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TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS

SHUN-XIANG OUYANG AND MICHAEL RÖCKNER

ABSTRACT. A time inhomogeneous generalized Mehler semigroup on a real separable Hilbert space H is defined through

ps,tf(x) =

∫ H f(U(t, s)x+ y)µt,s(dy), s, t ∈ R, t ≥ s, x ∈ H,

for every bounded measurable function f on H, where (U(t, s))t≥s is an evolution family of bounded operators on H and (µt,s)t≥s is a family of probability measures on (H,B(H)) satisfying the following time inhomogeneous skew convolution equations

µt,s = µt,r ∗ ( µr,s ◦ U(t, r)−1

) , t ≥ r ≥ s.

This kind of semigroups typically arise as the “transition semigroups” of non-autonomous (possibly non-continuous) Ornstein-Uhlenbeck processes driven by some proper additive process. Suppose that µt,s converges weakly to δ0 as t ↓ s or s ↑ t. We show that µt,s has further weak continuity properties in t and s. As a consequence, we prove that for every t ≥ s, µt,s is infinitely divisible. Natural stochastic processes associated with (µt,s)t≥s are constructed and are applied to get probabilistic proofs for the weak continuity and infinite divisibility. Then we analyze the structure, existence and uniqueness of the corresponding evolution systems of measures (=space-time invariant measures) of (ps,t)t≥s. We also estab- lish a dimension free Harnack inequality for (ps,t)t≥s and present some of its applications.

CONTENTS

1. Introduction 2 2. Generalized Mehler semigroups and skew convolution equations 6 3. Time inhomogeneous skew convolution equations 8 3.1. Preliminaries and motivations 8 3.2. Weak continuity 12 3.3. Infinite divisibility 17 3.4. Associated stochastic processes 22 4. Evolution systems of measures 27 4.1. Some properties 28 4.2. Existence and uniqueness 31 5. Harnack inequalities and applications 36

Date: January 3, 2015. 2010 Mathematics Subject Classification. 60J75, 47D07. Key words and phrases. Time inhomogeneous generalized Mehler semigroups, skew convolution equa-

tions, infinite divisibility, evolution system of measures, Harnack inequality. Supported in part by the DFG through SFB–701 and IRTG–1132. The support of Isaac Newton Institute

for Mathematical Sciences in Cambridge is also gratefully acknowledged where part of this work was done during the special semester on “Stochastic Partial Differential Equations”.

1

2 SHUN-XIANG OUYANG AND MICHAEL RÖCKNER

6. Appendix: null controllability 43 References 44

1. INTRODUCTION

We start the introduction of time inhomogeneous generalized Mehler semigroups and skew convolution equations from the well studied time homogeneous case.

Let H be a real separable Hilbert space with norm and inner product denoted by | · | and 〈·, ·〉 respectively. Let B(H) be the space of Borel measurable subsets of H, and let Bb(H) be the space of all bounded Borel measurable functions on H.

A time homogeneous generalized Mehler semigroup (pt)t≥0 on H is defined by

ptf(x) =

∫ H f(Ttx+ y)µt(dy), t ≥ 0, x ∈ H, f ∈ Bb(H). (1.1)

Here (Tt)t≥0 is a strongly continuous semigroup on H and (µt)t≥0 is a family of probability measures on (H,B(H)) satisfying the following skew convolution (semigroup) equation

µt+s = µs ∗ (µt ◦ T−1s ), s, t ≥ 0. (1.2)

Recall that for any two positive Borel measures µ and ν on H, the convolution µ ∗ ν of µ and ν is a Borel measure on (H,B(H)) such that

µ ∗ ν(B) := ∫ H

∫ H

1B(x+ y)µ(dx)ν(dy) = ∫ H µ(B − x) ν(dx), B ∈ B(H).

Condition (1.2) is necessary and sufficient for the semigroup property of (pt)t≥0 (and the Markov property of the corresponding stochastic process respectively) to hold. That is (see [22]),

(1.2) holds if and only if for all t, s ≥ 0, ptps = pt+s on Bb(H). The semigroup (1.1) is a generalization of the classical Mehler formula for the transi-

tion semigroup of an Ornstein-Uhlenbeck process driven by a Wiener process. The second named author and his coauthors studied this generalization for the Gaussian case in [8, 9] as well as the non-Gaussian case in [22]. Indeed, under some mild conditions there is a one- to-one correspondence between generalized Mehler semigroups and transition semigroups of Ornstein-Uhlenbeck processes driven by Lévy processes (cf. [9, 22]).

Generalized Mehler semigroups and skew convolution equations have been extensively studied. For instance, Schmuland and Sun [51] investigated the infinite divisibility of µt (t ≥ 0) and the continuity of t 7→ log µ̂t (here µ̂t denotes the Fourier transformation of µt, see also (2.11)); Lescot and Röckner considered in [33] and [34] the generator and perturbations of (pt)t≥0 respectively; Wang and Röckner established some useful functional inequalities for (pt)t≥0; van Neerven [53], Li and his coauthors [18, 19] studied the representation of µ̂t (t ≥ 0). For more literature on this topic we refer to [2, 3, 4, 30, 36] and the references therein.

