 = x = (x ) : ∞ x2 < , and
{ n n=1 n ∞}
A with the diagonal operator Ax = γ x for (x ) l2 and n > 1. Thus, under
n n n n
− ∈ P
assumptions (1.2) and (H), the equation (1.1) can be solved for each coordinate
separately, i.e.,
dXx,n = γ Xx,ndt+dZn, Xx,n = x , n N,
t − n t t 0 n ∈
with x = (x ) l2. Therefore, the unique solution to the equation (1.1) can be seen
n
as a stochastic∈process X = (Xx) taking values in RN with components
t t>0
t
(1.3) Xx,n = e−γntx + e−γn(t−s)dZn, n N,t > 0.
t n s ∈
Z0
In the following, we always assume that the process X = (Xx) takes values in
t t>0
H. According to Corollary 2.3 below, this is guaranteed if and only if
∞ 1
(1.4) e−2γns z2ν (dz)+ ν (dz) ds < ,
n n
∞
n=1Z0 (cid:18) Z{|z|6eγns} Z{|z|>eγns} (cid:19)
X
where for any n > 1, ν is the Lévy measure of the Lévy process (Zn) on R. We
n t t>0
first study the coupling property and explicit gradient estimates of the associated
Markov semigroup for the process X. Recall that the process X has successful
couplings (or has the coupling property) if and only if for any x,y H,
∈
lim P (x, ) P (y, ) = 0,
t t Var
t→∞k · − · k
where P (x,dz) is the transition kernel of the process X and stands for the
t Var
k·k
total variation norm. Let (P ) be the transition semigroup of the process X. The
t >0
uniform norm of its gradient is defined as follows:
Pt ∞ := sup zPtg(x) : z 6 1,x H,g Bb(H) with g H,∞ 6 1 ,
k∇ k |∇ | k k ∈ ∈ k k
(cid:26) (cid:27)
where
1
P g(x) := limsup P g(x+εz) P g(x) ,
z t t t
|∇ | ε −
ε→0
(cid:12) (cid:12)
and kgkH,∞ is denoted by the supremun(cid:12)(cid:12) norm, i.e. kgkH,∞ (cid:12)(cid:12)= supx∈H|g(x)|. A
continuous function f : [0, ) [0, ) is said to be a Bernstein function if
∞ → ∞
( 1)kf(k)(x) 6 0 for all x > 0 and k > 1. One of our main contributions is as
−
follows.
Theorem 1.1. Let X = (Xx) be the process with components (1.3) taking values
t t>0
in H, i.e. (1.4) holds. Suppose that for any n > 1, the Lévy measure ν of the Lévy
n
process (Zn) on R satisfies that
t t>0
(1.5) ν (dz) > z −1f (z−2)dz,
n n
| |
where f is a Bernstein function with f (0) = 0 and lim fn(r) = . Then, for
n n r→∞ log(1+r) ∞
any t > 0 and x,y H,
∈
(1.6) P 6 C ,
t ∞ t
k∇ k
LINEAR EVOLUTION EQUATIONS WITH CYLINDRICAL LÉVY NOISE 3
and
P (x, ) P (y, ) 6 2C x y ,
t t Var t
k · − · k k − k
where
Ct := 2sup ∞e−co2s1tfk(r)dr e−γkt ∞e−(cos1)(14−γke−2γkt)fk(r)dr , t > 0.
s k>1 (cid:20)(cid:18)Z0 (cid:19)∧(cid:18) Z0 (cid:19)(cid:21)
The coupling property and gradient estimates have been intensively studied for
linear stochastic differential equations driven by Lévy processes on Rd, see e.g. [5,
33, 34, 35, 36, 37]. Among them, gradient estimates for linear evolution equations
on Rd have been obtained in [37, Theorem 1.1] by using a similar bound condition
(1.6) on the Lévy measure, and then they are improved in [33, Theorem 1.3] and [35,
Theorem 3.1] via the symbol of Lévy processes. Recently, by using the lower bound
conditionsfortheLévymeasure withrespect toanicereference probability measure,
wehavesuccessfully obtainedthecoupling propertyandgradientestimates forlinear
stochastic differential equations driven by non-cylindrical Lévy processes on Banach
spaces, see [38, Theorem 1.2]. However, as mentioned in the paragraph before [38,
Theorem1.1], thesituationforcylindrical Lévy processes is essentially different from
that for non-cylindrical Lévy processes. This implies that the framework adopted
in [38] does not apply to the present settings. We stress that Theorem 1.1 improves
[23, Theorem 4.14] for cylindrical symmetric α-stable noise, and also points out the
difference form non-cylindrical Lévy processes.
