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TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS ANDSKEW CONVOLUTION EQUATIONS

SHUN-XIANG OUYANG AND MICHAEL ROCKNER

ABSTRACT A time inhomogeneous generalized Mehler semigroup on a real separableHilbert space H is defined through

pstf(x) =

intHf(U(t s)x+ y)microts(dy) s t isin R t ge s x isin H

for every bounded measurable function f on H where (U(t s))tges is an evolution familyof bounded operators on H and (microts)tges is a family of probability measures on (HB(H))

satisfying the following time inhomogeneous skew convolution equations

microts = microtr lowast(micrors U(t r)minus1

) t ge r ge s

This kind of semigroups typically arise as the ldquotransition semigroupsrdquo of non-autonomous(possibly non-continuous) Ornstein-Uhlenbeck processes driven by some proper additiveprocess Suppose that microts converges weakly to δ0 as t darr s or s uarr t We show that microts hasfurther weak continuity properties in t and s As a consequence we prove that for everyt ge s microts is infinitely divisible Natural stochastic processes associated with (microts)tges areconstructed and are applied to get probabilistic proofs for the weak continuity and infinitedivisibility Then we analyze the structure existence and uniqueness of the correspondingevolution systems of measures (=space-time invariant measures) of (pst)tges We also estab-lish a dimension free Harnack inequality for (pst)tges and present some of its applications

CONTENTS

1 Introduction 22 Generalized Mehler semigroups and skew convolution equations 63 Time inhomogeneous skew convolution equations 831 Preliminaries and motivations 832 Weak continuity 1233 Infinite divisibility 1734 Associated stochastic processes 224 Evolution systems of measures 2741 Some properties 2842 Existence and uniqueness 315 Harnack inequalities and applications 36

Date January 3 20152010 Mathematics Subject Classification 60J75 47D07Key words and phrases Time inhomogeneous generalized Mehler semigroups skew convolution equa-

tions infinite divisibility evolution system of measures Harnack inequalitySupported in part by the DFG through SFBndash701 and IRTGndash1132 The support of Isaac Newton Institute

for Mathematical Sciences in Cambridge is also gratefully acknowledged where part of this work was doneduring the special semester on ldquoStochastic Partial Differential Equationsrdquo

1

2 SHUN-XIANG OUYANG AND MICHAEL ROCKNER

6 Appendix null controllability 43References 44

1 INTRODUCTION

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

Let H be a real separable Hilbert space with norm and inner product denoted by | middot | and〈middot middot〉 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 HA time homogeneous generalized Mehler semigroup (pt)tge0 on H is defined by

ptf(x) =

intHf(Ttx+ y)microt(dy) t ge 0 x isin H f isin Bb(H) (11)

Here (Tt)tge0 is a strongly continuous semigroup on H and (microt)tge0 is a family of probabilitymeasures on (HB(H)) satisfying the following skew convolution (semigroup) equation

microt+s = micros lowast (microt Tminus1s ) s t ge 0 (12)

Recall that for any two positive Borel measures micro and ν on H the convolution micro lowast ν of microand ν is a Borel measure on (HB(H)) such that

micro lowast ν(B) =

intH

1B(x+ y)micro(dx)ν(dy) =

intHmicro(B minus x) ν(dx) B isin B(H)

Condition (12) is necessary and sufficient for the semigroup property of (pt)tge0 (and theMarkov property of the corresponding stochastic process respectively) to hold That is (see[22])

(12) holds if and only if for all t s ge 0 ptps = pt+s on Bb(H)

The semigroup (11) is a generalization of the classical Mehler formula for the transi-tion semigroup of an Ornstein-Uhlenbeck process driven by a Wiener process The secondnamed author and his coauthors studied this generalization for the Gaussian case in [8 9] aswell 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 semigroupsof Ornstein-Uhlenbeck processes driven by Levy processes (cf [9 22])

Generalized Mehler semigroups and skew convolution equations have been extensivelystudied For instance Schmuland and Sun [51] investigated the infinite divisibility of microt (t ge0) and the continuity of t 7rarr log microt (here microt denotes the Fourier transformation of microt seealso (211)) Lescot and Rockner considered in [33] and [34] the generator and perturbationsof (pt)tge0 respectively Wang and Rockner established some useful functional inequalitiesfor (pt)tge0 van Neerven [53] Li and his coauthors [18 19] studied the representation ofmicrot (t ge 0) For more literature on this topic we refer to [2 3 4 30 36] and the referencestherein

Recently much work for instance [13 14 23 32 57] has been devoted to the studyof non-autonomous Ornstein-Uhlenbeck processes which are solutions to linear stochasticpartial 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 Levy process To get

MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS 3

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

Let (A(t)D(A(t)))tisinR 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 ge s

xs = x

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

Typical examples forA(t) t isin R are partial differential operators eg of divergence typeon H = L2(U) with U an open bounded domain in Rd So for example

A(t) =dsum

ij=1

part

partxi

(aij(x t)

part

partxj

)

where aij U times Rrarr R are bounded measurable functions such that for some c isin (0infin)

dsumij=1

aij(x t)ξiξj ge c|ξ|2Rd ξ isin Rd

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

ous) evolution family if

(1) For every s isin R U(s s) is the identity operator and for all t ge r ge s

U(t r)U(r s) = U(t s)

(2) For every x isin H the map (t s) 7rarr U(t s)x is strongly continuous on (t s) isinR2 t ge s

An evolution family is also called evolution system propagator etc For more details werefer eg to [17 21 44]

Let (Zt)tisinR be an additive process taking values in H ie an H-valued stochasticallycontinuous stochastic process with independent increments We shall consider stochasticintegrals

int ts

Φ(r)dZr on [s t] of non-random functions Φ(t) t isin R which takes values inthe space of all linear operators on H

A direct way to define the stochastic integral is to regard it as the limit (eg in the senseof convergence in probability) of ldquoRiemann sumsrdquo We refer to [38] to get some idea Fora more general definition (using so called independently scattered random measures) werefer to [46] for the infinite dimensional case and [48 49] for the finite dimensional case Awork [41] is in preparation to extend the main results in [48 49] by the first named author

Consider the following stochastic differential equationdXt = A(t)Xtdt+ dZt

Xs = x(13)

Suppose that U(t middot) is integrable on [s t] for every s le t with respect to Zt Hence thefollowing stochastic convolution integrals

Xts =

int t

s

U(t σ) dZσ t ge s (14)

are well defined Then

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

int t

s

U(t r) dZr t ge s x isin H (15)

is called the mild solution of (13) For all t ge s let microts denote the distribution of XtsObviously the transition semigroup of X(t s x) is given by

Pstf(x) = Ef(X(t s x)) =

intHf (U(t s)x+ y) microts(dy) (16)

for all x isin H f isin Bb(H) and t ge s It is easy to check that (microts)tges satisfies the followingtime inhomogeneous skew convolution equation

) t ge r ge s

The aim of the present paper is to adopt the axiomatic approach in [9] and [22] to studythe non-autonomous process (15) through its transition semigroup (16) That is we shallstart from (16) and define for a given strongly continuous evolution family (U(t s))tges

pstf(x) =

intHf(U(t s)x+ y)microts(dy) x isin H f isin Bb(H) t ge s (17)

Here (microts)tges is a family of probability measures on (HB(H)) In order that the family(pst)tges satisfies the Chapman-Kolmogorov equations (flow property) we shall assume that(microts)tges satisfies the flowing time inhomogeneous skew convolution equations

) t ge r ge s (18)

In this case we call (pst)tges a time inhomogeneous generalized Mehler semigroup Clearly(17) and (18) are time inhomogeneous analogs of (11) and (12)

We would like to point out that (pst)tges is of course not a ldquotransition semigrouprdquo althoughwe call it (time inhomogeneous generalized Mehler) semigroup In this two parameter casesome authors call it hemigroup see for example [6 25] and [10 27] where (17) and (18)have been studied respectively in some special or different situations So one may also call(17) a (time inhomogeneous) generalized Mehler hemigroup Similarly one may call (18)a (time inhomogeneous) skew convolution hemigroup

As a toy example one may consider the following time inhomogeneous one dimensionalstochastic equation

dxt = a(t)xtdt+ dbt

xs = x(19)

where (bt)tisinR is a standard one-dimensional Brownian motion in R and a(t) is a continuousfunction on R Then (exp(

int tsa(r) dr))tges is the evolution family associated with a(t) The

mild solution of (19) is given by

x(t s x) = exp

(int t

s

a(r) dr

)x+

int t

s

exp

(int t

u

a(r) dr

)dbu t ge s x isin H (110)

Clearly int t

s

exp

(int t

u

a(r) dr

)dbu sim N(0 σts)

ie a Gaussian measure with mean 0 and variance σts given by

σts =

int t

s

exp

(2

int t

u

a(r) dr

)d u

It is easy to check that (cf (352))

σts = σtr + σrs middot exp

(2

int t

r

a(r) dr

) t ge r ge s

This means that (18) holds The corresponding time inhomogeneous generalized Mehlersemigroup is given by

pstf(x) =

intHf

(exp

(int t

s

a(r) dr

)x+ y

)N(0 σts)(dy) x isin H f isin Bb(H) t ge s

For further examples we refer to Section 5 where we consider time inhomogeneousSPDEs driven by Wiener processes in infinite dimension (in particular see Example 53)For corresponding examples where the Wiener process is replaced by a Levy process werefer eg to [32]

The present paper is organized as followsIn Section 2 the main result is Proposition 21 which shows that the flow property for

(pst)tges holds if and only if we have (18) for (microts)tgesIn Section 3 we concentrate on the skew convolution equations (18) In Subsection 31

we give some preliminaries and motivations In Subsection 32 we introduce Assumption37 and show some results on the weak continuity of microts in t and s In Subsection 33we prove that for every t ge s microts is infinitely divisible In Subsection 34 we show thatthere exists a natural stochastic process associated with (microts)tges This allows us to getprobabilistic proofs of the results on the weak continuity and infinite divisibility of microtsproved in previous subsections We shall study the spectral representation of microts in anotherwork

In Section 4 we study evolution systems of measures (ie space-time invariant measures)for the semigroup (pst)tges We first show some basic properties of the evolution systems ofmeasures Then we give sufficient and necessary conditions for the existence and unique-ness of evolution systems of measures

In Section 5 we prove a (dimension independent) Harnack inequality for (pst)tges usinga simple argument As applications of the Harnack inequality we prove that null controlla-bility implies the strong Feller property and that for the Gaussian case null controllabilityHarnack inequality and strong Feller property are in fact equivalent to each other as in thetime homogeneous case

The semigroup (pst)tges is called strongly Feller if for every bounded measurable functionf and every t ge s pstf is a bounded continuous function In the time homogeneous casethe strong Feller property has been investigated frequently One of the reasons is that if atransition semigroup is strong Feller and irreducible then it has a unique invariant measuresee eg [11]

As an application of the Harnack inequality we eg look at the hypercontractivity of(pst)tges Hypercontractivity of semigroup such as (16) is closely related to functionalinequalities spectral theory etc for the Kolmogorov operator (or generator) correspondingto SDE (13) see eg [54]

In Section 6 we append a brief introduction to the control theory of non-autonomouslinear control systems and null controllability The minimal energy representation is usefulfor the estimate of the constant in the Harnack inequality obtained in Section 5

2 GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS

First we characterize when the Chapman-Kolmogorov equations hold for (pst)tges in(17)

Proposition 21 For all s le r le t

psrprt = pst (ldquoChapman-Kolmogorov equationsrdquo) (21)

holds on Bb(H) if and only if (18) holds for all s le r le t

We shall prove this proposition in a more general framework This is inspired by thefact that a generalized Mehler semigroup is a special case of the so called skew convolutionsemigroups (see [35])

Let (ust)tges be a family of Borel Markov transition functions on H ie each ust is aprobability kernel on B(H) and the following Chapman-Kolmogorov equations

ustf(x) = usr(urtf)(x) x isin H f isin Bb(H) (22)

hold for all t ge r ge s Here Bb(H) denote the space of all bounded Borel measurablefunctions on H Writing (22) in integral form we haveint

H2

f(z)urt(y dz)usr(x dy) =

intHf(z)ust(x dz) (23)

Assume thatust(x+ y middot) = ust(x middot) lowast ust(y middot) (24)

for every t ge s and x y isin H Clearly Equation (24) implies

ust(0 middot) = δ0 (25)

For every probability measure micro on (HB(H)) we associate with ust (t ge s) a newprobability measure microust by

microust(A) =

intHust(xA)micro(dx) A isin B(H)

for every t ge sLet (microts)tges be a family of probability measures on (HB(H)) For all t ge s define a

family of functionsqst(middot middot) HtimesB(H)rarr R

byqst(x middot) = ust(x middot) lowast microts(middot) x isin H

Associated with qst(middot middot) we define an operator qst on B(H) by

qstf(x) =

intHf(y) qst(x dy) x isin H f isin Bb(H)

We have the following characterization of the Chapman-Kolmogorov equations for (qst)tges

Proposition 22 The family of operators (qst)tges satisfies

qst = qsrqrt t ge r ge s (26)

if and only ifmicrots = microtr lowast (microrsurt) t ge r ge s (27)

Proof For every f isin Bb(H) x isin H we have

qsrqrtf(x)

=

intHqrtf(y)qsr(x dy)

=

intH2

qrtf(y1 + y2)usr(x dy1)micrors(dy2)

=

intH4

f(z)qrt(y1 + y2 dz)usr(x dy1)micrors(dy2)

=

intH4

f(z1 + z2)urt(y1 + y2 dz1)microtr(dz2)usr(x dy1)micrors(dy2)

=

intH5

f(z11 + z12 + z2)urt(y1 dz11)urt(y2 dz12)microtr(dz2)usr(x dy1)micrors(dy2)

=

intH4

f(z11 + z12 + z2)ust(x dz11)urt(y2 dz12)microtr(dz2)micrors(dy2)

=

intH3

f(z11 + z12 + z2)ust(x dz11)(microrsurt)(dz12)microtr(dz2)

=

intHf(z)(ust(x middot) lowast (microrsurt) lowast microtr)(dz)

(28)

In the calculation above we have used (23) and (24) to get the fifth and sixth identityrespectively If (27) holds then by (28) we obtain

qsrqrtf(x) =

intHf(z)[ust(x middot) lowast microts](dz) = qstf(x)

That is (26) holdsConversely if (26) holds then we have

qstf(0) = qsrqrtf(0) (29)

for all f isin Bb(H) By taking x = 0 in (28) and using (25) we get

qsrqrtf(0) =

intHf(z) ((microrsurt) lowast microtr) (dz)

On the other hand we have

qstf(0) =

intHf(z)microts(dz)

Hence intHf(z) ((microrsurt) lowast microtr) (dz) =

intHf(z)microts(dz)

for all f isin Bb(H) This implies (27) So the proof is complete

Proof of Proposition 21 Letust(x middot) = δU(ts)x(middot)

for every t ge s and x isin H Then by property (1) of the evolution family (U(t s))tges(ust)tges is a Markov transition function satisfying (22) Note that we can rewrite the inho-mogeneous generalized Mehler semigroup (17) as

pstf(x) = (δU(ts)x lowast microts)f x isin H f isin Bb(H) (210)

Hence it is clear that (qst)tges coincides with (pst)tges Therefore the equivalence of (26)and (27) in Proposition 22 is exactly the equivalence of (21) and (18) in Proposition 21The latter is thus proved

For a linear bounded operator U on H let Ulowast denote its adjoint Clearly Ulowast is alsobounded Let micro denote the Fourier transform (or the characteristic functional) of a proba-bility measure micro on H ie

micro(ξ) =

intH

ei〈xξ〉 micro(dx) ξ isin H (211)

Probability measures on Hilbert spaces are determined by their characteristic functionals(see eg [52 Section IV22 Theorem 22]) In particular (18) holds if and only if

microts(ξ) = microtr(ξ)micrors(U(t r)lowastξ) ξ isin H (212)

3 TIME INHOMOGENEOUS SKEW CONVOLUTION EQUATIONS

In this section we concentrate on (18) We shall study the weak continuity infinitedivisibility and stochastic process associated with (microts)tges

31 Preliminaries and motivations In this subsection we fix some notations and presentsome basic results and motivations In particular we give some results on the weak conver-gence of measures satisfying convolution equations such as (18)

Convergence of probability measures We recall that a sequence of probability mea-sures (micron)nge1 on H converges weakly to a probability measure micro on H written as

micron rArr micro as nrarrinfin

if for every f isin Cb(H)

limnrarrinfin

intHf(x)micron(dx) =

intHf(x)micro(dx)

Here Cb(H) denotes the space of all bounded continuous functions on H Sometimes wealso write

limnrarrinfin

micron = micro

A sequence of H-valued random variables (Xn)nge1 converges stochastically or convergesin probability to an H-valued random variable X written as

XnPrminusrarr X

if for each ε gt 0limnrarrinfin

P(|Xn minusX| gt ε) = 0

Let micron and micro denote the distributions ofXn andX respectively Then it is well known thatas nrarrinfin Xn

Prminusrarr X implies micron rArr micro (in other words Xn converges to X in distribution)On the other hand if in particular X = x isin H is deterministic then micron rArr δx impliesXn

Prminusrarr X Therefore we have XnPrminusrarr x if and only if micron rArr δx

Additive processes Levy processes and convolution equations Let (Xt)tisinR be a sto-chastic process taking values in H Assume that X0 = 0 The process (Xt)tisinR is called anadditive process if it has independent increments ie if for any t gt s Xt minusXs is indepen-dent of σ(Xr r le s) Let microts denote the distribution of Xt minusXs For all t ge r ge s wehave

Xt minusXs = (Xt minusXr) + (Xr minusXs)

This impliesmicrots = microtr lowast micrors t ge r ge s (31)

since Xt minusXr Xr minusXs are independentUsually one requires that an additive process is stochastically continuous ie for every

t isin R and ε gt 0limhrarr0

P(|Xt+h minusXt| ge ε) = 0 (32)

This condition means thatmicrots rArr δ0 as s uarr tmicrots rArr δ0 as t darr s

(33)

If in addition (Xt)tisinR has stationary increments ie if for any t gt s the distribution ofXt minusXs only depends on t minus s then it is called a Levy process In this case we shall onlyconsider Xt for t ge 0 For each t ge 0 let microt denote the distribution of Xt Then obviouslywe have the following convolution equations

microt+s = microt lowast micros t s ge 0 (34)

The stochastic continuity condition (32) is reduced to

limtdarr0

P(|Xt| ge ε) = 0 (35)

This is equivalent tomicrot rArr δ0 as t darr 0

Infinitely divisible probability measures A probability measure micro on (HB(H)) issaid to be infinitely divisible if for any n isin N there exists a probability measure micron on(HB(H)) such that

micro = microlowastnn = micron lowast micron lowast middot middot middot lowast micron︸︷︷︸n times

Let (microt)tge0 be a family of probability measures If it satisfies (34) then obviously forevery t ge 0 microt is infinitely divisible If (microt)tge0 satisfies the skew convolution equations(12)

microt+s = micros lowast (microt Tminus1s ) s t ge 0

then it is proved in [51] that for every t ge 0 microt is also infinitely divisibleNow we look at the two-parameter convolution equations (31) First we consider the

finite dimensional case when H = Rd It is known (see [29] or [47 Theorem 91 andTheorem 97]) that if (microts)tges satisfies (33) then for any t ge s microts is infinitely divisibleThe idea of the proof given in [47] can be described as follows

First one shows that (microts)tges is locally uniformly weakly continuous by using (33) Thatis for every ε gt 0 and η gt 0 there is δ gt 0 such that for all s and t in [s0 t0] satisfying0 le tminus s le δ we have (cf Lemma 38)

microts(|x| gt ε) lt η

Then by the celebrated Kolmogorov-Khintchine limit theorems on sums of independentrandom variables (see [47 Theorem 93]) we obtain that microt0s0 is infinitely divisible