Recently, much work, for instance [13, 14, 23, 32, 57], has been devoted to the study of non-autonomous Ornstein-Uhlenbeck processes which are solutions to linear stochastic partial differential equations (SPDE) with time-dependent drifts. The noise in these equa- tions is modeled by a stationary process such as a Wiener process or Lévy process. To get

MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS 3

fully inhomogeneous Ornstein-Uhlenbeck processes, it is natural to consider more general noise modeled by (time) non-stationary processes such as additive processes.

Let (A(t),D(A(t)))t∈R be a family of linear operators on the space H with dense do- mains. Suppose that the non-autonomous Cauchy problem{

dxt = A(t)xtdt, t ≥ s, xs = x

is well posed (cf. [44]). That is, there exists an evolution family of bounded operators (U(t, s))t≥s on H associated with A(t) such that for every x ∈ D(A(s)), x(t) = U(t, s)x is a unique classical solution of this Cauchy problem.

Typical examples forA(t), t ∈ R, are partial differential operators e.g. of divergence type on H = L2(U) with U an open bounded domain in Rd. So, for example

A(t) = d∑

i,j=1

∂

∂xi

( aij(x, t)

∂

∂xj

) ,

where ai,j : U × R→ R are bounded measurable functions such that for some c ∈ (0,∞),

d∑ i,j=1

aij(x, t)ξiξj ≥ c|ξ|2Rd , ξ ∈ R d.

For details we refer to [44] and [32]. A family of bounded linear operators (U(t, s))t≥s on H is said to be a (strongly continu-

ous) evolution family if:

(1) For every s ∈ R, U(s, s) is the identity operator and for all t ≥ r ≥ s,

U(t, r)U(r, s) = U(t, s).

(2) For every x ∈ H, the map (t, s) 7→ U(t, s)x is strongly continuous on {(t, s) ∈ R2 : t ≥ s}.

An evolution family is also called evolution system, propagator etc.. For more details we refer e.g. to [17, 21, 44].

Let (Zt)t∈R be an additive process taking values in H, i.e. an H-valued stochastically continuous stochastic process with independent increments. We shall consider stochastic integrals

∫ t s

Φ(r)dZr on [s, t] of non-random functions Φ(t), t ∈ R, which takes values in the space of all linear operators on H.

A direct way to define the stochastic integral is to regard it as the limit (e.g. in the sense of convergence in probability) of “Riemann sums”. We refer to [38] to get some idea. For a more general definition (using so called independently scattered random measures), we refer to [46] for the infinite dimensional case and [48, 49] for the finite dimensional case. A work [41] is in preparation to extend the main results in [48, 49] by the first named author.

Consider the following stochastic differential equation{ dXt = A(t)Xtdt+ dZt,

Xs = x. (1.3)

4 SHUN-XIANG OUYANG AND MICHAEL RÖCKNER

Suppose that U(t, ·) is integrable on [s, t] for every s ≤ t with respect to Zt. Hence the following stochastic convolution integrals

Xt,s :=

∫ t s

U(t, σ) dZσ, t ≥ s, (1.4)

are well defined. Then

X(t, s, x) = U(t, s)x+

∫ t s

U(t, r) dZr, t ≥ s, x ∈ H (1.5)

is called the mild solution of (1.3). For all t ≥ s, let µ̄t,s denote the distribution of Xt,s. Obviously the transition semigroup of X(t, s, x) is given by

Ps,tf(x) = Ef(X(t, s, x)) = ∫ H f (U(t, s)x+ y) µ̄t,s(dy) (1.6)

for all x ∈ H, f ∈ Bb(H) and t ≥ s. It is easy to check that (µ̄t,s)t≥s satisfies the following time inhomogeneous skew convolution equation

µ̄t,s = µ̄t,r ∗ ( µ̄r,s ◦ U(t, r)−1

) , t ≥ r ≥ s

The aim of the present paper is to adopt the axiomatic approach in [9] and [22] to study the non-autonomous process (1.5) through its transition semigroup (1.6). That is, we shall start from (1.6) and define for a given strongly continuous evolution family (U(t, s))t≥s,

ps,tf(x) :=

∫ H f(U(t, s)x+ y)µt,s(dy), x ∈ H, f ∈ Bb(H), t ≥ s. (1.7)

Here (µt,s)t≥s is a family of probability measures on (H,B(H)). In order that the family (ps,t)t≥s satisfies the Chapman-Kolmogorov equations (flow property), we shall assume that (µt,s)t≥s satisfies the flowing time inhomogeneous skew convolution equations

µt,s = µt,r ∗ ( µr,s ◦ U(t, r)−1

) , t ≥ r ≥ s. (1.8)

In this case we call (ps,t)t≥s a time inhomogeneous generalized Mehler semigroup. Clearly, (1.7) and (1.8) are time inhom