As an application of Theorem 1.1, we will study the exponential ergodicity for the
stochastic process X = (Xx) with components (1.3). Let (f ) be a sequence
t t>0 n n>1
of Bernstein functions, and for any t > 0, let C be the constant defined in Theorem
t
1.1.
Theorem 1.2. Suppose that for any n > 1, (Zn) in (1.3) is a symmetric pure
t t>0
jump Lévy process associated with the (symmetric) Lévy measure ν . Assume that
n
the following two assumptions hold:
(i) There exists a constant α (0,1] such that
∈
∞ ∞
e−2γns z2ν (dz)+e−αγns z αν (dz) ds < ;
n n
| | ∞
n=1Z0 (cid:18) Z{|z|6eγns} Z{|z|>eγns} (cid:19)
X
(ii) For any n > 1,
ν (dz) > z −1f (z−2)dz,
n n
| |
where f is a Bernstein function such that f (0) = 0, lim fn(r) = and
n n r→∞ log(1+r) ∞
lim C = 0.
t
t→∞
Then, the process X = (Xx) with components (1.3) takes values in H, and there
t t>0
exist the unique invariant measure µ and a constant C > 0 such that for any t > 0
and x H,
∈
P (x, ) µ 6 C(1+ x α)Cα.
k t · − kVar k k t
The remaining part of this paper is organized as follows. In Section 2 we present
some preliminaries on cylindrical Lévy processes. Section 3 is devoted to the proof
of Theorem 1.1. Here, the most important estimates for the density function of one-
dimensional subordinateBrownian motionsareestablished inProposition3.1, which
iskeytotheproofofTheorem1.1andindicatesthedifferencefromfinitedimensional
4 JIAN WANG
situations. We also obtain the gradient estimate and the coupling property for the
linear evolution equation driven by cylindrical subordinate Brownian motions, see
Proposition 3.2 below. In Section 4, we will study the exponential ergodicity for the
linear evolution equations (1.1). The proof of Theorem 1.2 is presented.
2. Preliminarily: Cylindrical Lévy Processes on Hilbert Spaces
In the section, H will denote a real separable Hilbert space with the inner product
, and the norm . We will fix an orthonormal basis (e ) in H. Through the
n
bh·as·iis (e ) we will okft·ekn identify H with l2. More generally, for a given sequence
n
ρ = (ρ ) of real numbers, we set
n
∞
(2.7) l2 := (x ) RN : x2ρ2 < .
ρ n ∈ n n ∞
(cid:26) n=1 (cid:27)
X
It is clear that l2 becomes a separable Hilbert space with the inner product x,y =
ρ h i
∞
x y ρ2 for x = (x ),y = (y ) l2.
n n n n n ∈ ρ
n=1
PLet us recall that a Lévy process Z = (Zt)t>0 with values in H is an H-valued
process defined on some stochastic basis (Ω,F,(F ),P), continuous in probability,
t
havingstationaryindependent increments, càdlàgtrajectories, andsuchthatZ = 0,
0
P-a.s. It is well known that
EeihZt,xi = exp tΦ(x) , x H,
− ∈
where the characteristic exponent (or (cid:0)the symb(cid:1)ol) Φ can be expressed by the fol-
lowing infinite dimensional Lévy-Khintchine formula
1
Φ(x) = Qx,x i b,x + 1 eihx,zi +i x,z 1 Π(dz), x H.
{kzk61}
2h i− h i − h i ∈
H
Z (cid:18) (cid:19)
Here Q is a non-negative, self-adjoint trace class operator on H, b H and Π is the
Lévy measure on H 0 such that (1 z 2)Π(dz) < . W∈e call the triple
\{ } H\{0} ∧ k k ∞
(Q,b,Π) the characteristics of the Lévy process Z, e.g. see [20, Chapter VI] and [21,
R
Chapter 4].