The uniform weak continuity of (microts)tges on [s0 t0] can be proved by constructing astochastically continuous additive process (Xt)tisinR such that for any t ge s the incrementXt minusXs has the distribution microts (see [47 Theorem 97 (ii)] and [47 Lemma 96])

In Subsection 33 we shall modify the arguments above to study the infinite divisibilityof (microts)tges satisfying equation (18)

Weak convergence of measures satisfying convolution equations The results in thispart will be used in Section 4 First of all we include here two results from [43] Recall thata set M of probability measures on H is said to be shift (relatively) compact if for everysequence (micron)nge1 in M there is a sequence (νn)nge1 such that

(1) (νn)nge1 is a translate of (micron)nge1 That is there exists a sequence (xn)nge1 in H suchthat νn = micron lowast δxn for all n ge 1

(2) (νn)nge1 has a convergent subsequence

Theorem 31 Let (σn)nge1 (micron)nge1 and (νn)nge1 be three sequences of measures on H suchthat σn = micron lowast νn for all n isin N

(1) ([43 Theorem III21]) If the sequences (σn)nge1 and (micron)nge1 both are relativelycompact then so is the sequence (νn)nge1

(2) ([43 Theorem III22]) If the sequence (σn)nge1 is relatively compact then the se-quences (micron)nge1 and (νn)nge1 are shift compact respectively

Now we can show the following lemma

Lemma 32 Let micron νn σn with n ge 1 and micro ν σ be measures on (HB(H)) such thatσn = micron lowast νn

(1) If micron rArr micro and νn rArr ν as nrarrinfin then σn rArr micro lowast ν as nrarrinfin(2) Suppose that σn rArr σ and micron rArr micro as n rarr infin Then there exists a probability

measure ν such thatσ = micro lowast ν (36)

If ν is the unique measure such that (36) holds then νn rArr ν as nrarrinfin

Proof The first conclusion says that the convolution operation preserves weak continuityThe proof can be found for example in [28 Proposition 23] or [43 Theorem III11]

Now we show the second assertion Take an arbitrary subsequence (νni)ige1 from (νn)nge1

and considerσni = microni lowast νni i ge 1

Since σn rArr σ and micron rArr micro as n rarr infin both (σni)ige1 and (microni)ige1 are relatively compactBy Theorem 31 the sequence (νni)ige1 is also relatively compact Let (νnprimei)ige1 be a weaklyconvergent subsequence of (νni)ige1 with limit ν prime Then by the first assertion of this theoremwe have

σnprimei = micronprimei lowast νnprimei rArr micro lowast ν prime nprimei rarrinfinSince we also have σnprimei rArr σ as nprimei rarr infin we get σ = micro lowast ν prime This shows that there existsa probability measure ν = ν prime such that (36) holds If there is only one measure ν suchthat (36) holds then the discussion above shows that any subsequence of (νn)nge1 containsa further subsequence converging weakly to ν This is sufficient to conclude that (νn)nge1

converges weakly to ν (cf [7 Theorem 26]) Hence the proof is complete

Remark 33 In the second part of the previous theorem the assumption that ν is the uniquesolution to the convolution equation σ = micro lowast ν amounts to saying that the following cancel-lation law for convolution operation holds Let ν ν prime be two measures on H if

micro lowast ν = micro lowast ν prime (37)

then ν = ν prime It is obvious that this cancellation law holds provided micro has no zeros Indeedfrom (37) we get microν = microν prime If micro 6= 0 then ν = ν prime So ν = ν prime It is well known that if micro isan infinitely divisible distribution then micro has no zeros

Remark 34 After the proof of the second assertion of Lemma 32 we found that there isa similar result in [26 Corollary 224] where the condition that micro has no zeros is used Ourproof is different The example after [37 Theorem 511] shows that there exist probabilitymeasures micro ν and ν prime on R with ν 6= ν prime such that (37) holds It is called Khintchine phe-nomenon in the literature So if (37) holds then it is necessary to require ν = ν prime for thesecond assertion of Lemma 32 Otherwise if ν 6= ν prime then (micro lowast νn)nge1 with ν2kminus1 = ν andν2k = ν prime for all k ge 1 converges weakly but (νn)nge1 does not converge weakly

As a summary of the discussion above we have the following result

Corollary 35 Let micron νn σn with n ge 1 and σ micro be measures on H with the followingproperties

(1) For all n ge 1 σn = micron lowast νn(2) As nrarrinfin σn rArr σ and micron rArr micro

If micro is an infinitely divisible distribution then the sequence (νn)nge1 converges weakly tosome measure ν on H such that σ = micro lowast ν as nrarrinfin

In particular we have the following result

Corollary 36 Let micron νn σn with n ge 1 and σ be measures on H Suppose that for alln ge 1 σn = micron lowast νn If σn rArr σ and micron rArr δ0 as nrarrinfin then νn rArr σ as nrarrinfin

32 Weak continuity We shall use the following assumption

Assumption 37 For all s isin R

limtdarrs

microts = limtuarrs

microst = δ0 (38)

Let us explain that the weak limit δ0 in (38) is natural Taking s = r = t in (18) weobtain

micrott = micrott lowast micrott t isin R (39)

That is for every t isin R micrott is an idempotent probability measure on (HB(H)) By[52 Section I43 Proposition 47 Page 67 see also Section IV22 Corollary 1 Page 203](or [43 Section III3 Theorem 31 Page 62] and noting that there is no nontrivial compactsubgroup in H) the trivial measure δ0 is the only idempotent measure on (HB(H)) Hence

micrott = δ0 (310)

Now Assumption 37 means that microts is weakly continuous on the diagonal (t s) (t s) isinR2 t = s in two directions We aim to show that combined with the skewed convolutionequation (18) this assumption in fact implies more about the weak continuity of microts in tand s

We need the following simple fact

Lemma 38 Let (micron)nge1 be a sequence of probability measures on H Then micron rArr δ0 asnrarrinfin if and only if for all ε gt 0

limnrarrinfin

micron(x isin H |x| gt ε) = 0 (311)

Proof Suppose that micron rArr δ0 as n rarr infin Then by the Portmanteau theorem (see forinstance [7 Theorem 21])

lim supnrarrinfin

micron(F ) le δ0(F )

for all closed sets F in H Obviously x isin H |x| ge ε is closed Hence

0 le lim supnrarrinfin

micron(x isin H |x| gt ε)

le lim supnrarrinfin

micron(x isin H |x| ge ε) le δ0(x isin H |x| ge ε) = 0

So (311) holdsNow we assume that (311) holds for all ε gt 0 Let f be a continuous bounded function

on H DefineM = sup |f |+ 1

We are going to show micron(f)rarr δ0(f) as nrarrinfinSince f is continuous for any η gt 0 there exists a constant ε0 gt 0 such that for all|x| le ε0

|f(x)minus f(0)| lt η

2 (312)

By (311) there exists a constant N gt 0 such that for all n gt N

micron(x isin H |x| gt ε0) ltη

4M (313)

Combining (312) and (313) we obtain∣∣∣∣intHf dmicron minus

intHf dδ0

∣∣∣∣leintH|f(x)minus f(0)| dmicron

=

intxisinH |x|geε0

|f(x)minus f(0)| dmicron +

intxisinH |x|ltε0

|f(x)minus f(0)| dmicron

le2Mmicron(x isin H |x| gt ε0) + sup|x|ltε0

|f(x)minus f(0)| middot micron(x isin H |x| le ε0)

lt2M middot η

4M+η

2= η

This completes the proof

By Lemma 38 we have the following equivalent description of Assumption 37

Proposition 39 Equation (38) is equivalent to

limtdarrs

microts(x isin H |x| gt ε) = limsuarrt

microts(x isin H |x| gt ε) = 0 (314)

for all ε gt 0 More precisely they are equivalent to the following two conditions for everyε η gt 0 and for every u isin R there exists a constant δu such that

(1) For every t isin (u u+ δu)

microtu(x isin H |x| gt ε) lt η (315)

(2) For every s isin (uminus δu u)

microus(x isin H |x| gt ε) lt η (316)

We shall use the following two lemmas

Lemma 310 For every s0 lt t0 there exists some constant c ge 1 such that for all s0 les le t le t0

|U(t s)x| le c|x| x isin H s0 le s le t le t0 (317)

Proof For every x isin H |U(t s)x| is a continuous function of (t s) on Λt0s0 = (t s) s0 les le t le t0 Hence |U(t s)x| is uniformly bounded on Λt0s0 for every x isin H By theBanach-Steinhaus theorem we have

sup(ts)isinΛt0s0

U(t s) ltinfin

That is there exists some c gt 0 such that (317) holds

Lemma 311 Let T be a bounded linear operator on H hence for all x isin H |Tx| le c|x|for some constant c gt 0 Let micro be a measure on H and ε gt 0 be any constant Then wehave

micro Tminus1(x isin H |x| gt ε) le micro(x isin H |x| gt εc) (318)

Proof By assumption we have

micro Tminus1(x isin H |x| gt ε) =micro(x isin H |Tx| gt ε)lemicro(x isin H |x| gt εc)

Now we can prove the following result

Theorem 312 Suppose that Assumption 37 and (18) hold for a family of probabilitymeasures (microts)tges Then

(1) For every t isin R the map s 7rarr microts with s le t is weakly continuous(2) For every t s isin R with t ge s we have

microt+εs rArr microts as ε darr 0 (319)

where ε darr 0 means ε ge 0 and εrarr 0(3) For every t0 t s isin R with t0 ge t gt s

microtminusεs U(t0 tminus ε)minus1 rArr microts U(t0 t)minus1 as ε darr 0 (320)

Proof (1) Let s lt t We need to show

microtsminusε rArr microts as ε darr 0 (321)

andmicrots+ε rArr microts as ε darr 0 (322)

Equation (18) implies that for every ε isin (0 tminus s)

microtsminusε = microts lowast(microssminusε U(t s)minus1

)(323)

andmicrots = microts+ε lowast

(micros+εs U(t s+ ε)minus1

) (324)

By Lemma 310 there exists some constant c ge 1 such that for all ε isin (0 tminus s) we have

U(t s+ ε) le c

Hence by Lemma 311 we have for all η gt 0

microssminusε U(t s)minus1(x isin H |x| gt η) le microssminusε(x isin H |x| gt ηc) (325)

Because microssminusε rArr δ0 as ε darr 0 by Lemma 38 we get

limεdarr0

microssminusε(x isin H |x| gt ηc) = 0

Hence it follows from (325) that

limεdarr0

microssminusε U(t s)minus1(x isin H |x| gt η) = 0

By Lemma 38 we obtain

microssminusε U(t s)minus1 rArr δ0 ε darr 0

Therefore applying the first result of Lemma 32 to (323) we get (321)By the same arguments it is easy to show that

micros+εs U(t s+ ε)minus1 rArr δ0 as ε darr 0

Then by Corollary 36 (322) follows from (324)

(2) According to (18) we have for all t ge s ε ge 0

microt+εs = microt+εt lowast (microts U(t+ ε t)minus1)

By assumption we have microt+εt rArr δ0 as ε darr 0 Hence by applying the first assertion ofLemma 32 we get (319) by proving

microts U(t+ ε t)minus1 rArr microts as ε darr 0 (326)

Now we show (326) Let f be a continuous and bounded function on H For every ε gt 0

setfε(x) = f (U(t+ ε t)x) x isin H

It is clear that fε converges to f pointwise as ε darr 0 Moreover since f is bounded we knowthat fε is bounded Hence by Lebesguersquos dominated convergence theorem we have

limεdarr0

intHf(x) dmicrots U(t+ ε t)minus1(x) = lim

εdarr0

intHfε(x) dmicrots(x) =

intHf(x) dmicrots(x)

This proves (326)

(3) We first show in particular the following result

microtminusεs U(t tminus ε)minus1 rArr microts as ε darr 0 (327)

By (18) we have for all t ge tminus ε gt s

microts = microttminusε lowast (microtminusεs U(t tminus ε)minus1)

Since microttminusε rArr δ0 as ε darr 0 by Corollary 36 we get (327)Now let us show (320) By (327) we have for any bounded continuous function f on H

limεdarr0

intHf(x) dmicrotminusεs U(t0 tminus ε)minus1(x)

= limεdarr0

intHf (U(t0 tminus ε)x) dmicrotminusεs(x)

= limεdarr0

intHf (U(t0 t)U(t tminus ε)x) dmicrotminusεs(x)

= limεdarr0

intHf (U(t0 t)y) dmicrotminusεs U(t tminus ε)minus1(y)

=

intHf (U(t0 t)y) dmicrots(y)

=

intHf(y) dmicrots U(t0 t)

minus1(y)

This proves (320)

Concerning the space-homogeneous case we have the following result

Theorem 313 Let (microts)tges be a family of probability measures on (HB(H)) satisfying

microts = microtr lowast micrors t ge r ge s (328)

andlimtdarrs

microts = limsuarrt

microts = δ0

Then microts is weakly continuous in t and s with t ge s

Proof Below we always assume that we have tprime ge sprime when we write microtprimesprime Let εn and δn betwo nonnegative sequences converging to 0 as nrarrinfin Recall that by Theorem 312

microtsplusmnδn rArr microts as nrarrinfin (329)

By (328) we havemicrotsplusmnδn = microttminusεn lowast microtminusεnsplusmnδn (330)

Note that as n rarr infin we have microttminusεn rArr δ0 by assumption Hence by (329) and byapplying Corollary 36 to Equation (330) we obtain

microtminusεnsplusmnδn rArr microts as nrarrinfin (331)

Now by (328) we also have

microt+εnsplusmnδn = microt+εnt lowast microtsplusmnδn (332)

By assumption we have microt+εnt rArr δ0 as nrarrinfin Hence by using the weak continuity ofthe convolution operator and by applying (329) to (332) we get

microt+εnsplusmnδn rArr microts as nrarrinfin (333)

Combining (331) and (333) we get

microtplusmnεnsplusmnδn rArr microts as nrarrinfin

and hence the proof is complete

Remark 314 We provide also probabilistic proofs of Theorem 312 and Theorem 313 inSubsection 34 below

For every t ge s it is clear that pst is Feller ie pst(Cb(H)) sub Cb(H) Now we look atthe continuity of the map (s x) 7rarr pstf(x) for every f in Cb(H) The proposition below isa direct generalization of [9 Lemma 21] The proof is similar to the proof in [9]

Proposition 315 Let sn tn isin R xn isin H sn le tn with n ge 1 such that (sn tn)rarr (s t) isinR2 and xn rarr x isin H as n rarr infin If microtnsn rArr microts as n rarr infin then for any f isin Cb(H)psntnf(xn)rarr pstf(x) as nrarrinfin

Proof Since microtnsn rArr microts as nrarrinfin by Prohorovrsquos theorem for every ε gt 0 there existsa compact set K sub H such that

microrσ(K) ge 1minus ε for all (r σ) isin (t s) (tn sn) n isin N (334)

For abbreviation we set zn = U(tn sn)xn and z = U(t s)x By the strong continuity ofthe evolution family (U(t s))tges the set Z = z zn n isin N is compact Hence Z +K isalso compact So there exists an N isin N such that for any n gt N and for any y isin K

|f(zn + y)minus f(z + y)| lt ε (335)

since f is uniformly continuous on compactsBecause microtnsn rArr microts as nrarrinfin (taking N larger if necessary) we have for all n gt N∣∣∣∣int

Hf(z + y)microtnsn(dy)minus

intHf(z + y)microts(dy)

∣∣∣∣ lt ε (336)

From (334) (335) and (336) we get∣∣∣∣intHf(zn + y)microtnsn(dy)minus

∣∣∣∣le∣∣∣∣int

∣∣∣∣+

intK

|f(zn + y)minus f(z + y)| microtnsn(dy) + 2finfinmicro(H K)

lt2ε(1 + finfin)

Hence the result is proved since ε was arbitrary

33 Infinite divisibility The main result of this section is the following theorem

Theorem 316 Suppose that Assumption 37 and (18) hold for (microts)tges Then for everyt ge s microts is infinitely divisible

We shall use a similar method as indicated in Subsection 31 The main difficulty isproving the following lemma ie showing (microts U(t0 t)

minus1)tges is uniformly weakly con-tinuous on [s0 t0] In this subsection we prove it analytically In Subsection 34 we presenta probabilistic proof of it by constructing an associated stochastically continuous additiveprocess

Lemma 317 Suppose that (microts)tges satisfies (18) and Assumption 37 Then on everycompact interval [s0 t0] for all ε η gt 0 there exists a constant δ gt 0 such that for alls t isin [s0 t0] with 0 le tminus s lt δ

microts U(t0 t)minus1(x isin H |x| gt ε) lt η (337)

Proof It is trivial to see that (337) holds for the case when t = s So we shall assumet gt s By Lemma 310 there exists a constant c ge 1 such that

|U(t s)x| le c|x| x isin H s0 le s lt t le t0 (338)

Let us setεprime = εc

andA(r) = x isin H |x| gt r r gt 0

By Assumption 37 and Equations (315) (316) in Proposition 39 for every ε η gt 0t isin [s0 t0] there exists a constant δt ge 0 such that

microts

(A

(εprime

2c

))lt η2 s isin (tminus δt t) (339)

and

micrort

(A

(εprime

2c

))lt η2 r isin (t t+ δt) (340)

Since c ge 1 we have εprime

2cle εprime

2 Hence from estimates (339) and (340) it follows that

microts (A (εprime2)) lt η2 s isin (tminus δt t) (341)

and

micrort (A (εprime2)) lt η2 r isin (t t+ δt) (342)

Moreover according to Lemma 311 and (338) from estimates (339) and (340) weobtain

microts U(tprime t)minus1 (A (εprime2)) lt η2 tminus δt le s le t t le tprime le t0 (343)

and

micrort U(rprime r)minus1 (A (εprime2)) lt η2 t le r le t+ δt r le rprime le t0 (344)

For every t isin [s0 t0] let

It = (tminus δt t+ δt)

Obviously It t isin [s0 t0] covers the interval [s0 t0] Hence there is a finite subndashcoveringItj j = 1 2 middot middot middot n of [s0 t0] Then for every t isin [s0 t0] we have t isin Itj for somej isin 1 2 middot middot middot n Let

δ = min

δtj2

j = 1 2 n

For every s isin [s0 t0] such that 0 lt tminus s lt δ we have

Therefore both t and s are in the same sub-interval Itj We need to consider the followingthree cases respectively 1 s le tj lt t 2 s lt t le tj 3 tj lt s lt t

Case 1 (s le tj lt t) Note that for all x y isin H if |x + y| gt εprime then either |x| gt εprime2

or |y| gt εprime2 That is the following inequality holds

1A(εprime)(x+ y) le 1A(εprime2)(x) + 1A(εprime2)(y) (345)

By (18) (342) (343) and (345) we have

microts(A(εprime)) = microttj lowast (microtj s U(t tj)minus1)(A(εprime))

=

intH

1A(εprime)(x+ y)microttj(dx)(microtj s U(t tj)minus1)(dy)

leintH

intH

(1A(εprime2)(x) + 1A(εprime2)(y))microttj(dx)(microtj s U(t tj)minus1)(dy)

= microttj(A(εprime2)) + (microtj s U(t tj)minus1)(A(εprime2))

ltη

2+η

2= η

Therefore by Lemma 311 and (338) we have

microts U(t0 t)minus1(x isin H |x| gt ε) lt η

Case 2 (s lt t le tj) We first show that

(microts U(tj t)minus1)(A(εprime)) lt η

by contradiction If otherwise the following inequality

(microts U(tj t)minus1)(A(εprime)) ge η (346)

holds Then by (18) (341) and (346) we obtainη

2gt microtj s(A(εprime2)) = microtj t lowast (microts U(tj t)

minus1)(A(εprime2))

=

intH

1A(εprime2)(x+ y)microtj t(dx)(microts U(tj t)minus1)(dy)

geintH

intH

1A(εprime2)c(x) middot 1A(εprime)(y) microtj t(dx)(microts U(tj t)minus1)(dy)

= microtj t(A(εprime2)c) middot (microts U(tj t)minus1)(A(εprime))

ge η(

1minus η

2

)= η minus η2

2

Here we have used the fact that for all x y isin H |x + y| ge |y| minus |x| gt εprime

2if |y| gt εprime and

|x| le εprime2 Now we haveη

2gt η minus η2

2

consequently η gt 1 By (346) this means that

(microts U(tj t)minus1)(A(εprime)) gt 1

This is impossible since microts is a probability measureThen by Lemma 311 and (338) we have

microts U(t0 t)minus1(x isin H |x| gt ε)

=(microts U(tj t)

minus1) U(t0 tj)

minus1(x isin H |x| gt ε)lemicrots U(tj t)

minus1(x isin H |x| gt εc)=microts U(tj t)

minus1(A(εprime)) lt η

Case 3 (tj lt s lt t) Similar to Case 1 we only need to show microts(A(εprime)) lt η whoseproof turns out to be similar to the proof in Case 2 Indeed if

microts(A(εprime)) ge η (347)

then by (18) (342) and (344)η

2gt microttj(A(εprime2)) = microts lowast (microstj U(t s)minus1)(A(εprime2))

=

intH

1A(εprime2)(x+ y)microts(dx)(microstj U(t s)minus1)(dy)

ge microts(A(εprime)) middot (microstj U(t s)minus1)(A(εprime2)c)

ge η(

1minus η

2

)

This implies η gt 1 which contradicts (347) because microts is a probability measureCombining the three cases discussed above we obtain (337) and hence the proof is

complete

Now we are ready to prove Theorem 316

Proof of Theorem 316 For simplicity we only show that micro10 is infinitely divisible Theproof for the case microts with arbitrary t ge s is similar

First of all we shall show by induction that for every m isin N

micro10 =

2mminus1prodlowast

j=0

micro j+12m

j2m U

(1j + 1

2m

)minus1

(348)

Hereprodlowast denotes the convolution product

By (18) we have

micro10 = micro1 12lowast

(micro 1

20 U

(1

1

2

)minus1)

So (348) holds for m = 1 Now we assume that (348) holds for some m ge 1 Then by(18) we have for all j = 0 1 middot middot middot 2m minus 1

micro j+12m

j2m

= micro (2j+1)+1

2m+1 2j+1

2m+1lowast

(micro 2j+1

2m+1 2j

2m+1 U

((2j + 1) + 1

2m+12j + 1

2m+1

)minus1)

For any probability measures micro ν on H and measurable map T on H it is easy to checkthat

(micro lowast ν) Tminus1 = (micro Tminus1) lowast (ν Tminus1) (349)

So

micro j+12m

j2m U

(1j + 1

2m

)minus1

=micro (2j+1)+1

2m+1 2j+1

2m+1 U

(1

(2j + 1) + 1

2m+1

)minus1

lowast

(micro 2j+1

2m+1 2j

2m+1 U

(1

2j + 1

2m+1

)minus1)

=

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

Therefore by assumption we have

micro10 =

2mminus1prodlowast

j=0

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

=

2m+1minus1prodlowast

k=0

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

This proves (348) for all m ge 1By (348) and Lemma 317 micro10 is the limit of an infinitesimal triangular array Hence

micro10 is infinitely divisible according to [43 Corollary VI62]

Now we assume that for every t ge s the measure microts is infinitely divisible Then by theLevy-Khintchine theorem [43 Theorem VI410] there exists a negative definite Sazonovcontinuous function ψts on H such that

microts(ξ) = exp(minusψts(ξ)) ξ isin H

and ψts has the following form

ψts(ξ) = minusi〈ats ξ〉+1

2〈ξ Rtsξ〉

minusintH

(ei〈ξx〉minus1minus i〈ξ x〉

1 + |x|2

)mts(dx) ξ isin H

(350)

where ats isin H Rts is a nonnegative definite symmetric trace class operator on H and mts

is a Levy measure on H We shall write

microts = [ats Rtsmts] t ge s

In terms of the characteristic exponent ψts of microts condition (18) is equivalent to

ψts(ξ) = ψtr(ξ) + ψrs(U(t r)lowastξ) ξ isin H (351)

for every t ge r ge sAccording to (350) the right hand side of (351) is given by

ψtr(ξ) + ψrs(U(t r)lowastξ)

=minus i〈atr ξ〉+1

2〈ξ Rtrξ〉 minus

intH

1 + |x|2

)mtr(dx)

minus i〈U(t r)ars ξ〉+1

2〈ξ U(t r)RrsU(t r)lowastξ〉

minusintH

(ei〈ξU(tr)x〉minus1minus i〈ξ U(t r)x〉

1 + |x|2

)mrs(dx)

=minus i〈atr + U(t r)ars ξ〉+1

2〈ξ (Rtr + U(t r)RrsU(t r)lowast)ξ〉

minusintH

1 + |x|2

)(mtr + mrs U(t r)minus1)(dx)

+

intH

i〈ξ U(t r)x〉1 + |x|2

mrs(dx)minusintH

i〈ξ U(t r)x〉1 + |U(t r)x|2

mrs(dx)

Therefore by the uniqueness of the canonical representation for infinitely divisible dis-tributions we have the following identities (cf also the proof of [40 Corollary 1411]) forevery t ge r ge s

ats = atr + U(t r)ars

+

intHU(t r)x

(1

1 + |U(t r)x|2minus 1

1 + |x|2

)mrs(dx)

Rts = Rtr + U(t r)RrsU(t r)lowast

mts = mtr + mrs U(t r)minus1

(352)

In particular from (352) (or directly from (310)) we have

att = 0 Rtt = 0 mtt = 0 t isin R (353)

34 Associated stochastic processes There are natural Levy processes and additive pro-cesses associated with the convolution equations (34) and (31) respectively We refer to[47 Theorem 710 (ii) and Theorem 97 (ii)] for details The following theorem shows thatin some sense there is also a natural additive process associated with the family of measuressatisfying the skewed convolution equations (18)

Theorem 318 Let t0 isin R and (microts)t0getges be a system of probability measures on H suchthat for all s le r le t le t0

microts = microtr lowast (micrors U(t r)minus1) (354)

andmicrots rArr δ0 as s uarr tmicrots rArr δ0 as t darr s

(355)

Set for all s le t le t0microts = microts U(t0 t)

minus1

Then

(1) For all s le r le t le t0microts = microtr lowast micrors (356)

micross = δ0 (357)

(358)

(2) There is a stochastically continuous additive process (Xt)t0get satisfying the follow-ing conditions(a) For all t le t0 Xt has the distribution microt0t In particular Xt0 = 0 almost

surely(b) For all t0 ge t ge s the increment Xs minusXt has the distribution microts(c) For all t0 ge t1 gt t2 gt middot middot middot gt tn the incrementsXtjminusXtjminus1

with j = 1 2 middot middot middot nare independent

Proof (1) By (354) we have

microts = microts U(t0 t)minus1

=(microtr lowast

(micrors U(t r)minus1

)) U(t0 t)

minus1

=(microtr U(t0 t)

minus1)lowast(micrors U(t0 r)

minus1)

= microtr lowast micrors

This proves (356) Hence for all s le t0 we have micross = micross lowast micross Since the uniqueidempotent measure on a Hilbert space is the Dirac measure δ0 (357) follows immediately

Fix some s0 lt t0 By Lemma 310 there exists some constant c gt 0 such that for allx isin H and s0 le s le t le t0 |U(t s)x| le c|x| Hence for any ε gt 0 as in Lemma 311 wehave

microts(x isin H |x| gt ε) le microts(x isin H |x| gt εc)Therefore by Lemma 38 we obtain (358) from (355)

(2) For any n isin N t0 ge t1 gt t2 gt middot middot middot gt tn let Υt1t2middotmiddotmiddot tn be the probability measuredefined on (HotimesnB(Hotimesn)) in the following way

Υt1t2middotmiddotmiddot tn(B1 timesB2 times middot middot middot timesBn)

=

intH

1B1(y1)microt0t1(dy1)

intH

1B2(y1 + y2)microt1t2(dy2)

times middot middot middot timesintH

1Bn(y1 + y2 + middot middot middot+ yn)microtnminus1tn(dyn)

(359)

where Bj isin B(H) for j = 1 2 middot middot middot nBy using (356) the family of probability measures (Υt1t2middotmiddotmiddot tn)t0get1gtt2gtmiddotmiddotmiddotgttn satisfies

the consistency condition Therefore by Kolmogorovrsquos extension theorem there is a uniqueprobability measure P on the path space (ΩF ) =

(H(minusinfint0]B(H(minusinfint0])

)such that for

all Bj isin B(H) j = 1 2 middot middot middot n

P(Xt1 isin B1 Xt2 isin B2 middot middot middot Xtn isin Bn) = Υt1t2middotmiddotmiddot tn(B1 timesB2 times middot middot middot timesBn) (360)

Here Xt is the canonical process on (ΩF ) defined by Xt(ω) = ω(t) t le t0Note that for any f isin Bb(Hotimesn) (359) and (360) imply

E[f(Xt1 Xt2 middot middot middot Xtn)]

=

intHotimesn

f(y1 y1 + y2 middot middot middot y1 + y2 + middot middot middot+ yn) microt0t1(dy1)

times microt1t2(dy2)times middot middot middot times microtnminus1tn(dyn)

(361)

In particular from (361) we get that for every t le t0 Xt is distributed as microt0t = microt0tHence P(Xt0 = 0) = 1 since Xt0 sim microt0t0 = δ0

Let z1 middot middot middot zn isin H It follows from (361) that

E

[exp

(i

nsumj=1

langzj Xtj minusXtjminus1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

)microt0t1microt1t2(dy2) middot middot middot microtnminus1tn(dyn)

=nprodj=1

intH

exp(i〈zj yj〉) microtjminus1tj(dyj)

This implies for every j = 1 2 n

E[exp

(i〈zj Xtj minusXtjminus1

〉)]

=

intH

exp (i〈zj yj〉) microtj tjminus1(dyj) (362)

and

E

[exp

(i

nsumj=1

〈zj Xtj minusXtjminus1〉

)]=

nprodj=1

E[exp

〉)] (363)

Equation (362) shows thatXtjminusXtjminus1has distribution microtj tjminus1

while Equation (363) showsthat (Xt)t0get has independent increments

For any t0 ge t ge s the increment Xs minus Xt has distribution microts It follows from (358)that Xs minus Xt converges to 0 in probability as t tends to s or s tends to t This proves that(Xt)t0get is stochastically continuous

There is another way to construct a stochastic process associated with (microts)tges satisfying(18)

Theorem 319 Let s0 isin R and (microts)tgesges0 be a system of probability measures on H suchthat for all s0 le s le r le t

Then there is a stochastically continuous process (Ys)sges0 satisfying the following condi-tions

(1) For every s0 le s Ys has the distribution micross0 In particular Ys0 = 0 almost surely(2) For every s0 le s le t the increment Yt minus U(t s)Ys has the distribution microts More-

over Yt minus U(t s)Ys and Ys are independent

Suppose that for every s ge s0 micros+εs rArr δ0 as ε darr 0 Then Ys+ε converges in probability toYs as ε darr 0 So micros+εs0 converges weakly to micross0 as ε darr 0

Proof Let (ΩF ) =(H[s0minusinfin)B(H[s0minusinfin))

)and let (Ys)sges0 be the canonical process on

(ΩF ) defined by

Ys(ω) = ω(s) s ge s0

For any n isin N s0 le s1 lt s2 lt middot middot middot lt sn let τs1s2middotmiddotmiddot sn be the probability measuredefined on (HotimesnB(Hotimesn)) by

τs1s2middotmiddotmiddot sn (B1 timesB2 times middot middot middot timesBn)

=

intH

1B1(y1)micros1s0(dy1)

intH

1B1 (U(s2 s1)y1 + y2)micros2s1(dy2)

timesintH

1B3(U(s3 s1)y1 + U(s3 s2)y2 + y3)micros3s2(dy3)

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

)microsnsnminus1(dyn)

(365)

where Bj isin B(H) j = 1 2 middot middot middot nEquation (364) implies that the family of probability measures (τs1s2middotmiddotmiddot sn)s0les1lts2ltmiddotmiddotmiddotltsn

satisfies the consistency condition Hence by Kolmogorovrsquos extension theorem there is aunique probability measure P on (ΩF ) such that for all Bj isin B(H) j = 1 2 middot middot middot n ands0 le s1 lt s2 lt middot middot middot lt sn

P(Ys1 isin B1 Ys2 isin B2 middot middot middot Ysn isin Bn) = τs1s2middotmiddotmiddot sn(B1 timesB2 times middot middot middot timesBn) (366)

Hence for every f isin Bb(Hotimes2) s0 le s lt t we have

E[f(Ys Yt)] =

intHtimesH

f (y1 U(t s)y1 + y2) micross0(dy1)microts(dy2) (367)

Therefore for every z1 z2 isin H s0 le s le t we have

E [exp (i 〈z1 Xs〉+ i〈z2 Xt minus U(t s)Xs〉]

=

intH

exp(i 〈z1 y1〉)micross0(dy1)

intH

exp(i 〈z2 y2〉)microts(dy2)

This implies that Xs and Xt minus U(t s)Xs are independent and they have distributions micross0and microts respectively In particular the distribution of Xs0 is given by δ0 So Xs0 = 0

almost surely Thus (1) and (2) are provedNow for every s ge s0 and ε gt 0 we have

Xs+ε minusXs = (Xs+ε minus U(s+ ε s)Xs) + (U(s+ ε s)minus I)Xs

Since Xs and Xs+εminusU(s+ ε s)Xs are independent the distribution of Xs+εminusXs is givenby

micros+εs lowast(micross0 (U(s+ ε s)minus I)minus1

)

It is obvious that as ε darr 0 micross0 (U(s+ ε s)minus I)minus1 converges weakly to δ0 Indeed forevery continuous bounded function f on H we have

limεdarr0

intHf(x) dmicross0 (U(s+ ε s)minus I)minus1 (x)

=

intH

limεdarr0

f ((U(s+ ε s)minus I)x) dmicross0(x)

=

intHf(0) dmicross0(x) = f(0) = δ0(f)

Suppose in addition that micros+εs converges weakly to δ0 as ε darr 0 Then we have

)rArr δ0 as ε darr 0

Hence Xs+ε converges in probablity to Xs as ε darr 0 This implies that Xs+ε converges indistribution to Xs as ε darr 0 That is micros+εs0 converges weakly to micross0 as ε darr 0

In the following example we construct concrete stochastic processes (Xt)tlet0 and (Ys)sles0that satisfy the conditions in Theorem 318 and Theorem 319 respectively

Example 320 Let (U(t s))tges be an evolution system of bounded operators on H and let(Zt)tisinR be a stochastically continuous additive process on some probability space (ΩP)

taking values in H As in (14) we assume that the following stochastic convolution inte-grals

Xts =

int t

s

U(t σ) dZσ t ge s

are well defined Let microts be the distribution of Xts(1) For fixed t0 isin R consider the stochastic process (Xt)t0get = (Xt0t)t0get It is clear

that for all t le t0 Xt has distribution microt0t Moreover (microts)t0getges fulfills condition (354)Note that for all t0 ge t gt s the increment Xt0s minusXt0t has distribution

First we note that (cf [41])

Xt0s minusXt0t =

int t0

s

U(t0 σ) dZσ minusint t0

t

U(t0 σ) dZσ = U(t0 t)

int t

s

U(t σ) dZσ

So for all B isin B(H) we have

P(Xt0s minusXt0t isin B) = P(int t

s

U(t σ) dZσ isin U(t0 t)minus1B

)= microts(U(t0 t)

minus1B) = microts(B)

Clearly the increments of (Xt0t)t0get are independent Hence

Xt0s minusXt0t = (Xt0s minusXt0r) + (Xt0r minusXt0t) s le r le t le t0

implies (356)Suppose that (355) holds Then it follows that the process (Xt0t)t0get is stochastically

continuous Hence (358) follows(2) For fixed s0 isin R consider the stochastic process (Ys)sges0 = (Xss0)sges0 Clearly for

every s ge s0 Ys has distribution micross0 Moreover for every t ge s ge s0 we have

Yt minus U(t s)Ys =

int t

s0

U(t σ) dZσ minus U(t s)

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ minusint s

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

This shows that Yt minus U(t s)Ys has distribution microts and that Yt minus U(t s)Ys is independentof Ys

Let s ge s0 and ε gt 0 By (368) it is clear that

Ys+ε minus Ys = Xs+εs + (U(s+ ε s)minus I)Xss0

Note that Xs+εs and Xss0 are independent So the distribution of Ys+ε minus Ys is given by

micros+εs lowast(micross0 (U(s+ ε s)minus I)minus1)

Suppose that Xs+εs converges in probability to 0 (equivalently micros+εs converges weaklyto δ0) as ε darr 0 Since we also know that micross0 (U(s+ ε s)minus I)minus1 converges weakly to δ0we obtain that Ys+ε converges in probability to Ys as ε darr 0 Therefore Ys+ε converges indistribution to Ys as ε darr 0 ie micros+εs0 converges weakly to micross0 as ε darr 0

Using the stochastic processes constructed in Theorem 318 and Theorem 319 we haveprobabilistic proofs of Theorem 312 Theorem 313 and Lemma 317 as we shall shownow

Another proof of Theorem 312 Since Part (2) has been shown in Theorem 319 it remainsto show (1) and (3) By Theorem 318 there is a stochastically continuous additive process(Xt)t0get such that for all t0 ge t the distribution ofXt is given by microt0t and for all t0 ge t ge sthe distribution of the increment Xs minusXt is given by microts = microts U(t0 t) Hence for everyδ gt 0 t0 gt s and ε isin R we have

limεrarr0

P(|Xs+ε minusXs| ge δ) = 0

This means that Xs+ε converges in probability to Xs as ε rarr 0 This implies that Xs+ε

converges in distribution to Xs as ε rarr 0 That is microt0s+ε converges weakly to microt0s as

ε rarr 0 Since t0 was arbitrary we have microts+ε converges weakly to microts as ε rarr 0 So (1) isproved

Now for every t0 ge t ge s we have

limεdarr0

P(|(Xs minusXtminusε)minus (Xs minusXt)| ge δ) = 0

So Xs minusXtminusε converges in distribution to Xs minusXt as ε darr 0 This proves (3)

Another proof of Theorem 313 Similar to the proof of Theorem 318 there is a stochasti-cally continuous additive process (Xt)tisinR such that microts is the distribution of the incrementXt minusXs for all t ge s Hence for every δ gt 0 we have

limεrarr0ηrarr0

P(|(Xt+ε minusXs+η)minus (Xt minusXs)| ge δ) = 0

This implies that Xt+ε minus Xs+η converges in distribution to Xt minus Xs Hence microt+εs+η con-verges weakly to microts as εrarr 0 and η rarr 0

To give a probabilistic proof of Lemma 317 let us recall the following lemma which hasbeen proved in [47 Lemma 96] for the finite dimensional case The proof for the infinitedimensional case is the same

Lemma 321 A stochastically continuous process (Xt)tisinR taking values in H is uniformlystochastically continuous on any finite interval That is for every s0 lt t0 for every ε gt 0

and η gt 0 there is δ gt 0 such that for all s and t in [s0 t0] with |tminus s| lt δ we have