For cylindrical Lévy process Z given by (1.2), we assume that (Zn) are de-
t t>0,n>1
fined on the same stochastic basis (Ω,F,(F ),P) satisfying the usual assumptions.
t
Since for any n > 1, (Zn) is a pure jump Lévy process on R, we have, for each
t t>0
n > 1 and t > 0,
EeihZtn = e−tψn(h), h R,
∈
where
(2.8) ψ (h) = 1 eihz +ihz1 ν (dz), h R,n > 1,
n {|z|61} n
− ∈
R
Z (cid:16) (cid:17)
and ν is the Lévy measure on R 0 with (1 z2)ν (dz) < , e.g. see [29].
n \{ } R\{0} ∧ n ∞
The following result essentially follows from [21, Theorem 4.40], which gives us an
R
if and only if condition such that the cylindrical Lévy process Z = (Z ) given by
t t>0
(1.2) takes values in H.
LINEAR EVOLUTION EQUATIONS WITH CYLINDRICAL LÉVY NOISE 5
Proposition 2.1. ([21, Theorem 4.40]) The cylindrical Lévy process Z given by
(1.2) takes values in H if and only if
∞
(2.9) (1 z2)ν (dz) < .
n
∧ ∞
n=1Z
X
As a direct consequence, we have the following example.
Example 2.2. Let Z = (Z ) be a cylindrical Lévy process with the following
t t>0
form
∞
(2.10) Z = β Zne , t > 0,
t n t n
n=1
X
where (e ) is an orthonormal basis of H, (Zn) are independent, real valued,
n t t>0,n>1
identically distributed Lévy processes defined on a fixed stochastic basis, and (β )
n
is a given possibly unbounded sequence of positive numbers. Let ν be the common
Lévymeasure corresponding to(Zn) , e.g.see(2.8). Then, thecylindrical Lévy
t t>0,n>1
process Z given by (2.10) takes values in H if and only if
∞
β2 z2ν(dz)+ ν(dz) < .
n ∞
n=1(cid:18) Z{|z|61/βn} Z{|z|>1/βn} (cid:19)
X
Example 2.2 covers [23, Proposition 3.3] for cylindrical symmetric α-stable pro-
cesses, and also extends [25, Proposition 2.4] where for any n > 1, the symmetry of
the Lévy process (Zn) is required. On the other hand, according to Proposition
t t>0
2.1, we know that the cylindrical Lévy process Z given by (2.10) is a Lévy process
with values in the space l2, see (2.7), where (ρ ) is a sequence of positive numbers
ρ n
such that
∞
(ρ β )2 z2ν(dz)+ ν(dz) < .
n n
∞
n=1(cid:18) Z{|z|61/(ρnβn)} Z{|z|>1/(ρnβn)} (cid:19)
X
That is, with thesequence (ρ ) above, thecylindrical Lévy process Z given by (2.10)
n
is an honest l2-valued Lévy process, see [23, Remark 4.1] and [25, Remark 2.7].
ρ
Furthermore, as an application of Proposition 2.1, we can provide a criteria such
that the process X = (Xx) with components (1.3) takes values in H.
t t>0
Corollary 2.3. The process X = (Xx) with components (1.3) takes values in H
t t>0
if and only if (1.4) holds. Moreover, under (1.4), X = (Xx) is a Markov process,
t t>0
and
∞
Xx = Xx,ne = etAx+Z (t),
t t n A
n=1
X
where
t ∞ t
Z (t) = e(t−s)AdZ = e−γn(t−s)dZn e .
A s s n
Z0 n=1(cid:18)Z0 (cid:19)
X
When the Lévy process (Zn) is square integrable with zero mean, Corollary
t t>0
2.3 has been proved in [21, Proposition 9.7]; when the Lévy process (Zn) is
t t>0
symmetric, Corollary 2.3 is just [25, Theorem 2.8]. For the general case, we refer to
[28, Corollary 6.3] for the recent study. Below we include the proof of Corollary 2.3
for the sake of completeness.