P(|Xt minusXs| gt ε) lt η

Another proof of Lemma 317 By Theorem 318 there is a stochastically continuous addi-tive process (Xt)t0getges0 such that microts is the distribution of the increment Xs minus Xt for allt0 ge t ge s ge s0 By Lemma 321 we obtain that (Xt)t0getges0 is uniformly stochasticallycontinuous This means that for every ε gt 0 and η gt 0 there is a δ gt 0 such that for alls t isin [s0 t0] satisfying |tminus s| lt δ we have

P(|Xs minusXt| gt ε) lt η

In other wordsmicrots(|x| gt ε) lt η

This proves (337) since microts = microts U(t0 t)minus1 by definition

4 EVOLUTION SYSTEMS OF MEASURES

In general we cannot expect a stationary invariant measure for the time inhomogeneousgeneralized Mehler semigroup (pst)tges defined in (17) So we shall look for a family ofprobability measures (νt)tisinR on H such thatint

Hpstf(x) νs(dx) =

intHf(x) νt(dx) s le t (41)

for all f isin Bb(H) Such a family of probability measures is called an evolution system ofmeasures for (pst)tges in [14] (and entrance law in [20]) It can be regarded as a space-timeinvariant measure for (pst)tges

We shall first show some properties of evolution systems of measures for (pst)tges andthen study their existence and uniqueness

41 Some properties

Lemma 41 A family of probability measures (νt)tisinR on H is an evolution system of mea-sures for (pst)tges if and only if for every t ge s

microts lowast(νs U(t s)minus1

)= νt (42)

or equivalently for every t ge s

microts(ξ)νs(U(t s)lowastξ

)= νt(ξ) ξ isin H (43)

Proof Note that for all f isin Bb(H) and s le t we haveintHpstf(x) νs(dx)

=

intH

intHf(U(t s)x+ y)microts(dy)νs(dx)

=

intH

intHf(x+ y)microts(dy)(νs U(t s)minus1)(dx)

=

intHf(z) (microts lowast (νs U(t s)minus1)(dz)

So (41) holds if and only if for all f isin Bb(H)intHf(z) (microts lowast (νs U(t s)minus1)(dz) =

intHf(z) νt(dz) (44)

Thus the proof is complete by noting that (44) holds if and only if (42) holds

Remark 42 A probability measure micro on H is said to be operator self-decomposable if

micro = (micro Tminus1t ) lowast microt t ge 0 (45)

holds for a family of semigroups (Tt)tge0 and measures (microt)tge0 Operator self-decomposabilityhas been studied very well see for example [2 5 31 50 56] and the references thereinIn the setting of (11) any solution micro to the convolution equation (45) is just an invari-ant measure for the generalized Mehler semigroup (11) Obviously Equation (45) is thehomogeneous version of Equation (42)

Proposition 43 Let (νt)tisinR and (microts)tges be families of probability measures on H Let(U(t s))tges be an evolution family of operators on H Suppose that (42) holds and for allξ isin H and all t s isin R t ge s νt(U(t s)lowastξ) 6= 0 Then (microts)tges satisfies (18)

Proof For any t ge r ge s by (42) and (349) we have

)= νt

= microtr lowast(νr U(t r)minus1

)= microtr lowast

([micrors lowast (νs U(r s)minus1)] U(t r)minus1

)= microtr lowast

([micrors U(t r)minus1] lowast [(νs U(r s)minus1) U(t r)minus1]

)= microtr lowast

)lowast(νs U(t s)minus1

)

So for all ξ isin H we have

microts(ξ) middot νs(U(t s)lowastξ) = microtr(ξ) middot micrors(U(r s)lowastξ) middot νs(U(t s)lowastξ)

Since νt(U(t s)lowastξ) 6= 0 by assumption we have

microts(ξ) = microtr(ξ) middot micrors(U(r s)lowastξ)

This proves (18)

Similar as in Theorem 319 there exists a stochastic process associated with (νt)tisinR

Theorem 44 Let (νt)tisinR and (microts)tges be families of probability measures on H Let(U(t s))tges be an evolution family of operators on H Suppose that (18) and (42) holdThen there is a stochastic process (Xtminusinfin)tisinR such that for every t ge sXtminusinfinminusU(t s)Xsminusinfinand Xsminusinfin are independent and have distributions microts and νs respectively

Proof Let Ω = H(minusinfininfin) be the collection of all functions ω = (ω(t))tisin(minusinfininfin) from(minusinfininfin) into H Let

Xtminusinfin(ω) = ω(t) t isin (minusinfininfin)

be the canonical process on Ω Let F be the Borel σ-algebra generated by cylinder sets onΩ For any n isin N minusinfin lt t1 le t2 le middot middot middot le tn ltinfin Bj isin B(H) j = 1 2 middot middot middot n define

νt1t2middotmiddotmiddot tn(B1 timesB2 times middot middot middot timesBn)

=

intH

1B1(y1)νt1(dy1)

intH

1B2(U(t2 t1)y1 + y2)microt2t1(dy2)

timesintH

1B3(U(t3 t1)y1 + U(t3 t2)y2 + y3)microt3t2(dy3)times middot middot middot

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

)microtntnminus1(dyn)

(46)

Then we extend νt1t2middotmiddotmiddot tn to a probability measure on (HotimesnB(Hotimesn)) Then it is easyto check that the family of probability measures νt1t2middotmiddotmiddot tnt1let2lemiddotmiddotmiddotletn satisfies the consis-tency condition Therefore by Kolmogorovrsquos extension theorem there is a unique probabil-ity measure P on (ΩF ) such that for allminusinfin lt t1 le t2 le middot middot middot le tn ltinfin andBj isin B(H)j = 1 2 middot middot middot n we have

P(Xt1minusinfin isin B1 Xt2minusinfin isin B2 middot middot middot Xtnminusinfin isin Bn)

=νt1t2middotmiddotmiddot tn(B1 timesB2 times middot middot middot timesBn)(47)

Similar to the proof of Theorem 319 for every z1 z2 isin H s le t we have

E [exp (i 〈z1 Xsminusinfin〉+ i〈z2 Xtminusinfin minus U(t s)Xsminusinfin〉]

=

intH

exp(i 〈z1 y1〉)νs(dy1)

intH

This implies that Xsminusinfin and XtminusU(t s)Xsminusinfin are independent and they have distributionsνs and microts respectively So the proof is complete

Example 45 As in Example 320 let (U(t s))tges be an evolution system of bounded op-erators on H and let (Zt)tisinR be an additive process taking values in H Suppose that for alls lt t U(t middot) is integrable with respect to (Zt)tisinR on [s t] That is the following stochasticconvolution integrals

Xts =

int t

s

U(t σ) dZσ t ge s

are well defined Suppose that for all t isin R Xts converges in probability to Xtminusinfin assrarr minusinfin That is we assume that the following improper integral

Xtminusinfin =

int t

minusinfinU(t σ) dZσ t isin R

exists (cf [41]) For every t ge s let microts be the distribution of Xts For every t isin R let νtdenote the distribution of Xtminusinfin

Note that for all t ge s

Xtminusinfin = U(t s)Xsminusinfin +Xts (48)

So Xtminusinfin minus U(t s)Xsminusinfin has distribution microts Since Xsminusinfin and Xts are independentwe obtain that Xsminusinfin and Xtminusinfin minus U(t s)Xsminusinfin are independent By (48) we get νt =

) This proves that (νt)tisinR is an evolution system of measures for the

transition function (pst)tges of X(t s x) = U(t s)x+Xts x isin H t ge s

Concerning the infinite divisibility of νt we have the following simple result

Proposition 46 Let (νt)tisinR be an evolution system of measure for (pst)tges Suppose thatfor some s0 isin R the measures νs0 and (microts0)tges0 are infinitely divisible then for all t ge s0νt is infinitely divisible

Proof According to (42) for all t ge s0 νt is the convolution of microts0 and νs0 U(t s)minus1Since microts0 is infinitely divisible we only need to show that νs0 U(t s0)minus1 is also infinitelydivisible

Because νs0 is infinitely divisible for any n isin N there is some probability measure ν(n)s0

on (HB(H)) such that νs0 = (ν(n)s0 )lowastn So by (349) we have

νs0 U(t s0)minus1 = (ν(n)s0 U(t s0)minus1)lowastn

for all t ge s0 This proves that νs0 U(t s0)minus1 is also infinitely divisible

Theorem 47 Let (ν(1)t )tisinR be an evolution system of measures for (pst)tges Let (ν

(2)t )tisinR

and (σt)tisinR be two families of probability measures on H such that

ν(2)t = ν

(1)t lowast σt t isin R (49)

and

σt = σs U(t s)minus1 t ge s

Then (ν(2)t )tisinR is also an evolution system of measures for (pst)tges

Proof For every ξ isin H by (43) and (49) we have

ν(2)t (ξ) = ν

(1)t (ξ)σt(ξ) = microts(ξ)ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

)σs(U(t s)lowastξ) = microts(ξ)ν

(2)s

(U(t s)lowastξ

)

Hence the assertion follows by Lemma 41

42 Existence and uniqueness For special cases of our (pst)tges in this paper existenceand uniqueness of evolution systems of measures have been studied in [13 32 57] etcin different settings Our framework is more general We emphasize that Theorem 49Theorem 410 and Corollary 412 below not only generalize the corresponding results in[57] for finite dimensional Levy driven non-autonomous Ornstein-Uhlenbeck processesbut also contain some new results even in the finite dimension case

From now on we always assume that

(1) The family (microts)tges satisfies (18) That is for all s le r le t

)

(2) For every t ge s microts is infinitely divisible with representation (as eg in the case itsatisfies Assumption 37)

microts = [ats Rtsmts]

where ats isin H Rts is a nonnegative definite self-adjoint trace class operator on Hand mts is a Levy measure on H

By (352) for every fixed t isin R (mts)tges is a family of Levy measures decreasing in sin the sense that for all A isin B(H 0) and all s le r le t

mts(A) ge mtr(A)

which allows us to define mtminusinfin for every t isin R by setting mtminusinfin(0) = 0 and

mtminusinfin(A) = limsrarrminusinfin

mts(A) A isin B(H 0)

Conditions under which mtminusinfin is a Levy measure will be given later in Theorem 49

Lemma 48 Suppose that for every t isin R

supslet

trRts ltinfin (410)

Then there is a trace class operator Rtminusinfin on H such that for all x y isin H

〈Rtminusinfinx y〉 = limsrarrminusinfin

〈Rtsx y〉 (411)

Proof By (352) for every x isin H and t isin R 〈Rtsx x〉 is decreasing in s More preciselyfor every x isin H and s le r le t we have

〈Rtsx x〉 = 〈Rtrx x〉+ 〈RrsU(t r)lowastx U(t r)lowastx〉 ge 〈Rtrx x〉

It follows that for every s le r le t

trRtr le trRts

Therefore as s tends to minusinfin the limit of 〈Rtsx x〉 existsBy (410) for every t isin R there exists a constant Ct gt 0 such that

supslet〈Rtsx x〉 le Ct|x|2

holds for every x isin H By the polarization identity for every t isin R x y isin H limsrarrminusinfin〈Rtsx y〉exists Fixing x isin H and letting y isin H vary we get a functional limsrarrminusinfin〈Rtsx middot〉 So by

Rieszrsquos representation theorem for every x isin H there exists an element xlowastt isin H for everyt isin R such that for all y isin H

limsrarrminusinfin

〈Rtsx y〉 = 〈xlowastt y〉

Let Rtminusinfin denote the map x 7rarr xlowastt By (410) it is clear that Rtminusinfin is a trace class operatorThis proves (411)

Now we are ready to show the following result on the existence of evolution system ofmeasures for (pst)tges

Theorem 49 Suppose that for every t isin R the following three hypotheses hold

(H1) supslet trRts ltinfin(H2) supslet

intH(1 and |x|2) mts(dx) ltinfin

(H3) For every t isin R atminusinfin = limsrarrminusinfin ats exists and is finite

Then for every t isin R mtminusinfin is a Levy measure Rtminusinfin is a nonnegative definite self-adjointtrace class operator such that

trRtminusinfin = supslet

Rts ltinfin

Moreover the system of measures (νt)tisinR given by

νt = [atminusinfin Rtminusinfinmtminusinfin] t isin R

is an evolution system of measures for (pst)tges

Proof Suppose that (H1) (H2) and (H3) hold By Lemma 48 for every t isin R Rtminusinfin is anonnegative definite self-adjoint trace class operator satisfying

Rts ltinfin

For each t isin R intH

(1 and |y|2) mtminusinfin(dy) = supslet

intH

(1 and |x|2) mts(dx) ltinfin

This shows that mtminusinfin is a Levy measureNow we show that (νt)tisinR is an evolution system of measures for (pst)tges By (18) for

every t ge s ge rmicrots lowast

(microsr U(t s)minus1

)= microtr (412)

Note that microts = [ats Rtsmts] converges weakly to [atminusinfin Rtminusinfinmtminusinfin] = νt as s rarrminusinfin (cf [22 Lemma 34]) Hence letting r rarr minusinfin on both sides of (412) we obtain

)= νt

By Lemma 41 this proves that (νt)tisinR is an evolution system of measures for (pst)tges

The following theorem is the converse to Theorem 49 It also gives some sufficientconditions for the uniqueness of the evolution system of measures

Theorem 410 Suppose that there is an evolution system of measures (νt)tisinR for (pst)tgesThen

(1) Hypotheses (H1) and (H2) hold and for every t isin R

[0 Rtsmts]rArr [0 Rtminusinfinmtminusinfin] as srarr minusinfin (413)

(2) There exists a family of probability measures (σt)tisinR such that for every t isin R

δats lowast(νs U(t s)minus1

)rArr σt as srarr minusinfin (414)

andνt = [0 Rtminusinfinmtminusinfin] lowast σt t isin R (415)

(3) Suppose that the following hypothesis holds(H4) For every t isin R as srarr minusinfin νs U(t s)minus1 converges weakly to some proba-

bility measure σt on HIf for each t isin R σt is infinitely divisible then the limit in (H3) exists Moreoverfor all t isin R we have

σt = δatminusinfin lowast σt (416)

νt = νt lowast σt (417)

andσt = σs U(t s)minus1 t ge s (418)

Here (νt)tisinR is as defined in Theorem 49(4) If the limit in (H3) exists then the limit in (H4) exists Hence (416) (417) and

(418) hold

Proof (1) For every t ge s we set

NRts = [0 NRts 0] Mts = [0 0mts]

Since (νt)tisinR is an evolution system of measures for (pst)tges by Lemma 41 we have forall t ge s

νt = microts lowast(νs U(t s)minus1

)= [ats Rtsmts] lowast

(νs U(t s)minus1

)= δats lowastNRts lowastMts lowast

(νs U(t s)minus1

)

(419)

Applying Theorem 31 to (419) the sequence of probability measure (δatminusn lowast NRtminusn lowastMtminusn)nge1 is shift compact That is for every t isin R there exists a sequence (ytminusn)nge1 inH such that the following sequence of probability measures

δytminusn lowast (δatminusn lowastNRtminusn lowastMtminusn) = [ytminusn + atminusn Rtminusnmtminusn] n ge 1

is weakly relatively compact This implies (see [43 Theorem VI53])

supnisinN

mtminusn(x isin H |x| ge 1) ltinfin (420)

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

It follows from (421) thatsupslet

trRts ltinfin

Combining (420) and (421) we obtain

supslet

intH

Therefore for every t isin R we obtain by taking limits as before a Levy measure mtminusinfin anda trace class operator Rtminusinfin Hence (413) follows

(2) By (419) we have for all t ge s

νt =(δats lowast

(νs U(t s)minus1

))lowastNRts lowastMts

We have shown in (413) thatNRtslowastMts converges weakly to an infinitely divisible measure[0 Rtminusinfinmtminusinfin] as s rarr minusinfin Therefore by Corollary 35 the measures δats lowast

(νs

U(t s)minus1) converges weakly as srarr minusinfin So (414) and (415) are proved(3) Applying Corollary 35 to (414) using (H4) and the assumption that σt is infinitely

divisible for every t isin R the limit of ats as s rarr minusinfin exists So (H3) and hence (416)hold By (415) and (416) we have for every t isin R

νt = [0 Rtminusinfinmtminusinfin] lowast σt = [atminusinfin Rtminusinfinmtminusinfin] lowast σt = νt lowast σt

This proves (417)Now we show (418) For every u le s le t and ξ isin H we have

ˆνu(U(t u)lowastξ) = ˆνu(U(s u)lowastU(t s)lowastξ

) (422)

Letting urarr minusinfin in (422) and using (H4) we get σt(ξ) = σs(U(t s)lowastξ

) This is equivalent

to (418)(4) If (H3) holds then δats rArr δatminusinfin as s rarr minusinfin Note that any Dirac measure is

infinitely divisible So by applying Corollary 35 to (414) we see that the limit in (H4)exists and hence (417) and (418) hold

Remark 411 As is known any invariant measure ν for a time homogeneous GaussianOrnstein-Uhlenbeck semigroup is of the form ν lowast microinfin where ν is a measure on H that isinvariant under the action of a semigroup and microinfin is a Gaussian measure We refer to [24Theorem 522] for details We emphasize that the structure of ν lowastmicroinfin is analogous to (417)

By Theorem 410 we have the following result on the uniqueness

Corollary 412 Let (νt)tisinR be an evolution system of measures for (pst)tges Suppose thatfor every t isin R there is a sequence (sn)nge1 bounded above by t such that the followingconditions are fulfilled

(1) As nrarrinfin sn rarr minusinfin(2) There exist some constants Mω gt 0 such that

U(t sn) leM eminusω(tminussn) (423)

(3) The sequence of probability measures (νsn)nge1 is uniformly tightThen (H1) (H2) and (H3) hold Hence ([atminusinfin Rtminusinfinmtminusinfin])tisinR exists and it is the uniqueevolution system of measures for (pst)tges hence equal to (νt)tisinR

Proof By the proof of Theorem 410 it is sufficient to showˆνsn U(t sn)minus1 rArr δ0 as nrarrinfin (424)

Let ε η gt 0 Since (νsn)nge1 is uniformly tight there is a compact set Kη sub H such that forall n ge 1

νsn(H Kη) lt η (425)

Set for all n ge 1 Cn = Mminus1 eω(tminussn) Since sn rarr minusinfin as n rarr infin also Cn rarr infinas n rarr infin So there exists some N0 gt 0 such that the compact set Kη is contained inx isin H |x| le εCn for all n ge N0 Therefore for all n ge N0 we have

νsn(x isin H |x| gt εCn) le νsn(H Kη) lt η (426)

Because

U(t sn) leM eminusω(tminussn) = 1Cn

by (426) we have for all n ge N0

νsn U(t sn)minus1(x isin H |x| gt ε) le νsn(x isin H |x| gt εCn) lt η

By Lemma 38 this implies (424) Thus the proof is complete

As an application of Corollary 412 we consider the uniqueness of periodic evolutionsystems of measures

Corollary 413 Suppose that (H1) (H2) and (H3) hold and that (microts)tges is periodic withperiod T gt 0 ie for every t ge s microt+Ts+T = microts Then (νt)tisinR = ([atminusinfin Rtminusinfinmtminusinfin])tisinRexists and is a periodic evoluton system of measures for (pst)tges with period T That is forevery t isin R νt+T = νt Suppose in addition that there exist some constants Mω gt 0 suchthat

U(t s) leM eminusω(tminuss)

Then (νt)tisinR is the unique periodic evolution system of measures with period T for (pst)tges

Proof By Theorem 49 (νt)tisinR = ([atminusinfin Rtminusinfinmtminusinfin])tisinR exists For every t ge s wehave

microt+Ts+T = microts (427)

By letting srarr minusinfin on both sides of (427) we obtain νt+T = νt This shows that (νt)tisinR isperiodic with period T

Now it remains to show the uniqueness Take any s0 le t and set sn = s0 minus nT for alln ge 1 Then νsn equiv νs0 for all n ge 1 So it is obvious that (νsn)nge1 is uniformly tight ByCorollary 412 (νt)tisinR is the unique evolution system of measures for (pst)tges