6 JIAN WANG
Proof of Corollary 2.3. For any n > 1 and t > 0, let us consider the stochastic
convolution
t
Yn = e−γn(t−s)dZn.
t s
Z0
The law of Yn is an infinitely divisible probability distribution, and its characteristic
t
exponent is
t
log(E(eihYtn)) = ψ (e−γnsh)ds, t > 0,h R,
n
− ∈
Z0
where ψ is given by (2.8), e.g. see [21, Corollary 4.29] or [35, Lemma 3.1]. Since the
n
driving Lévyprocess (Zn) isapurejumpLévy process, theLévy triplet(0,b ,ν )
t t>0 t n,t
of Yn is given by, e.g. see [30, Theorem 3.1] and [19, Proposition 2.1],
t
t
ν (D) = ν (e−sγnD)ds, D B(R 0 ),
n,t n
∈ \{ }
Z0
t
b = esγnz 1 esγnz 1 (z) dsν (dz),
t {|z|61} {|z|61} n
−
Zz6=0Z0
(cid:16) (cid:17)
(cid:0) (cid:1)
where ν is the Lévy measure of ψ given in (2.8). Therefore,
n n
∞
(1 z2)ν (dz)
n,t
∧
n=1Z
X
∞ t
(2.11) = (1 (e−sγnz)2)ν (dz) ds
n
∧
n=1Z0 (cid:18)ZR (cid:19)
X
∞ t
= e−2γns z2ν (dz)+ ν (dz) ds.
n n
n=1Z0 (cid:18) Z{|z|6eγns} Z{|z|>eγns} (cid:19)
X
Assumption (1.4) gives us that P(Xx H) = 1, due to (2.11) and (2.9). So,
in order to complete the proof, it rema1in∈s to show that P(Xx H) = 1 implies
P(Xx H) = 1 for any t > 0. Since (1.4) verifies that 1 ∈
t ∈
∞ t
e−2γns z2ν (dz)+ ν (dz) ds < , 0 < t 6 1,
n n
∞
n=1Z0 (cid:18) Z{|z|6eγns} Z{|z|>eγns} (cid:19)
X
also by (2.9), for any 0 < t 6 1, P(Xx H) = 1.
t ∈
For any t > 0, set Y = Yn . Then,
t t n>1
(cid:0) (cid:1)
∞ t t
Y = e−γn(t−u)dZn e =: e(t−u)AdZ .
t u n u
n=0(cid:18)Z0 (cid:19) Z0
X
LINEAR EVOLUTION EQUATIONS WITH CYLINDRICAL LÉVY NOISE 7
For any t > 1, we have the following identity on the product space RN:
t 1
Y e(t−1)AY = e(t−u)AdZ e(t−1)A e(1−u)AdZ
t 1 u u
− −
Z0 Z0
t 1
= e(t−u)AdZ e(t−u)AdZ
u u
−
Z0 Z0
t
= e(t−u)AdZ
u
Z1
t−1
= e(t−1−u)AdZ1,
u
Z0
where Z1 = Z Z , u > 0, is still a Lévy process with values in RN. Note
u 1+u − 1
that t−1e(t−1−u)AdZ1 has the same law as Y . Since P(Y H) = 1 for any
1 < t 60 2 and P(Y u H) = 1, we have, by thte−1identity abovt−e,1P∈(Y H) = 1 for
1 t
any 1R< t 6 2. Using∈an iteration produce, we furthermore infer that P∈(Y H) = 1
t
∈
for any t > 0. This proves the required assertion.
The Markov property of X follows from the identity below
t−s
Y e(t−s)AY = e(t−s−u)AdZs, 0 < s 6 t,
t − s u
Z0
where Zs = Z Z . (cid:3)
u s+u − s
We close this section with two remarks and one example for Corollary 2.3.
Remark 2.4. (1) According to [9, Theorem 2.3] or [1, Theorem 7], if the Lévy
process Z takes values in H, then the linear equation (1.1) has the unique mild
solution X which also takes values in H. Thus, it is not surprising to see that (2.9)
implies (1.4).
(2) If the cylindrical Lévy process Z given by (1.2) takes values in the Hilbert
space H, i.e. the condition (2.9) holds, then, by the Kotelenez regularity result (see
[21, Theorem 9.20]), trajectories of the process X which solves (1.1) are càdlàg with
values in H, e.g. see [23, the first paragraph in Section 4.1] and [25, Remark 2.10].