Remark 414 Existence and uniqueness of evolution systems of measures have been stud-ied for stochastic evolution equations with time dependent periodic coefficients driven byGaussian and Levy processes in [13] and [32] respectively Clearly Corollary 413 appliesto these cases More generally one can apply it to study stochastic evolution equations withtime dependent periodic coefficients driven by so called semi-Levy processes A stochasticprocess (Zt)tisinR is called a semi-Levy process with period T gt 0 if it is an additive processsuch that for all t ge s Zt+T minus Zs+T has the same distribution as Zt minus Zs as in [39]

In [39] it is shown that for the finite dimensional case under some conditions ν0 is semi-self-decomposable Moreover this is closely related to the so called semi-selfsimilar andsemi-stationary processes One may study similar self-decomposibility and semi-stationarityin the infinite dimensional case as in [1 39]

5 HARNACK INEQUALITIES AND APPLICATIONS

Harnack inequalities for generalized Mehler semigroups or Ornstein-Uhlenbeck semi-group driven by Levy processes were proved in [32 40 42 45] The method in [32] and[45] relies on taking the derivative of a proper functional the method in [40 42] is basedon coupling of stochastic processes and Girsanov transformation Here we shall establish aHarnack inequality for (pst)tges defined by (17) by a much simper method

Suppose that for all t ge s microts = [ats Rtsmts] is an infinitely divisible measure on Hsatisfying (18)

For each t ge s set

microgts = [0 Rts 0] microjts = [ats 0mts]

and for every f isin Bb(H) x isin H set

pgstf(x) = (microgts lowast δU(ts)x)(f) =

intHf(U(t s)x+ y)microgts(dy)

pjstf(x) = (microjts lowast δx)(f) =

intHf(x+ y)microjts(dy)

With these notations we have the following decomposition for pst which plays a keyrole

Proposition 51 For every t ge s x isin H and f isin Bb(H) we have

pstf(x) = pgst(pjst)f(x)

Proof Since microts = microgts lowast microjts we get

pstf(x) =(microts lowast δU(ts)x)(f) = (microgts lowast microjts lowast δU(ts)x)(f)

=((microgts lowast δU(ts)x) lowast microjts

)(f)

=

intHmicrogts lowast δU(ts)x(dy)

intHf(y + z)microjts(dz)

=(microgts lowast δU(ts)x

)(pjstf) = pgst(p

jstf)(x)

Define for every t ge s

Γts = Rminus12ts U(t s) (51)

with domain

D(Γts) = x isin H U(t s)x isin R12ts (H)

If x isin D(Γts) then we set |Γtsx| = infin Let B+b (H) denote the space of all bounded

positive measurable functions on H

Theorem 52 For every α gt 1 t ge s and f isin B+b (H) we have

(pstf(x))α le exp

(α|Γts(xminus y)|2

2(αminus 1)

)pstf

α(y) x y isin H (52)

Proof We only need to consider the case when U(t s)(H) isin R12ts (H) Otherwise the

right hand side of (52) is infinite by the definition of |Γts(middot)| and hence the inequality (52)becomes trivial

Let us first show that it is sufficient to have the following Harnack inequality for pgst

(pgstf(x))α le exp

2(αminus 1)

)pgstf

Indeed by Proposition 51 we have pst = pgstpjst So by applying inequality (53) to pgst

and Jensenrsquos inequality to pjst we obtain

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

This proves (53)Now it remains to show (53) The method is the same as in the case of a one dimensional

(time homogeneous) Ornstein-Uhlenbeck process see [40 Page 69]Let N(mQ) denote the Gaussian measure on H with mean m isin H and covariance

operatorQ on H By the Cameron-Martin formula for Gaussian measures (see [15 Theorem221]) we have

ρts(xminus y z) =dN(U(t s)(xminus y) Rts)

dN(0 Rts)(z)

= exp

(langRminus12ts U(t s)(xminus y) R

minus12ts z

rangminus 1

2|Rminus12

ts U(t s)(xminus y)|2)

(54)

Moreover for any h isin H we have (see [16 Proposition 125])

intH

exp(〈h x〉) dmicrogts(x) = exp

(1

2|R12

ts h|2) (55)

By changing variables and using (54) we obtain

pgstf(x)

=

intHf(U(t s)x+ z)microgts(dz)

=

intHf(U(t s)y + U(t s)(xminus y) + z)N(0 Rts)(dz)

=

intHf(U(t s)y + zprime)

dN(U(t s)(xminus y) Rts)

dN(0 Rts)(zprime)N(0 Rts)(dz

prime)

=

intHf(U(t s)y + zprime)ρts(xminus y zprime)microgts(dzprime)

= exp

(minus1

2|Γts(xminus y)|2

)middotintHf(U(t s)y + zprime) exp

minus12ts z

rang)microgts(dz

prime)

(56)

By Holderrsquos inequality and (55) we haveintHf(U(t s)y + zprime) exp

minus12ts z

rang)microgts(dz

prime)

le(int

Hfα(U(t s)y + zprime)microgts(dz

prime)

)1α

middot(int

Hexp

(α

αminus 1〈Rminus12

ts Rminus12ts U(t s)(xminus y) zprime〉

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

(αminus 1)2|Rminus12

ts U(t s)(xminus y)|2))(αminus1)α

=(pgstf

α(y))1α

exp

(α

2(αminus 1)|Γts(xminus y)|2

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

This proves (53) Hence the proof is complete as we have argued at the beginning that(53) is sufficient

Example 53 Suppose that H = L2(0 1) Let ∆ denote the Dirichlet Laplacian on H Leten n ge 1 be the eigenbasis of ∆ with respective eigenvalues minusn2π2 n ge 1 Let a(t) be acontinuous function on R Then

a(t)∆en = minusa(t)n2π2en n = 1 2 middot middot middot

The evolution family (U(t s))tges associated with a(t)∆ is given by

U(t s)x =infinsumn=1

exp

(minusn2π2

int t

s

a(r) dr

)〈x en〉en x isin H

Let (Wt)tisinR be a R-Wiener process taking values in H We assume that there exist asequence of nonnegative real numbers (λn)nge1 such that

Ren = λnen

and a sequence (bn)nge1 of real independent Brownian motion such that

〈Wt x〉 =infinsumn=1

radicλn〈x en〉bn(t) x isin H t isin R

Now let us consider the following SPDE on HdXt = a(t)∆Xtdt+ dWt

Xs = x(58)

Assume that for all t ge s

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

Then the following stochastic convolution

WU(t s) =

int t

s

U(t r) dWr

is well defined The distribution of WU(t s) is given by microts sim N(0 Rts) where

Rts =

int t

s

U(t r)RU(t r)lowast dr

Clearly trRts = rts By [15 Chapter 5] the mild solution of (58) is given by

X(t s x) = U(t s)x+WU(t s) x isin H t ge s

The generalized Mehler semigroup of X(t s x) is given by

pstf(x) =

intHf (U(t s)x+ y)microts(dy) (59)

for all x isin H f isin Bb(H) and t ge sWe are going to look for the Harnack inequality for pstLet Γts = R

minus12ts U(t s) Note that for all n ge 1

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

Rminus12ts U(t s)en =

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

Therefore for every x isin H we have

Γtsx2 = 〈Rminus12ts U(t s)xR

minus12ts U(t s)x〉

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

Hence we obtain the following Harnack inequality for pst

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

for every α gt 1 x y isin H t ge s and f isin B+b (H)

Applying Theorem 52 we have the following result

Theorem 54 Fix t ge s The implications (1)rArr (2)rArr (3)rArr (4)rArr (5) of the followingstatements hold

(1) one hasU(t s)(H) sub R

12ts (H) (510)

(2) Γts ltinfin and for every α gt 1 f isin B+b (H)

(pstf(x))α le exp

(α(Γts middot |xminus y|)2

2(αminus 1)

)pstf

(3) Γts ltinfin and there exists some α gt 1 such that (511) holds for all f isin B+b (H)

(4) Γts ltinfin and for every f isin B+b (H) with f gt 1

pst log f(x) le log pstf(y) +Γts2

2|xminus y|2 x y isin H (512)

(5) pst is strong Feller ie for every f isin Bb(H) pstf isin Cb(H)

In particular if mts equiv 0 then all these statements are equivalent

Proof If (510) holds then Γts is a bounded linear operator on H Hence by Theorem52 we get (2) from (1) That (2) implies (3) is trivial The implications (3)rArr(4)rArr(5) areconsequences of Harnack inequalities as proved in [55]

Now we show that if mts equiv 0 then (5) implies (1) Note that

pstf(x) =

intHf(y)N(U(t s)xRts)(dy)

If (510) does not hold then there exists an x0 isin H such that U(t s)x0 isin R12ts (H) Take

xn =1

nx0 isin H n = 1 2

By the Cameron-Martin theorem (see eg [15]) for each n = 1 2 middot middot middot the Gaussianmeasure micron = N(U(t s)xn Rts) is orthogonal to micro0 = N(0 Rts) since U(t s)xn isinR

12ts (H) It means that for every n = 1 2 middot middot middot there exists a set An isin B(H) such that

micron(An) = 1 and micro0(An) = 0 Let A = cupnge1An Then we have micro0(A) = 0 micron(A) = 1

since micro0(A) lesuminfin

n=1 micro0(An) = 0 and micron(A) ge micron(An) = 1

Take f = 1A Because xn tends to 0 as n rarr infin and pst is strong Feller pstf(xn)

converges to pstf(0) as nrarrinfin But this is impossible because we have pstf(xn) = 1 forall n ge 1 and pstf(0) = 0 So (510) must hold

Remark 55 IfRts has the form (62) then (510) is equivalent to the null controllability ofa non-autonomous control system (61) (see Section 6 for details) For this reason condition(510) is also called null-controllability condition This gives an equivalent description ofthe strong Feller property in the Gaussian case

Remark 56 In [12] it is shown that the null controllability implies the strong Feller prop-erty for autonomous Ornstein-Uhlenbeck processes with deterministic perturbation drivenby a Wiener process Obviously our result generalizes this result

In fact (510) implies more Let UCinfin(H) denote the space of all infinitely Frechet dif-ferentiable functions with uniform continuous derivatives on H

Proposition 57 Suppose that (510) holds Then for every f isin Bb(H) and every t gt spstf isin UCinfin(H)

Proof In view of the decomposition pst = pgstpjst shown in Proposition 51 we only need

to show that pgst isin UCinfin(H) for every g isin Bb(H) The rest of the proof is the same as in[16 Theorem 622]

We have the following quantitative estimate for the strong Feller property This result isshown in [42] for Levy driven Ornstein-Uhlenbeck process by a coupling method

Proposition 58 Let t gt s and x y isin H Then

|pstf(x)minus pstf(y)|2

le(

e|Γts(xminusy)|2 minus1)

minpstf

2(z)minus (pstf(z))2 z = x y

(513)

Proof Let h = pjstf Then by Proposition 51 we have pstf = pgsth Moreover

h2 = (pjstf)2 le pjstf2

by Jensenrsquos inequality So for every z isin H we have

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

2(z)minus (pstf(z))2(514)

Note also that (513) is symmetric in x and y So according to (514) we only need to showthe following inequality

|pgsth(x)minus pgsth(y)|2 le(

e|Γts(xminusy)|2 minus1) (pgsth

2(y)minus (pgsth(y))2) (515)

Using the notation in (54) we have

pgsth(x) =

intHh(U(t s)x+ z)microgts(dz) =

intHρts(xminus y z)h(U(t s)y + z)microgts(dz)

Therefore we obtain

|pgsth(x)minus pgsth(y)|2

=

(intH

[ρts(xminus y z)minus 1] middot [h(U(t s)y + z)minus pgsth(y)]microgts(dz)

)2

leintH

(ρts(xminus y z)minus 1)2 microgts(dz)

intH

[h(U(t s)y + z)minus pgsth(y)

]2microgts(dz)

=

(intHρ2ts(xminus y z)micro

gts(dz)minus 1

)middot(int

Hh2(U(t s)y + z)microgts(dz)minus (pgsth(y))2

)=(

2(y)minus (pgsth(y))2)

Note that here we have used (55) to obtainintHρ2ts(xminus y z)micro

gts(dz) = e|Γts(xminusy)|2

Now we apply the Harnack inequality (52) to study the hyperboundedness of the transi-tion function pst In [23] hypercontractivity is studied for the Gaussian case via log-Sobolevinequality

Theorem 59 Let (νt)tisinR be an evolution system of measures for (pst)tges For every s le tα gt 1 and ε ge 0 let

Cst(α ε) =

intH

[intH

exp

(minusα|Γts(xminus y)|2

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

Then for all f isin Lα(H νt)

pstfLα(1+ε)(Hνs) le Cst(α ε)minusα(1+ε)fLα(Hνt) (516)

Proof From the Harnack inequality (52) we have

(pstf(x))α exp

[minusα|Γts(xminus y)|2

2(αminus 1)

]le pstf

α(y) x y isin H

Integrating both sides of the inequality above with respect to νs(dy) and using the fact that(νt)tisinR is an evolution system of measures we obtain

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

intH|f |α(y) νt(dy)

Hence

(pst|f |)α(1+ε)(x) le[int

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

Integrating both sides of the inequality above with respect to νs(dx) we get (516)

6 APPENDIX NULL CONTROLLABILITY

Consider the following non-autonomous linear control systemdz(t) = A(t)z(t)dt+ C(t)u(t) dt

z(s) = x(61)

where (A(t))tisinR is a family of linear operators on H with dense domains and (C(t))tisinR isa family of bounded linear operators on H Let (U(t s))tges be an evolution family on Hassociated with (A(t))tisinR Consider the mild solution of (61)

z(t s x) = U(t s)x+

int t

s

U(t r)C(r)u(r) dr x isin H t ge s

In control theory z(t s x) is interpreted as the state of the system and u as a strategy tocontrol the system If there exists some u isin L2([s t]H) such that z(t s x) = 0 then wesay the system (61) can be transferred to 0 at time t from the initial state x isin H at times If for every initial state x isin H the system (61) can be transferred to 0 then we say thesystem (61) is null controllable at time t We refer to [58] (see also [15 Appendix B]) forthe details on null controllability of autonomous control systems

Set for every t ge s

Πtsx =

int t

s

U(t r)C(r)C(r)lowastU(t r)lowast dr x isin H (62)

Proposition 61 Let x isin H and t ge s The system (61) can be transferred to 0 at timet from x if and only if U(t s)x isin Π

12ts (H) Moreover the minimal energy among all

strategies transferring x to 0 at time t is given by |Πminus12ts U(t s)x|2 ie

|Πminus12ts U(t s)x|2

= inf

int t

s

|u(r)|2 dr z(t s x) = 0 z(s s x) = x u isin L2([s t]H)

(63)

Proof For every t ge s define a linear operator

Lts L2([s t]H)rarr H u 7rarr Ltsu =

int t

s

U(t r)C(r)u(r) dr

The adjoint Llowastts of Lts is given by

(Llowasttsx)(r) = Clowast(r)U(t r)lowastx x isin H r isin [s t]

It is easy to check thatΠts = LtsL

lowastts

Then by [15 Corollary B4] Lts(L2([s t]H) = Πts(H) Hence the first assertion of thetheorem is proved since the initial state x can be transferred to 0 if and only if U(t s)x iscontained in the image space of Lts due to the fact that z(t s x) = U(t s)x+ Ltsu

By [15 Corollary B4] we also get

|Πminus12ts y| = |Lminus1

ts y| y isin Lts(L2([s t]H)) (64)

Here the inverse is understood as a pseudondashinverse Taking y = U(t s)x in (64) we obtain(63)

From Proposition 61 we get the following corollary

Corollary 62 The system (61) is null controllable at time t if and only if

U(t s)(H) sub Π12ts (H) (65)

From (63) it is easy to get upper bounds of |Πminus12ts U(t s)x|2 by choosing proper null

control functions u The following proposition is analogous to [42 Proposition 21]

Proposition 63 Let t gt s Assume that for every r isin [s t] the operator C(r) is invertibleThen for every strictly positive function ξ isin C([s t])

|Πminus12t U(t s)x|2 le

int ts|C(r)minus1U(r s)x|2 ξ2

r dr(int tsξr dr

)2 x isin H (66)

In particular if C(r) equiv C and |Cminus1U(r s)x|2 le h(r)|Cminus1x|2 for every x isin H then

|Πminus12t U(t s)x|2 le |Cminus1x|2int t

sh(r)minus1 dr

x isin H (67)

Proof We only need to consider the case when U(t s)x isin Π12ts (H) and the function [s t] 3

r 7rarr ξrC(r)minus1U(r s)x belongs to L2([0 t]H) Then the following function

u(r) = minus ξrint tsξr dr

C(r)minus1U(r s)x r isin [s t]

is a null control of the system (61) And hence the estimate (66) follows from (63) Thesecond estimate (67) follows by taking ξ(r) = h(r)minus1 for all r isin [s t]

Acknowledgment The authors thank Alexander Grigorprimeyan and Zenghu Li for usefuldiscussions Furthermore we thank the referee for his careful reading of the paper andvaluable suggestions for improving this paper

REFERENCES

[1] David Applebaum Levy processes in stochastic differential geometry in Levy Processes pp 111ndash137Birkhauser Boston Boston MA 2001

[2] Martingale-valued measures Ornstein-Uhlenbeck processes with jumps and operator self-decomposability in Hilbert space In memoriam Paul-Andre Meyer Seminaire de Probabilites XXXIXin Lecture Notes in Mathematics vol 1874 pp 171ndash196 Springer Berlin 2006

[3] Covariant Mehler semigroups in Hilbert space Markov Process Related Fields 13 (2007)no 1 159ndash168

[4] On the infinitesimal generators of Ornstein-Uhlenbeck processes with jumps in Hilbert spacePotential Anal 26 (2007) no 1 79ndash100

[5] O E Barndorff-Nielsen J L Jensen and M Soslashrensen Some stationary processes in discrete and con-tinuous time Adv in Appl Probab 30 (1998) no 4 989ndash1007

[6] P Becker-Kern and W Hazod Mehler hemigroups and embedding of discrete skew convolution semi-groups on simply connected nilpotent Lie groups in Infinite Dimensional Harmonic Analysis IVpp 32ndash46 World Sci Publ Hackensack NJ 2009

[7] Patrick Billingsley Convergence of probability measures second ed Wiley Series in Probability andStatistics Probability and Statistics John Wiley amp Sons Inc New York 1999 A Wiley-IntersciencePublication

[8] Vladimir I Bogachev and Michael Rockner Mehler formula and capacities for infinite-dimensionalOrnstein-Uhlenbeck processes with general linear drift Osaka J Math 32 (1995) no 2 237ndash274

[9] Vladimir I Bogachev Michael Rockner and Byron Schmuland Generalized Mehler semigroups andapplications Probab Theory Related Fields 105 (1996) no 2 193ndash225

[10] Eike Born Hemigroups of probability measures on a locally compact group Math Ann 287 (1990)no 4 653ndash673

[11] G Da Prato and J Zabczyk Ergodicity for infinite-dimensional systems London Mathematical SocietyLecture Note Series vol 229 Cambridge University Press Cambridge 1996

[12] Giuseppe Da Prato Null controllability and strong Feller property of Markov transition semigroupsNonlinear Anal 25 (1995) no 9-10 941ndash949

[13] Giuseppe Da Prato and Alessandra Lunardi Ornstein-Uhlenbeck operators with time periodic coeffi-cients J Evol Equ 7 (2007) no 4 587ndash614

[14] Giuseppe Da Prato and Michael Rockner A note on evolution systems of measures for time-dependentstochastic differential equations in Seminar on Stochastic Analysis Random Fields and ApplicationsV Progr Probab vol 59 pp 115ndash122 Birkhauser Basel 2008

[15] Giuseppe Da Prato and Jerzy Zabczyk Stochastic equations in infinite dimensions Encyclopedia ofMathematics and its Applications vol 44 Cambridge University Press Cambridge 1992

[16] Second order partial differential equations in Hilbert spaces London Mathematical SocietyLecture Note Series vol 293 Cambridge University Press Cambridge 2002

[17] Ju L Dalecprimekiı and M G Kreın Stability of solutions of differential equations in Banach space Ameri-can Mathematical Society Providence RI 1974 Translated from the Russian by S Smith Translationsof Mathematical Monographs Vol 43

[18] Donald A Dawson and Zenghu Li Non-differentiable skew convolution semigroups and relatedOrnstein-Uhlenbeck processes Potential Anal 20 (2004) no 3 285ndash302

[19] Donald A Dawson Zenghu Li Byron Schmuland and Wei Sun Generalized Mehler semigroups andcatalytic branching processes with immigration Potential Anal 21 (2004) no 1 75ndash97

[20] Eugene Borisovich Dynkin Three classes of infinite-dimensional diffusions J Funct Anal 86 (1989)no 1 75ndash110

[21] Klaus-Jochen Engel and Rainer Nagel One-parameter semigroups for linear evolution equations Grad-uate Texts in Mathematics vol 194 Springer-Verlag New York 2000 With contributions by S BrendleM Campiti T Hahn G Metafune G Nickel D Pallara C Perazzoli A Rhandi S Romanelli and RSchnaubelt

[22] Marco Fuhrman and Michael Rockner Generalized Mehler semigroups the non-Gaussian case Poten-tial Anal 12 (2000) no 1 1ndash47

[23] Matthias Geissert and Alessandra Lunardi Asymptotic behavior and hypercontractivity in non-autonomous Ornstein-Uhlenbeck equations J Lond Math Soc (2) 79(1)85ndash106 2009

[24] Martin Hairer An introduction to stochastic PDEs 2009 httparxivorgpdf09074178[25] Wilfried Hazod On Mehler semigroups stable hemigroups and selfdecomposability in Infinite Dimen-

sional Harmonic Analysis III pp 83ndash97 World Sci Publ Hackensack NJ 2005[26] Herbert Heyer Structural aspects in the theory of probability second ed Series on Multivariate Analy-

sis vol 8 World Scientific Publishing Co Pte Ltd Hackensack NJ 2010 With an additional chapterby Gyula Pap

[27] Herbert Heyer and Gyula Pap Gaussian hemigroups on a locally compact group Acta Math Hungar103 (2004) no 3 197ndash224

[28] Goran Hognas and Arunava Mukherjea Probability measures on semigroups Probability and Its Appli-cations Springer Verlag 2011 2nd ed

[29] Kiyoshi Ito Essentials of stochastic processes Translations of Mathematical Monographs vol 231American Mathematical Society Providence RI 2006 Translated from the 1957 Japanese original byYuji Ito (See also the 1961 Chinese version)

[30] Zbigniew J Jurek Measure valued cocycles from my papers in 1982 and 1983 and Mehler semigroupsavailabe at httpwwwmathuniwrocpl˜zjjurek 2004

[31] Zbigniew J Jurek and J David Mason Operator-limit distributions in probability theory Wiley Seriesin Probability and Mathematical Statistics Probability and Mathematical Statistics John Wiley amp SonsInc New York 1993 A Wiley-Interscience Publication

[32] Florian Knable Ornstein-Uhlenbeck equations with time-dependent coefficients and Levy noise in finiteand infinite dimensions J Evol Equ 11 (2011) no 4 959ndash993

[33] Paul Lescot and Michael Rockner Generators of Mehler-type semigroups as pseudo-differential opera-tors Infin Dimens Anal Quantum Probab Relat Top 5 (2002) no 3 297ndash315

[34] Perturbations of generalized Mehler semigroups and applications to stochastic heat equationswith Levy noise and singular drift Potential Anal 20 (2004) no 4 317ndash344

[35] Zenghu Li Branching processes with immigration and related topics Front Math China 1 (2006) no 173ndash97

[36] Measure-valued branching markov processes Spriger Verlag 2011[37] Eugene Lukacs Characteristic functions Hafner Publishing Co New York 1970 Second edition re-

vised and enlarged[38] Eugene Lukacs Stochastic convergence second ed vol 30 Academic Press [Harcourt Brace Jo-

vanovich Publishers] New York 1975 Probability and Mathematical Statistics Vol 30[39] Makoto Maejima and Ken-iti Sato Semi-Levy processes semi-selfsimilar additive processes and semi-

stationary Ornstein-Uhlenbeck type processes J Math Kyoto Univ 43 (2003) no 3 609ndash639[40] Shun-Xiang Ouyang Harnack inequalities and applications for stochastic equations PhD thesis Biele-

feld University 2009 Available on httpbiesonubuni-bielefelddevolltexte

20091463pdfouyangpdf[41] Infinite dimensional additive processes and stochastic integrals in preparation 2014[42] Shun-Xiang Ouyang Michael Rockner and Feng-Yu Wang Harnack inequalities and applications for

Ornstein-Uhlenbeck semigroups with jump Potential Anal 36 (2012) no 2 301ndash315[43] K R Parthasarathy Probability measures on metric spaces AMS Chelsea Publishing Providence RI

2005 Reprint of the 1967 original[44] Amnon Pazy Semigroups of linear operators and applications to partial differential equations Applied

Mathematical Sciences vol 44 Springer-Verlag New York 1983[45] Michael Rockner and Feng-Yu Wang Harnack and functional inequalities for generalized Mehler semi-

groups J Funct Anal 203 (2003) no 1 237ndash261[46] Jan Rosinski Bilinear random integrals Dissertationes Math (Rozprawy Mat) 259 (1987)[47] Ken-iti Sato Levy processes and infinitely divisible distributions Cambridge Studies in Advanced Math-

ematics vol 68 Cambridge University Press Cambridge 1999 Translated from the 1990 Japaneseoriginal Revised by the author

[48] Stochastic integrals in additive processes and application to semi-Levy processes Osaka JMath 41 (2004) no 1 211ndash236

[49] Additive processes and stochastic integrals Illinois J Math 50 (2006) no 1-4 825ndash851 (elec-tronic)

[50] Ken-iti Sato and Makoto Yamazato Operator-self-decomposable distributions as limit distributions ofprocesses of Ornstein-Uhlenbeck type Stochastic Process Appl 17 (1984) no 1 73ndash100

[51] Byron Schmuland and Wei Sun On the equation microt+s = micros lowast Tsmicrot Statist Probab Lett 52 (2001)no 2 183ndash188

[52] N N Vakhania V I Tarieladze and S A Chobanyan Probability distributions on Banach spacesMathematics and its Applications (Soviet Series) vol 14 D Reidel Publishing Co Dordrecht 1987Translated from the Russian and with a preface by Wojbor A Woyczynski

[53] J M A M van Neerven Continuity and representation of Gaussian Mehler semigroups Potential Anal13 (2000) no 3 199ndash211

[54] Feng-Yu Wang Functional inequalities Markov processes and spectral theory Science Press Bei-jingNew York 2004

[55] Feng-Yu Wang Harnack inequalities on manifolds with boundary and applications J Math Pures Appl(9) 94 (2010) no 3 304ndash321

[56] Stephen James Wolfe On a continuous analogue of the stochastic difference equation Xn = ρXnminus1 +

Bn Stochastic Process Appl 12 (1982) no 3 301ndash312[57] Robert Wooster Evolution systems of measures for non-autonomous Ornstein-Uhlenbeck processes with

Levy noise Comm Stoch Anal 5 (2011) no 2 353ndash370[58] Jerzy Zabczyk Mathematical control theory Modern Birkhauser Classics Birkhauser Boston Inc

Boston MA 2008 An introduction Reprint of the 1995 edition

(Shun-Xiang Ouyang) DEPARTMENT OF MATHEMATICS UNIVERSITY OF BIELEFELD 33615 BIELE-FELD GERMANY

E-mail address Corresponding author souyangmathuni-bielefeldde

(Michael Rockner) DEPARTMENT OF MATHEMATICS UNIVERSITY OF BIELEFELD 33615 BIELEFELDGERMANY

E-mail address roecknermathuni-bielefeldde

1 Introduction
2 Generalized Mehler semigroups and skew convolution equations
3 Time inhomogeneous skew convolution equations
- 31 Preliminaries and motivations
- 32 Weak continuity
- 33 Infinite divisibility
- 34 Associated stochastic processes
- - 4 Evolution systems of measures
  - - 41 Some properties
    - 42 Existence and uniqueness
    - - 5 Harnack inequalities and applications
      - 6 Appendix null controllability
      - References

Page 2: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

6 Appendix null controllability 43References 44

1 INTRODUCTION

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

Let H be a real separable Hilbert space with norm and inner product denoted by | middot | and〈middot middot〉 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 HA time homogeneous generalized Mehler semigroup (pt)tge0 on H is defined by

ptf(x) =

intHf(Ttx+ y)microt(dy) t ge 0 x isin H f isin Bb(H) (11)

Here (Tt)tge0 is a strongly continuous semigroup on H and (microt)tge0 is a family of probabilitymeasures on (HB(H)) satisfying the following skew convolution (semigroup) equation

microt+s = micros lowast (microt Tminus1s ) s t ge 0 (12)

Recall that for any two positive Borel measures micro and ν on H the convolution micro lowast ν of microand ν is a Borel measure on (HB(H)) such that

micro lowast ν(B) =

intH

1B(x+ y)micro(dx)ν(dy) =

intHmicro(B minus x) ν(dx) B isin B(H)

Condition (12) is necessary and sufficient for the semigroup property of (pt)tge0 (and theMarkov property of the corresponding stochastic process respectively) to hold That is (see[22])

(12) holds if and only if for all t s ge 0 ptps = pt+s on Bb(H)

The semigroup (11) is a generalization of the classical Mehler formula for the transi-tion semigroup of an Ornstein-Uhlenbeck process driven by a Wiener process The secondnamed author and his coauthors studied this generalization for the Gaussian case in [8 9] aswell 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 semigroupsof Ornstein-Uhlenbeck processes driven by Levy processes (cf [9 22])

Generalized Mehler semigroups and skew convolution equations have been extensivelystudied For instance Schmuland and Sun [51] investigated the infinite divisibility of microt (t ge0) and the continuity of t 7rarr log microt (here microt denotes the Fourier transformation of microt seealso (211)) Lescot and Rockner considered in [33] and [34] the generator and perturbationsof (pt)tge0 respectively Wang and Rockner established some useful functional inequalitiesfor (pt)tge0 van Neerven [53] Li and his coauthors [18 19] studied the representation ofmicrot (t ge 0) For more literature on this topic we refer to [2 3 4 30 36] and the referencestherein

Recently much work for instance [13 14 23 32 57] has been devoted to the studyof non-autonomous Ornstein-Uhlenbeck processes which are solutions to linear stochasticpartial 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 Levy process To get

xs = x

A(t) =dsum

ij=1

part

partxi

(aij(x t)

part

partxj

)

dsumij=1

int ts

Xs = x(13)

Xts =

int t

s

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

int t

s

) t ge r ge s

pstf(x) =

) t ge r ge s (18)

dxt = a(t)xtdt+ dbt

xs = x(19)

x(t s x) = exp

(int t

s

a(r) dr

)x+

int t

s

exp

(int t

u

a(r) dr

Clearly int t

s

exp

(int t

u

a(r) dr

)dbu sim N(0 σts)

σts =

int t

s

exp

(2

int t

u

a(r) dr

)d u

(2

int t

r

a(r) dr

) t ge r ge s

pstf(x) =

intHf

(exp

(int t

s

a(r) dr

)x+ y

H2

microust(A) =

qstf(x) =

qsrqrtf(x)

=

intH2

=

intH4

=

intH4

=

intH5

=

intH4

=

intH3

=

(28)

qsrqrtf(x) =

qsrqrtf(0) =

qstf(0) =

intHf(z)microts(dz)

micro(ξ) =

intH

limnrarrinfin

intHf(x)micro(dx)

limnrarrinfin

micron = micro

XnPrminusrarr X

(33)

limtdarr0

P(|Xt| ge ε) = 0 (35)

limtdarrs

microts = limtuarrs

microst = δ0 (38)

micrott = δ0 (310)

limnrarrinfin

lim supnrarrinfin

2 (312)

4M (313)

intHf dδ0

=

intxisinH |x|geε0

intxisinH |x|ltε0

lt2M middot η

4M+η

2= η

limtdarrs

sup(ts)isinΛt0s0

U(t s) ltinfin

)(323)

) (324)

U(t s+ ε) le c

limεdarr0

εdarr0

This proves (326)

limεdarr0

= limεdarr0

=

minus1(y)

This proves (320)

andlimtdarrs

microts = limsuarrt

microts = δ0

∣∣∣∣ lt ε (336)

∣∣∣∣le∣∣∣∣int

∣∣∣∣+

intK

lt2ε(1 + finfin)

microts

(A

(εprime

2c

and

micrort

(A

(εprime

2c

2cle εprime

and

δ = min

δtj2

j = 1 2 n

By (18) (342) (343) and (345) we have

=

intH

leintH

intH

ltη

2+η

2= η

=

intH

geintH

intH

ge η(

1minus η

2

)= η minus η2

2

2gt η minus η2

2

=(microts U(tj t)

minus1) U(t0 tj)

then by (18) (342) and (344)η

=

intH

ge η(

1minus η

2

)

complete

micro10 =

2mminus1prodlowast

j=0

micro j+12m

j2m U

(1j + 1

2m

)minus1

(348)

By (18) we have

(micro 1

20 U

(1

1

2

)minus1)

micro j+12m

j2m

= micro (2j+1)+1

2m+1 2j+1

2m+1lowast

(micro 2j+1

2m+1 2j

2m+1 U

((2j + 1) + 1

2m+12j + 1

2m+1

)minus1)

So

micro j+12m

j2m U

(1j + 1

2m

)minus1

=micro (2j+1)+1

2m+1 2j+1

2m+1 U

(1

(2j + 1) + 1

2m+1

)minus1

lowast

(micro 2j+1

2m+1 2j

2m+1 U

(1

2j + 1

2m+1

)minus1)

=

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

micro10 =

2mminus1prodlowast

j=0

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

=

k=0

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

2〈ξ Rtsξ〉

minusintH

1 + |x|2

)mts(dx) ξ isin H

(350)

intH

1 + |x|2

)mtr(dx)

minusintH

1 + |x|2

)mrs(dx)

minusintH

1 + |x|2

+

intH

i〈ξ U(t r)x〉1 + |x|2

mrs(dx)minusintH

i〈ξ U(t r)x〉1 + |U(t r)x|2

mrs(dx)

+

intHU(t r)x

(1

1 + |x|2

)mrs(dx)

(352)

(355)

minus1

Then

micross = δ0 (357)

(358)

=(microtr lowast

)) U(t0 t)

minus1

=(microtr U(t0 t)

minus1)

=

intH

(359)

)such that for

=

intHotimesn

(361)

E

[exp

(i

nsumj=1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

=nprodj=1

intH

E[exp

〉)]

=

intH

and

E

[exp

(i

nsumj=1

)]=

nprodj=1

E[exp

〉)] (363)

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 3: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

xs = x

A(t) =dsum

ij=1

part

partxi

(aij(x t)

part

partxj

)

dsumij=1

int ts

Xs = x(13)

Xts =

int t

s

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

int t

s

) t ge r ge s

pstf(x) =

) t ge r ge s (18)

dxt = a(t)xtdt+ dbt

xs = x(19)

x(t s x) = exp

(int t

s

a(r) dr

)x+

int t

s

exp

(int t

u

a(r) dr

Clearly int t

s

exp

(int t

u

a(r) dr

)dbu sim N(0 σts)

σts =

int t

s

exp

(2

int t

u

a(r) dr

)d u

(2

int t

r

a(r) dr

) t ge r ge s

pstf(x) =

intHf

(exp

(int t

s

a(r) dr

)x+ y

H2

microust(A) =

qstf(x) =

qsrqrtf(x)

=

intH2

=

intH4

=

intH4

=

intH5

=

intH4

=

intH3

=

(28)

qsrqrtf(x) =

qsrqrtf(0) =

qstf(0) =

intHf(z)microts(dz)

micro(ξ) =

intH

limnrarrinfin

intHf(x)micro(dx)

limnrarrinfin

micron = micro

XnPrminusrarr X

(33)

limtdarr0

P(|Xt| ge ε) = 0 (35)

limtdarrs

microts = limtuarrs

microst = δ0 (38)

micrott = δ0 (310)

limnrarrinfin

lim supnrarrinfin

2 (312)

4M (313)

intHf dδ0

=

intxisinH |x|geε0

intxisinH |x|ltε0

lt2M middot η

4M+η

2= η

limtdarrs

sup(ts)isinΛt0s0

U(t s) ltinfin

)(323)

) (324)

U(t s+ ε) le c

limεdarr0

εdarr0

This proves (326)

limεdarr0

= limεdarr0

=

minus1(y)

This proves (320)

andlimtdarrs

microts = limsuarrt

microts = δ0

∣∣∣∣ lt ε (336)

∣∣∣∣le∣∣∣∣int

∣∣∣∣+

intK

lt2ε(1 + finfin)

microts

(A

(εprime

2c

and

micrort

(A

(εprime

2c

2cle εprime

and

δ = min

δtj2

j = 1 2 n

By (18) (342) (343) and (345) we have

=

intH

leintH

intH

ltη

2+η

2= η

=

intH

geintH

intH

ge η(

1minus η

2

)= η minus η2

2

2gt η minus η2

2

=(microts U(tj t)

minus1) U(t0 tj)

then by (18) (342) and (344)η

=

intH

ge η(

1minus η

2

)

complete

micro10 =

2mminus1prodlowast

j=0

micro j+12m

j2m U

(1j + 1

2m

)minus1

(348)

By (18) we have

(micro 1

20 U

(1

1

2

)minus1)

micro j+12m

j2m

= micro (2j+1)+1

2m+1 2j+1

2m+1lowast

(micro 2j+1

2m+1 2j

2m+1 U

((2j + 1) + 1

2m+12j + 1

2m+1

)minus1)

So

micro j+12m

j2m U

(1j + 1

2m

)minus1

=micro (2j+1)+1

2m+1 2j+1

2m+1 U

(1

(2j + 1) + 1

2m+1

)minus1

lowast

(micro 2j+1

2m+1 2j

2m+1 U

(1

2j + 1

2m+1

)minus1)

=

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

micro10 =

2mminus1prodlowast

j=0

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

=

k=0

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

2〈ξ Rtsξ〉

minusintH

1 + |x|2

)mts(dx) ξ isin H

(350)

intH

1 + |x|2

)mtr(dx)

minusintH

1 + |x|2

)mrs(dx)

minusintH

1 + |x|2

+

intH

i〈ξ U(t r)x〉1 + |x|2

mrs(dx)minusintH

i〈ξ U(t r)x〉1 + |U(t r)x|2

mrs(dx)

+

intHU(t r)x

(1

1 + |x|2

)mrs(dx)

(352)

(355)

minus1

Then

micross = δ0 (357)

(358)

=(microtr lowast

)) U(t0 t)

minus1

=(microtr U(t0 t)

minus1)

=

intH

(359)

)such that for

=

intHotimesn

(361)

E

[exp

(i

nsumj=1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

=nprodj=1

intH

E[exp

〉)]

=

intH

and

E

[exp

(i

nsumj=1

)]=

nprodj=1

E[exp

〉)] (363)

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 4: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

Xts =

int t

s

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

int t

s

) t ge r ge s

pstf(x) =

) t ge r ge s (18)

dxt = a(t)xtdt+ dbt

xs = x(19)

x(t s x) = exp

(int t

s

a(r) dr

)x+

int t

s

exp

(int t

u

a(r) dr

Clearly int t

s

exp

(int t

u

a(r) dr

)dbu sim N(0 σts)