On the other hand, assume that the process X which solves (1.1) has H-càdlàg
modification, where the Lévy process Z is defined by (1.2). Then, according to (the
proof of) [18, Theorem 2.1], for any ε > 0,
∞
ν (dz) < ,
n
∞
n=1Z{|z|>ε}
X
where for any n > 1, ν is the Lévy measure corresponding to the pure jump Lévy
n
process (Zn) .
t t>0
Example 2.5 (Continuation of Example 2.2). Let X be the process with compo-
nents (1.3), where Z is the cylindrical Lévy process given by (2.10). Then, the
process X takes values in H if and only if
∞ 1
(2.12) β2e−2γns z2ν(dz)+ ν(dz) ds < .
n=1Z0 (cid:18) n Z{|z|6βn−1eγns} Z{|z|>βn−1eγns} (cid:19) ∞
X
8 JIAN WANG
3. Coupling Property and Gradient Estimates
3.1. One-dimensional Ornstein-Uhlenbeck Processes Driven by Subordi-
nate Brownian Motions. In this subsection, we will consider one-dimensional
Ornstein-Uhlenbeck processes driven by subordinate Brownian motions, which are
a class of special but important Lévy processes.
We first recall some facts and properties for subordinate Brownian motions. Sup-
pose that (B ) is a Brownian motion on R with
t t>0
E eih(Bt−B0) = e−th2, h R,t > 0,
∈
h i
and (S ) is a subordinator (that is, (S ) is a nonnegative Lévy process on
t t>0 t t>0
[0, ) such that S is increasing and right-continuous in t with S = 0) independent
t 0
∞
of (B ) . For any t > 0, let µS be the probability distribution of the subordinator
t t>0 t
S, i.e. µS(D) = P(S D) for any D B([0, )). It is well known that the
t t ∈ ∈ ∞
associated Laplace transformation of µS for each t > 0 is given by
t
∞
e−rsµS(ds) = e−tf(r), r > 0,
t
Z0
where f(r) is a Bernstein function, i.e. ( 1)kf(k)(r) 6 0 for all r > 0 and k > 1.
−
We refer to [32] for more details about Bernstein functions and subordinators. The
process Z = (Z ) on R defined by
t t>0
Z = B , t > 0
t St
is called subordinate Brownian motion, which is a symmetric Lévy process with
E eih(Zt−Z0) = e−tf(h2), h R,t > 0.
∈
h i
That is, the symbol or the characteristic exponent of the subordinate Brownian
motion(Z ) is f(h2), see [13]. The transition density function of the subordinated
t t>0
Brownian motion Z exists, and it is given by
∞ 1 x y 2
(3.13) pZ(x,y) = pZ(x y) = exp | − | µS(ds)
t t − √4πs − 4s t
Z0 (cid:18) (cid:19)
for t > 0 and x,y R. Examples of subordinate Brownian motions include sym-
∈
metric α-stable processes, relativistic α-stable processes and so on.
Now, we will consider the process X = (X ) given by
t t>0
t
(3.14) Xx = e−γtx+ e−γ(t−s)dZ , t > 0,x R,
t s ∈
Z0
where γ > 0 and (Z ) is a subordinated Brownian motion on R associated with
t t>0
the Bernstein function f.
The following result is the key to the proof of Theorem 1.1. Denote by C∞(R)
b
the set of all infinite differentiable functions on R such that all their derivatives are
bounded.
Proposition 3.1. If
f(r)
(3.15) lim = ,
r→∞ log(1+r) ∞
LINEAR EVOLUTION EQUATIONS WITH CYLINDRICAL LÉVY NOISE 9
then the Ornstein-Uhlenbeck process (Xx) given by (3.14) has a density function
t t>0
of the form p ( e−γntx). Here, for any t > 0, the function p enjoys the following
t t
·−
properties:
(i) p > 0 and p C∞(R) such that p is even on R, p′ is odd on R.
t t ∈ b t t
(ii)
(p′(x))2 ∞
(3.16) t dx 6 2 e−Ft(r)dr,
p (x)
ZR t Z0
where for any t,r > 0,
t
F (r) = f(e−2γsr)ds.
t
Z0
Proof. Let us first consider the stochastic convolution
t
Y = e−γ(t−s)dZ , t > 0.
t s
Z0
A direct calculation shows for any h R and t > 0,
∈
t
Ee−ihYt = exp f(e−2γsh2)ds = exp F (h2) ,
t
− −
(cid:18) Z0 (cid:19) (cid:18) (cid:19)
e.g. see [21, Corollary 4.29] or [35, Lemma 3.1].
Since r f(e−2γsr) is a Bernstein function for every s > 0 and γ > 0, it follows
7→
from [32, Corollary 3.7] that F is also a Bernstein function. Then, there exists a
t
probability measure µ on [0, ) such that its Laplace transformation
t
∞
∞
e−rsµ (ds) = e−Ft(r), r > 0.
t
Z0
Furthermore, due to (3.15), we have
F (r)
t
lim = for all t > 0.
r→∞ log(1+r) ∞
According to (the proof of) [16, Theorem 1], we know that for all t > 0 there exists
a density function p for Y such that p C∞(R), and p(n) exists and belongs to
L (R) C (R) for atll n >t 0. By (3.14)t,∈we bknow that fotr any t > 0, the density
1 ∞
∩
function of Xx exists and is equivalent to p ( e−γtx). On the other hand, as
t t · −
mentioned above, F (r) is a Bernstein function. Therefore, according to (3.13),
t
pt(x) = ∞ 1 e−x4s2 µt(ds), x R.
√4πs ∈
Z0
Clearly, p (x) > 0 for all x R, p (x) is even and p′(x) is odd. In what follows, we
write p (xt) = q (x2) for all x∈ R,twhere t
t t
∈
∞ 1
qt(x) = e−4xs µt(ds).
√4πs
Z0
It is easy to see that for any n > 1, q(n) exists such that for any x > 0, q(2n−1)(x) < 0
t t
(2n)
and q (x) > 0. Therefore,
t
(p′(x))2 ∞ (p′(x))2 ∞ √xq′(x)2
t dx = 2 t dx = 4 t dx.
p (x) p (x) q (x)
ZR t Z0 t Z0 t
10 JIAN WANG
Since, by q(+ ) = q′(+ ) = 0, q′(x) < 0 and q(3)(x) < 0 for all x > 0,
∞ ∞ t t
∞ ∞
q′(x)2 = 2 q′(u)q′′(u)du 6 2q′′(x) q′(u)du = 2q′′(x)q (x),
t − t t t − t t t
Zx (cid:18) Zx (cid:19)
we arrive at
(p′(x))2 ∞
t dx 6 8 √xq′′(x)dx.
p (x) t
ZR t Z0
Noting that
1 ∞ 1
qt′′(x) = 32√π s5/2e−4xs µt(ds),
Z0
we further get
(p′(x))2 1 ∞ ∞ 1
pt(x) dx 64√π √x s5/2e−4xs µt(ds)dx
ZR t Z0 Z0
2 ∞ ∞ 1
= √re−rdr µ (ds)
t
√π s
Z0 Z0
∞ 1
=2 µ (ds)
t
s
Z0
∞ ∞
=2 e−rsdrµ (ds)
t
Z0 Z0
∞ ∞
=2 dr e−rsµ (ds)
t
Z0 Z0
∞
=2 e−Ft(r)dr.
Z0
This proves (3.16). The proof is completed. (cid:3)
3.2. Ornstein-Uhlenbeck Processes on H Driven by Cylindrical Subordi-
nated Brownian Motions. In this part, let X be the process with components
(1.3), where for any n > 1, (Zn) is a subordinated Brownian motion associated
t t>0
with the Bernstein function f satisfying
n
f (r)
n
lim = .
r→∞ log(1+r) ∞
For simplicity, we call X the Ornstein-Uhlenbeck process driven by additive cylin-
drical subordinate Brownian motions.
We will drive explicit estimates about the uniform norm of the gradient for the
semigroup (P ) corresponding to linear evolution equations driven by additive
t >0
cylindrical subordinate Brownian motions. Recall that the uniform norm of the
gradient for the semigroup (P ) is defined by
t t>0
Pt ∞ := sup zPtg(x) : z 6 1,x H,g Bb(H) with g H,∞ 6 1 ,
k∇ k |∇ | k k ∈ ∈ k k
(cid:26) (cid:27)
where
1
P g(x) := limsup P g(x+εz) P g(x) .
z t t t
|∇ | ε| − |
ε→0