σts =

int t

s

exp

(2

int t

u

a(r) dr

)d u

(2

int t

r

a(r) dr

) t ge r ge s

pstf(x) =

intHf

(exp

(int t

s

a(r) dr

)x+ y

H2

microust(A) =

qstf(x) =

qsrqrtf(x)

=

intH2

=

intH4

=

intH4

=

intH5

=

intH4

=

intH3

=

(28)

qsrqrtf(x) =

qsrqrtf(0) =

qstf(0) =

intHf(z)microts(dz)

micro(ξ) =

intH

limnrarrinfin

intHf(x)micro(dx)

limnrarrinfin

micron = micro

XnPrminusrarr X

(33)

limtdarr0

P(|Xt| ge ε) = 0 (35)

limtdarrs

microts = limtuarrs

microst = δ0 (38)

micrott = δ0 (310)

limnrarrinfin

lim supnrarrinfin

2 (312)

4M (313)

intHf dδ0

=

intxisinH |x|geε0

intxisinH |x|ltε0

lt2M middot η

4M+η

2= η

limtdarrs

sup(ts)isinΛt0s0

U(t s) ltinfin

)(323)

) (324)

U(t s+ ε) le c

limεdarr0

εdarr0

This proves (326)

limεdarr0

= limεdarr0

=

minus1(y)

This proves (320)

andlimtdarrs

microts = limsuarrt

microts = δ0

∣∣∣∣ lt ε (336)

∣∣∣∣le∣∣∣∣int

∣∣∣∣+

intK

lt2ε(1 + finfin)

microts

(A

(εprime

2c

and

micrort

(A

(εprime

2c

2cle εprime

and

δ = min

δtj2

j = 1 2 n

By (18) (342) (343) and (345) we have

=

intH

leintH

intH

ltη

2+η

2= η

=

intH

geintH

intH

ge η(

1minus η

2

)= η minus η2

2

2gt η minus η2

2

=(microts U(tj t)

minus1) U(t0 tj)

then by (18) (342) and (344)η

=

intH

ge η(

1minus η

2

)

complete

micro10 =

2mminus1prodlowast

j=0

micro j+12m

j2m U

(1j + 1

2m

)minus1

(348)

By (18) we have

(micro 1

20 U

(1

1

2

)minus1)

micro j+12m

j2m

= micro (2j+1)+1

2m+1 2j+1

2m+1lowast

(micro 2j+1

2m+1 2j

2m+1 U

((2j + 1) + 1

2m+12j + 1

2m+1

)minus1)

So

micro j+12m

j2m U

(1j + 1

2m

)minus1

=micro (2j+1)+1

2m+1 2j+1

2m+1 U

(1

(2j + 1) + 1

2m+1

)minus1

lowast

(micro 2j+1

2m+1 2j

2m+1 U

(1

2j + 1

2m+1

)minus1)

=

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

micro10 =

2mminus1prodlowast

j=0

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

=

k=0

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

2〈ξ Rtsξ〉

minusintH

1 + |x|2

)mts(dx) ξ isin H

(350)

intH

1 + |x|2

)mtr(dx)

minusintH

1 + |x|2

)mrs(dx)

minusintH

1 + |x|2

+

intH

i〈ξ U(t r)x〉1 + |x|2

mrs(dx)minusintH

i〈ξ U(t r)x〉1 + |U(t r)x|2

mrs(dx)

+

intHU(t r)x

(1

1 + |x|2

)mrs(dx)

(352)

(355)

minus1

Then

micross = δ0 (357)

(358)

=(microtr lowast

)) U(t0 t)

minus1

=(microtr U(t0 t)

minus1)

=

intH

(359)

)such that for

=

intHotimesn

(361)

E

[exp

(i

nsumj=1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

=nprodj=1

intH

E[exp

〉)]

=

intH

and

E

[exp

(i

nsumj=1

)]=

nprodj=1

E[exp

〉)] (363)

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 5: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

Clearly int t

s

exp

(int t

u

a(r) dr

)dbu sim N(0 σts)

σts =

int t

s

exp

(2

int t

u

a(r) dr

)d u

(2

int t

r

a(r) dr

) t ge r ge s

pstf(x) =

intHf

(exp

(int t

s

a(r) dr

)x+ y

H2

microust(A) =

qstf(x) =

qsrqrtf(x)

=

intH2

=

intH4

=

intH4

=

intH5

=

intH4

=

intH3

=

(28)

qsrqrtf(x) =

qsrqrtf(0) =

qstf(0) =

intHf(z)microts(dz)

micro(ξ) =

intH

limnrarrinfin

intHf(x)micro(dx)

limnrarrinfin

micron = micro

XnPrminusrarr X

(33)

limtdarr0

P(|Xt| ge ε) = 0 (35)

limtdarrs

microts = limtuarrs

microst = δ0 (38)

micrott = δ0 (310)

limnrarrinfin

lim supnrarrinfin

2 (312)

4M (313)

intHf dδ0

=

intxisinH |x|geε0

intxisinH |x|ltε0

lt2M middot η

4M+η

2= η

limtdarrs

sup(ts)isinΛt0s0

U(t s) ltinfin

)(323)

) (324)

U(t s+ ε) le c

limεdarr0

εdarr0

This proves (326)

limεdarr0

= limεdarr0

=

minus1(y)

This proves (320)

andlimtdarrs

microts = limsuarrt

microts = δ0

∣∣∣∣ lt ε (336)

∣∣∣∣le∣∣∣∣int

∣∣∣∣+

intK

lt2ε(1 + finfin)

microts

(A

(εprime

2c

and

micrort

(A

(εprime

2c

2cle εprime

and

δ = min

δtj2

j = 1 2 n

By (18) (342) (343) and (345) we have

=

intH

leintH

intH

ltη

2+η

2= η

=

intH

geintH

intH

ge η(

1minus η

2

)= η minus η2

2

2gt η minus η2

2

=(microts U(tj t)

minus1) U(t0 tj)

then by (18) (342) and (344)η

=

intH

ge η(

1minus η

2

)

complete

micro10 =

2mminus1prodlowast

j=0

micro j+12m

j2m U

(1j + 1

2m

)minus1

(348)

By (18) we have

(micro 1

20 U

(1

1

2

)minus1)

micro j+12m

j2m

= micro (2j+1)+1

2m+1 2j+1

2m+1lowast

(micro 2j+1

2m+1 2j

2m+1 U

((2j + 1) + 1

2m+12j + 1

2m+1

)minus1)

So

micro j+12m

j2m U

(1j + 1

2m

)minus1

=micro (2j+1)+1

2m+1 2j+1

2m+1 U

(1

(2j + 1) + 1

2m+1

)minus1

lowast

(micro 2j+1

2m+1 2j

2m+1 U

(1

2j + 1

2m+1

)minus1)

=

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

micro10 =

2mminus1prodlowast

j=0

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

=

k=0

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

2〈ξ Rtsξ〉

minusintH

1 + |x|2

)mts(dx) ξ isin H

(350)

intH

1 + |x|2

)mtr(dx)

minusintH

1 + |x|2

)mrs(dx)

minusintH

1 + |x|2

+

intH

i〈ξ U(t r)x〉1 + |x|2

mrs(dx)minusintH

i〈ξ U(t r)x〉1 + |U(t r)x|2

mrs(dx)

+

intHU(t r)x

(1

1 + |x|2

)mrs(dx)

(352)

(355)

minus1

Then

micross = δ0 (357)

(358)

=(microtr lowast

)) U(t0 t)

minus1

=(microtr U(t0 t)

minus1)

=

intH

(359)

)such that for

=

intHotimesn

(361)

E

[exp

(i

nsumj=1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

=nprodj=1

intH

E[exp

〉)]

=

intH

and

E

[exp

(i

nsumj=1

)]=

nprodj=1

E[exp

〉)] (363)

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 6: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

H2

microust(A) =

qstf(x) =

qsrqrtf(x)

=

intH2

=

intH4

=

intH4

=

intH5

=

intH4

=

intH3

=

(28)

qsrqrtf(x) =

qsrqrtf(0) =

qstf(0) =

intHf(z)microts(dz)

micro(ξ) =

intH

limnrarrinfin

intHf(x)micro(dx)

limnrarrinfin

micron = micro

XnPrminusrarr X

(33)

limtdarr0

P(|Xt| ge ε) = 0 (35)

limtdarrs

microts = limtuarrs

microst = δ0 (38)

micrott = δ0 (310)

limnrarrinfin

lim supnrarrinfin

2 (312)

4M (313)

intHf dδ0

=

intxisinH |x|geε0

intxisinH |x|ltε0

lt2M middot η

4M+η

2= η

limtdarrs

sup(ts)isinΛt0s0

U(t s) ltinfin

)(323)

) (324)

U(t s+ ε) le c

limεdarr0

εdarr0

This proves (326)

limεdarr0

= limεdarr0

=

minus1(y)

This proves (320)

andlimtdarrs

microts = limsuarrt

microts = δ0

∣∣∣∣ lt ε (336)

∣∣∣∣le∣∣∣∣int

∣∣∣∣+

intK

lt2ε(1 + finfin)

microts

(A

(εprime

2c

and

micrort

(A

(εprime

2c

2cle εprime

and

δ = min

δtj2

j = 1 2 n

By (18) (342) (343) and (345) we have

=

intH

leintH

intH

ltη

2+η

2= η

=

intH

geintH

intH

ge η(

1minus η

2

)= η minus η2

2

2gt η minus η2

2

=(microts U(tj t)

minus1) U(t0 tj)

then by (18) (342) and (344)η

=

intH

ge η(

1minus η

2

)

complete

micro10 =

2mminus1prodlowast

j=0

micro j+12m

j2m U

(1j + 1

2m

)minus1

(348)

By (18) we have

(micro 1

20 U

(1

1

2

)minus1)

micro j+12m

j2m

= micro (2j+1)+1

2m+1 2j+1

2m+1lowast

(micro 2j+1

2m+1 2j

2m+1 U

((2j + 1) + 1

2m+12j + 1

2m+1

)minus1)

So

micro j+12m

j2m U

(1j + 1

2m

)minus1

=micro (2j+1)+1

2m+1 2j+1

2m+1 U

(1

(2j + 1) + 1

2m+1

)minus1

lowast

(micro 2j+1

2m+1 2j

2m+1 U

(1

2j + 1

2m+1

)minus1)

=

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

micro10 =

2mminus1prodlowast

j=0

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

=

k=0

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

2〈ξ Rtsξ〉

minusintH

1 + |x|2

)mts(dx) ξ isin H

(350)

intH

1 + |x|2

)mtr(dx)

minusintH

1 + |x|2

)mrs(dx)

minusintH

1 + |x|2

+

intH

i〈ξ U(t r)x〉1 + |x|2

mrs(dx)minusintH

i〈ξ U(t r)x〉1 + |U(t r)x|2

mrs(dx)

+

intHU(t r)x

(1

1 + |x|2

)mrs(dx)

(352)

(355)

minus1

Then

micross = δ0 (357)

(358)

=(microtr lowast

)) U(t0 t)

minus1

=(microtr U(t0 t)

minus1)

=

intH

(359)

)such that for

=

intHotimesn

(361)

E

[exp

(i

nsumj=1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

=nprodj=1

intH

E[exp

〉)]

=

intH

and

E

[exp

(i

nsumj=1

)]=

nprodj=1

E[exp

〉)] (363)

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 7: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

qstf(x) =

qsrqrtf(x)

=

intH2

=

intH4

=

intH4

=

intH5

=

intH4

=

intH3

=

(28)

qsrqrtf(x) =

qsrqrtf(0) =

qstf(0) =

intHf(z)microts(dz)

micro(ξ) =

intH

limnrarrinfin

intHf(x)micro(dx)

limnrarrinfin

micron = micro

XnPrminusrarr X

(33)

limtdarr0

P(|Xt| ge ε) = 0 (35)

limtdarrs

microts = limtuarrs

microst = δ0 (38)

micrott = δ0 (310)

limnrarrinfin

lim supnrarrinfin

2 (312)

4M (313)

intHf dδ0

=

intxisinH |x|geε0

intxisinH |x|ltε0

lt2M middot η

4M+η

2= η

limtdarrs

sup(ts)isinΛt0s0

U(t s) ltinfin

)(323)

) (324)

U(t s+ ε) le c

limεdarr0

εdarr0

This proves (326)

limεdarr0

= limεdarr0

=

minus1(y)

This proves (320)

andlimtdarrs

microts = limsuarrt

microts = δ0

∣∣∣∣ lt ε (336)

∣∣∣∣le∣∣∣∣int

∣∣∣∣+

intK

lt2ε(1 + finfin)

microts

(A

(εprime

2c

and

micrort

(A

(εprime

2c

2cle εprime

and

δ = min

δtj2

j = 1 2 n

By (18) (342) (343) and (345) we have

=

intH

leintH

intH

ltη

2+η

2= η

=

intH

geintH

intH

ge η(

1minus η

2

)= η minus η2

2

2gt η minus η2

2

=(microts U(tj t)

minus1) U(t0 tj)

then by (18) (342) and (344)η

=

intH

ge η(

1minus η

2

)

complete

micro10 =

2mminus1prodlowast

j=0

micro j+12m

j2m U

(1j + 1

2m

)minus1

(348)

By (18) we have

(micro 1

20 U

(1

1

2

)minus1)

micro j+12m

j2m

= micro (2j+1)+1

2m+1 2j+1

2m+1lowast

(micro 2j+1

2m+1 2j

2m+1 U

((2j + 1) + 1

2m+12j + 1

2m+1

)minus1)

So

micro j+12m

j2m U

(1j + 1

2m

)minus1

=micro (2j+1)+1

2m+1 2j+1

2m+1 U

(1

(2j + 1) + 1

2m+1

)minus1

lowast

(micro 2j+1

2m+1 2j

2m+1 U

(1

2j + 1

2m+1

)minus1)

=

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

micro10 =

2mminus1prodlowast

j=0

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

=

k=0

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

2〈ξ Rtsξ〉

minusintH

1 + |x|2

)mts(dx) ξ isin H

(350)

intH

1 + |x|2

)mtr(dx)

minusintH

1 + |x|2

)mrs(dx)

minusintH

1 + |x|2

+

intH

i〈ξ U(t r)x〉1 + |x|2

mrs(dx)minusintH

i〈ξ U(t r)x〉1 + |U(t r)x|2

mrs(dx)

+

intHU(t r)x

(1

1 + |x|2

)mrs(dx)

(352)

(355)

minus1

Then

micross = δ0 (357)

(358)

=(microtr lowast

)) U(t0 t)

minus1

=(microtr U(t0 t)

minus1)

=

intH

(359)

)such that for

=

intHotimesn

(361)

E

[exp

(i

nsumj=1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

=nprodj=1

intH

E[exp

〉)]

=

intH

and

E

[exp

(i

nsumj=1

)]=

nprodj=1

E[exp

〉)] (363)

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 8: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

intHf(z)microts(dz)

micro(ξ) =

intH

limnrarrinfin

intHf(x)micro(dx)

limnrarrinfin

micron = micro

XnPrminusrarr X

(33)

limtdarr0

P(|Xt| ge ε) = 0 (35)

limtdarrs

microts = limtuarrs

microst = δ0 (38)

micrott = δ0 (310)

limnrarrinfin

lim supnrarrinfin

2 (312)

4M (313)

intHf dδ0

=

intxisinH |x|geε0

intxisinH |x|ltε0

lt2M middot η

4M+η

2= η

limtdarrs

sup(ts)isinΛt0s0

U(t s) ltinfin

)(323)

) (324)

U(t s+ ε) le c

limεdarr0

εdarr0

This proves (326)

limεdarr0

= limεdarr0

=

minus1(y)

This proves (320)

andlimtdarrs

microts = limsuarrt

microts = δ0

∣∣∣∣ lt ε (336)

∣∣∣∣le∣∣∣∣int

∣∣∣∣+

intK

lt2ε(1 + finfin)

microts

(A

(εprime

2c

and

micrort

(A

(εprime

2c

2cle εprime

and

δ = min

δtj2

j = 1 2 n

By (18) (342) (343) and (345) we have

=

intH

leintH

intH

ltη

2+η

2= η

=

intH

geintH

intH

ge η(

1minus η

2

)= η minus η2

2

2gt η minus η2

2

=(microts U(tj t)

minus1) U(t0 tj)

then by (18) (342) and (344)η

=

intH

ge η(

1minus η

2

)

complete

micro10 =

2mminus1prodlowast

j=0

micro j+12m

j2m U

(1j + 1

2m

)minus1

(348)

By (18) we have

(micro 1

20 U

(1

1

2

)minus1)

micro j+12m

j2m

= micro (2j+1)+1

2m+1 2j+1

2m+1lowast

(micro 2j+1

2m+1 2j

2m+1 U

((2j + 1) + 1

2m+12j + 1

2m+1

)minus1)

So

micro j+12m

j2m U

(1j + 1

2m

)minus1

=micro (2j+1)+1

2m+1 2j+1

2m+1 U

(1

(2j + 1) + 1

2m+1

)minus1

lowast

(micro 2j+1

2m+1 2j

2m+1 U

(1

2j + 1

2m+1

)minus1)

=

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

micro10 =

2mminus1prodlowast

j=0

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

=

k=0

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

2〈ξ Rtsξ〉

minusintH

1 + |x|2

)mts(dx) ξ isin H

(350)

intH

1 + |x|2

)mtr(dx)

minusintH

1 + |x|2

)mrs(dx)

minusintH

1 + |x|2

+

intH

i〈ξ U(t r)x〉1 + |x|2

mrs(dx)minusintH

i〈ξ U(t r)x〉1 + |U(t r)x|2

mrs(dx)

+

intHU(t r)x

(1

1 + |x|2

)mrs(dx)

(352)

(355)

minus1

Then

micross = δ0 (357)

(358)

=(microtr lowast

)) U(t0 t)

minus1

=(microtr U(t0 t)

minus1)

=

intH

(359)

)such that for

=

intHotimesn

(361)

E

[exp

(i

nsumj=1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

=nprodj=1

intH

E[exp

〉)]

=

intH

and

E

[exp

(i

nsumj=1

)]=

nprodj=1

E[exp

〉)] (363)

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 9: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

XnPrminusrarr X

(33)

limtdarr0

P(|Xt| ge ε) = 0 (35)

limtdarrs

microts = limtuarrs

microst = δ0 (38)

micrott = δ0 (310)

limnrarrinfin

lim supnrarrinfin

2 (312)

4M (313)

intHf dδ0

=

intxisinH |x|geε0

intxisinH |x|ltε0

lt2M middot η

4M+η

2= η

limtdarrs

sup(ts)isinΛt0s0

U(t s) ltinfin

)(323)

) (324)

U(t s+ ε) le c

limεdarr0

εdarr0

This proves (326)

limεdarr0

= limεdarr0

=

minus1(y)

This proves (320)

andlimtdarrs

microts = limsuarrt

microts = δ0

∣∣∣∣ lt ε (336)

∣∣∣∣le∣∣∣∣int

∣∣∣∣+

intK

lt2ε(1 + finfin)

microts

(A

(εprime

2c

and

micrort

(A

(εprime

2c

2cle εprime

and

δ = min

δtj2

j = 1 2 n

By (18) (342) (343) and (345) we have

=

intH

leintH

intH

ltη

2+η

2= η

=

intH

geintH

intH

ge η(

1minus η

2

)= η minus η2

2

2gt η minus η2

2

=(microts U(tj t)

minus1) U(t0 tj)

then by (18) (342) and (344)η

=

intH

ge η(

1minus η

2

)

complete

micro10 =

2mminus1prodlowast

j=0

micro j+12m

j2m U

(1j + 1

2m

)minus1

(348)

By (18) we have

(micro 1

20 U

(1

1

2

)minus1)

micro j+12m

j2m

= micro (2j+1)+1

2m+1 2j+1

2m+1lowast

(micro 2j+1

2m+1 2j

2m+1 U

((2j + 1) + 1

2m+12j + 1

2m+1

)minus1)

So

micro j+12m

j2m U

(1j + 1

2m

)minus1

=micro (2j+1)+1

2m+1 2j+1

2m+1 U

(1

(2j + 1) + 1

2m+1

)minus1

lowast

(micro 2j+1

2m+1 2j

2m+1 U

(1

2j + 1

2m+1

)minus1)

=

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

micro10 =

2mminus1prodlowast

j=0

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

=

k=0

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

2〈ξ Rtsξ〉

minusintH

1 + |x|2

)mts(dx) ξ isin H

(350)

intH

1 + |x|2

)mtr(dx)

minusintH

1 + |x|2

)mrs(dx)

minusintH

1 + |x|2

+

intH

i〈ξ U(t r)x〉1 + |x|2

mrs(dx)minusintH

i〈ξ U(t r)x〉1 + |U(t r)x|2

mrs(dx)

+

intHU(t r)x

(1

1 + |x|2

)mrs(dx)

(352)

(355)

minus1

Then

micross = δ0 (357)

(358)

=(microtr lowast

)) U(t0 t)

minus1

=(microtr U(t0 t)

minus1)

=

intH

(359)

)such that for

=

intHotimesn

(361)

E

[exp

(i

nsumj=1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

=nprodj=1

intH

E[exp

〉)]

=

intH

and

E

[exp

(i

nsumj=1

)]=

nprodj=1

E[exp

〉)] (363)

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 10: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

limtdarrs

microts = limtuarrs

microst = δ0 (38)

micrott = δ0 (310)

limnrarrinfin

lim supnrarrinfin

2 (312)

4M (313)

intHf dδ0

=

intxisinH |x|geε0

intxisinH |x|ltε0

lt2M middot η

4M+η

2= η

limtdarrs

sup(ts)isinΛt0s0

U(t s) ltinfin

)(323)

) (324)

U(t s+ ε) le c

limεdarr0

εdarr0

This proves (326)

limεdarr0

= limεdarr0

=

minus1(y)

This proves (320)

andlimtdarrs

microts = limsuarrt

microts = δ0

∣∣∣∣ lt ε (336)

∣∣∣∣le∣∣∣∣int

∣∣∣∣+

intK

lt2ε(1 + finfin)

microts

(A

(εprime

2c

and

micrort

(A

(εprime

2c

2cle εprime

and

δ = min

δtj2

j = 1 2 n

By (18) (342) (343) and (345) we have

=

intH

leintH

intH

ltη

2+η

2= η

=

intH

geintH

intH

ge η(

1minus η

2

)= η minus η2

2

2gt η minus η2

2

=(microts U(tj t)

minus1) U(t0 tj)

then by (18) (342) and (344)η

=

intH

ge η(

1minus η

2

)

complete

micro10 =

2mminus1prodlowast

j=0

micro j+12m

j2m U

(1j + 1

2m

)minus1

(348)

By (18) we have

(micro 1

20 U

(1

1

2

)minus1)

micro j+12m

j2m

= micro (2j+1)+1

2m+1 2j+1

2m+1lowast

(micro 2j+1

2m+1 2j

2m+1 U

((2j + 1) + 1

2m+12j + 1

2m+1

)minus1)

So

micro j+12m

j2m U

(1j + 1

2m

)minus1

=micro (2j+1)+1

2m+1 2j+1

2m+1 U

(1

(2j + 1) + 1

2m+1

)minus1

lowast

(micro 2j+1

2m+1 2j

2m+1 U

(1

2j + 1

2m+1

)minus1)

=

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

micro10 =

2mminus1prodlowast

j=0

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

=

k=0

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

2〈ξ Rtsξ〉

minusintH

1 + |x|2

)mts(dx) ξ isin H

(350)

intH

1 + |x|2

)mtr(dx)

minusintH

1 + |x|2

)mrs(dx)

minusintH

1 + |x|2

+

intH

i〈ξ U(t r)x〉1 + |x|2

mrs(dx)minusintH

i〈ξ U(t r)x〉1 + |U(t r)x|2

mrs(dx)

+

intHU(t r)x

(1

1 + |x|2

)mrs(dx)

(352)

(355)

minus1

Then

micross = δ0 (357)

(358)

=(microtr lowast

)) U(t0 t)

minus1

=(microtr U(t0 t)

minus1)

=

intH

(359)

)such that for

=

intHotimesn

(361)

E

[exp

(i

nsumj=1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

=nprodj=1

intH

E[exp

〉)]

=

intH

and

E

[exp

(i

nsumj=1

)]=

nprodj=1

E[exp

〉)] (363)

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 11: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

limtdarrs

microts = limtuarrs

microst = δ0 (38)

micrott = δ0 (310)

limnrarrinfin

lim supnrarrinfin

2 (312)

4M (313)

intHf dδ0

=

intxisinH |x|geε0

intxisinH |x|ltε0

lt2M middot η

4M+η

2= η

limtdarrs

sup(ts)isinΛt0s0

U(t s) ltinfin

)(323)

) (324)

U(t s+ ε) le c

limεdarr0

εdarr0

This proves (326)

limεdarr0

= limεdarr0

=

minus1(y)

This proves (320)

andlimtdarrs

microts = limsuarrt

microts = δ0

∣∣∣∣ lt ε (336)

∣∣∣∣le∣∣∣∣int

∣∣∣∣+

intK

lt2ε(1 + finfin)

microts

(A

(εprime

2c

and

micrort

(A

(εprime

2c

2cle εprime

and

δ = min

δtj2

j = 1 2 n

By (18) (342) (343) and (345) we have

=

intH

leintH

intH

ltη

2+η

2= η

=

intH

geintH

intH

ge η(

1minus η

2

)= η minus η2

2

2gt η minus η2

2

=(microts U(tj t)

minus1) U(t0 tj)

then by (18) (342) and (344)η

=

intH

ge η(

1minus η

2

)

complete

micro10 =

2mminus1prodlowast

j=0

micro j+12m

j2m U

(1j + 1

2m

)minus1

(348)

By (18) we have

(micro 1

20 U

(1

1

2

)minus1)

micro j+12m

j2m

= micro (2j+1)+1

2m+1 2j+1

2m+1lowast

(micro 2j+1

2m+1 2j

2m+1 U

((2j + 1) + 1

2m+12j + 1

2m+1

)minus1)

So

micro j+12m

j2m U

(1j + 1

2m

)minus1

=micro (2j+1)+1

2m+1 2j+1

2m+1 U

(1

(2j + 1) + 1

2m+1

)minus1

lowast

(micro 2j+1

2m+1 2j

2m+1 U

(1

2j + 1

2m+1

)minus1)

=

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

micro10 =

2mminus1prodlowast

j=0

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

=

k=0

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

2〈ξ Rtsξ〉

minusintH

1 + |x|2

)mts(dx) ξ isin H

(350)

intH

1 + |x|2

)mtr(dx)

minusintH

1 + |x|2

)mrs(dx)

minusintH

1 + |x|2

+

intH

i〈ξ U(t r)x〉1 + |x|2

mrs(dx)minusintH

i〈ξ U(t r)x〉1 + |U(t r)x|2

mrs(dx)

+

intHU(t r)x

(1

1 + |x|2

)mrs(dx)

(352)

(355)

minus1

Then

micross = δ0 (357)

(358)

=(microtr lowast

)) U(t0 t)

minus1

=(microtr U(t0 t)

minus1)

=

intH

(359)

)such that for

=

intHotimesn

(361)

E

[exp

(i

nsumj=1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

=nprodj=1

intH

E[exp

〉)]

=

intH

and

E

[exp

(i

nsumj=1

)]=

nprodj=1

E[exp

〉)] (363)

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 12: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

limtdarrs

microts = limtuarrs

microst = δ0 (38)

micrott = δ0 (310)

limnrarrinfin

lim supnrarrinfin

2 (312)

4M (313)

intHf dδ0

=

intxisinH |x|geε0

intxisinH |x|ltε0

lt2M middot η

4M+η

2= η

limtdarrs

sup(ts)isinΛt0s0

U(t s) ltinfin

)(323)

) (324)

U(t s+ ε) le c

limεdarr0

εdarr0

This proves (326)

limεdarr0

= limεdarr0

=

minus1(y)

This proves (320)

andlimtdarrs

microts = limsuarrt

microts = δ0

∣∣∣∣ lt ε (336)

∣∣∣∣le∣∣∣∣int

∣∣∣∣+

intK

lt2ε(1 + finfin)

microts

(A

(εprime

2c

and

micrort

(A

(εprime

2c

2cle εprime

and

δ = min

δtj2

j = 1 2 n

By (18) (342) (343) and (345) we have

=

intH

leintH

intH

ltη

2+η

2= η

=

intH

geintH

intH

ge η(

1minus η

2

)= η minus η2

2

2gt η minus η2

2

=(microts U(tj t)

minus1) U(t0 tj)

then by (18) (342) and (344)η

=

intH

ge η(

1minus η

2

)

complete

micro10 =

2mminus1prodlowast

j=0

micro j+12m

j2m U

(1j + 1

2m

)minus1

(348)

By (18) we have

(micro 1

20 U

(1

1

2

)minus1)

micro j+12m

j2m

= micro (2j+1)+1

2m+1 2j+1

2m+1lowast

(micro 2j+1

2m+1 2j

2m+1 U

((2j + 1) + 1

2m+12j + 1

2m+1

)minus1)

So

micro j+12m

j2m U

(1j + 1

2m

)minus1

=micro (2j+1)+1

2m+1 2j+1

2m+1 U

(1

(2j + 1) + 1

2m+1

)minus1

lowast

(micro 2j+1

2m+1 2j

2m+1 U

(1

2j + 1

2m+1

)minus1)

=

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

micro10 =

2mminus1prodlowast

j=0

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

=

k=0

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

2〈ξ Rtsξ〉

minusintH

1 + |x|2

)mts(dx) ξ isin H

(350)

intH

1 + |x|2

)mtr(dx)

minusintH

1 + |x|2

)mrs(dx)

minusintH

1 + |x|2

+

intH

i〈ξ U(t r)x〉1 + |x|2

mrs(dx)minusintH

i〈ξ U(t r)x〉1 + |U(t r)x|2

mrs(dx)

+

intHU(t r)x

(1

1 + |x|2

)mrs(dx)

(352)

(355)

minus1

Then

micross = δ0 (357)

(358)

=(microtr lowast

)) U(t0 t)

minus1

=(microtr U(t0 t)

minus1)

=

intH

(359)

)such that for

=

intHotimesn

(361)

E

[exp

(i

nsumj=1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

=nprodj=1

intH

E[exp

〉)]

=

intH

and

E

[exp

(i

nsumj=1

)]=

nprodj=1

E[exp

〉)] (363)

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 13: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

intHf dδ0

=

intxisinH |x|geε0

intxisinH |x|ltε0

lt2M middot η

4M+η

2= η

limtdarrs

sup(ts)isinΛt0s0

U(t s) ltinfin

)(323)

) (324)

U(t s+ ε) le c

limεdarr0

εdarr0

This proves (326)

limεdarr0

= limεdarr0

=

minus1(y)

This proves (320)

andlimtdarrs

microts = limsuarrt

microts = δ0

∣∣∣∣ lt ε (336)

∣∣∣∣le∣∣∣∣int

∣∣∣∣+

intK

lt2ε(1 + finfin)

microts

(A

(εprime

2c

and

micrort

(A

(εprime

2c

2cle εprime

and

δ = min

δtj2

j = 1 2 n

By (18) (342) (343) and (345) we have

=

intH

leintH

intH

ltη

2+η

2= η

=

intH

geintH

intH

ge η(

1minus η

2

)= η minus η2

2

2gt η minus η2

2

=(microts U(tj t)

minus1) U(t0 tj)

then by (18) (342) and (344)η

=

intH

ge η(

1minus η

2

)

complete

micro10 =

2mminus1prodlowast

j=0

micro j+12m

j2m U

(1j + 1

2m

)minus1

(348)

By (18) we have

(micro 1

20 U

(1

1

2

)minus1)

micro j+12m

j2m

= micro (2j+1)+1

2m+1 2j+1

2m+1lowast

(micro 2j+1

2m+1 2j

2m+1 U

((2j + 1) + 1

2m+12j + 1

2m+1

)minus1)

So

micro j+12m

j2m U

(1j + 1

2m

)minus1

=micro (2j+1)+1

2m+1 2j+1

2m+1 U

(1

(2j + 1) + 1

2m+1

)minus1

lowast

(micro 2j+1

2m+1 2j

2m+1 U

(1

2j + 1

2m+1

)minus1)

=

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

micro10 =

2mminus1prodlowast

j=0

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

=

k=0

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

2〈ξ Rtsξ〉

minusintH

1 + |x|2

)mts(dx) ξ isin H

(350)

intH

1 + |x|2

)mtr(dx)

minusintH

1 + |x|2

)mrs(dx)

minusintH

1 + |x|2

+

intH

i〈ξ U(t r)x〉1 + |x|2

mrs(dx)minusintH

i〈ξ U(t r)x〉1 + |U(t r)x|2

mrs(dx)

+

intHU(t r)x

(1

1 + |x|2

)mrs(dx)

(352)

(355)

minus1

Then

micross = δ0 (357)

(358)

=(microtr lowast

)) U(t0 t)

minus1

=(microtr U(t0 t)

minus1)

=

intH

(359)

)such that for

=

intHotimesn

(361)

E

[exp

(i

nsumj=1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

=nprodj=1

intH

E[exp

〉)]

=

intH

and

E

[exp

(i

nsumj=1

)]=

nprodj=1

E[exp

〉)] (363)

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 14: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

)(323)

) (324)

U(t s+ ε) le c

limεdarr0

εdarr0

This proves (326)

limεdarr0

= limεdarr0

=

minus1(y)

This proves (320)

andlimtdarrs

microts = limsuarrt

microts = δ0

∣∣∣∣ lt ε (336)

∣∣∣∣le∣∣∣∣int

∣∣∣∣+

intK

lt2ε(1 + finfin)

microts

(A

(εprime

2c

and

micrort

(A

(εprime

2c

2cle εprime

and

δ = min

δtj2

j = 1 2 n

By (18) (342) (343) and (345) we have

=

intH

leintH

intH

ltη

2+η

2= η

=

intH

geintH

intH

ge η(

1minus η

2

)= η minus η2

2

2gt η minus η2

2

=(microts U(tj t)

minus1) U(t0 tj)

then by (18) (342) and (344)η

=

intH

ge η(

1minus η

2

)

complete

micro10 =

2mminus1prodlowast

j=0

micro j+12m

j2m U

(1j + 1

2m

)minus1

(348)

By (18) we have

(micro 1

20 U

(1

1

2

)minus1)

micro j+12m

j2m

= micro (2j+1)+1

2m+1 2j+1

2m+1lowast

(micro 2j+1

2m+1 2j

2m+1 U

((2j + 1) + 1

2m+12j + 1

2m+1

)minus1)

So

micro j+12m

j2m U

(1j + 1

2m

)minus1

=micro (2j+1)+1

2m+1 2j+1

2m+1 U

(1

(2j + 1) + 1

2m+1

)minus1

lowast

(micro 2j+1

2m+1 2j

2m+1 U

(1

2j + 1

2m+1

)minus1)

=

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

micro10 =

2mminus1prodlowast

j=0

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

=

k=0

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

2〈ξ Rtsξ〉

minusintH

1 + |x|2

)mts(dx) ξ isin H

(350)

intH

1 + |x|2

)mtr(dx)

minusintH

1 + |x|2

)mrs(dx)

minusintH

1 + |x|2

+

intH

i〈ξ U(t r)x〉1 + |x|2

mrs(dx)minusintH

i〈ξ U(t r)x〉1 + |U(t r)x|2

mrs(dx)

+

intHU(t r)x

(1

1 + |x|2

)mrs(dx)

(352)

(355)

minus1

Then

micross = δ0 (357)

(358)

=(microtr lowast

)) U(t0 t)

minus1

=(microtr U(t0 t)

minus1)

=

intH

(359)

)such that for

=

intHotimesn

(361)

E

[exp

(i

nsumj=1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

=nprodj=1

intH

E[exp

〉)]

=

intH

and

E

[exp

(i

nsumj=1

)]=

nprodj=1

E[exp

〉)] (363)

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 15: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

limεdarr0

εdarr0

This proves (326)

limεdarr0

= limεdarr0

=

minus1(y)

This proves (320)

andlimtdarrs

microts = limsuarrt

microts = δ0

∣∣∣∣ lt ε (336)

∣∣∣∣le∣∣∣∣int

∣∣∣∣+

intK

lt2ε(1 + finfin)

microts

(A

(εprime

2c

and

micrort

(A

(εprime

2c

2cle εprime

and

δ = min

δtj2

j = 1 2 n

By (18) (342) (343) and (345) we have

=

intH

leintH

intH

ltη

2+η

2= η

=

intH

geintH

intH

ge η(

1minus η

2

)= η minus η2

2

2gt η minus η2

2

=(microts U(tj t)

minus1) U(t0 tj)

then by (18) (342) and (344)η

=

intH

ge η(

1minus η

2

)

complete

micro10 =

2mminus1prodlowast

j=0

micro j+12m

j2m U

(1j + 1

2m

)minus1

(348)

By (18) we have

(micro 1

20 U

(1

1

2

)minus1)

micro j+12m

j2m

= micro (2j+1)+1

2m+1 2j+1

2m+1lowast

(micro 2j+1

2m+1 2j

2m+1 U

((2j + 1) + 1

2m+12j + 1

2m+1

)minus1)

So

micro j+12m

j2m U

(1j + 1

2m

)minus1

=micro (2j+1)+1

2m+1 2j+1

2m+1 U

(1

(2j + 1) + 1

2m+1

)minus1

lowast

(micro 2j+1

2m+1 2j

2m+1 U

(1

2j + 1

2m+1

)minus1)

=

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

micro10 =

2mminus1prodlowast

j=0

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

=

k=0

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

2〈ξ Rtsξ〉

minusintH

1 + |x|2

)mts(dx) ξ isin H

(350)

intH

1 + |x|2

)mtr(dx)

minusintH

1 + |x|2

)mrs(dx)

minusintH

1 + |x|2

+

intH

i〈ξ U(t r)x〉1 + |x|2

mrs(dx)minusintH

i〈ξ U(t r)x〉1 + |U(t r)x|2

mrs(dx)

+

intHU(t r)x

(1

1 + |x|2

)mrs(dx)

(352)

(355)

minus1

Then

micross = δ0 (357)

(358)

=(microtr lowast

)) U(t0 t)

minus1

=(microtr U(t0 t)

minus1)

=

intH

(359)

)such that for

=

intHotimesn

(361)

E

[exp

(i

nsumj=1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

=nprodj=1

intH

E[exp

〉)]

=

intH

and

E

[exp

(i

nsumj=1

)]=

nprodj=1

E[exp

〉)] (363)

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 16: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

∣∣∣∣ lt ε (336)

∣∣∣∣le∣∣∣∣int

∣∣∣∣+

intK

lt2ε(1 + finfin)

microts

(A

(εprime

2c

and

micrort

(A

(εprime

2c

2cle εprime

and

δ = min

δtj2

j = 1 2 n

By (18) (342) (343) and (345) we have

=

intH

leintH

intH

ltη

2+η

2= η

=

intH

geintH

intH

ge η(

1minus η

2

)= η minus η2

2

2gt η minus η2

2

=(microts U(tj t)

minus1) U(t0 tj)

then by (18) (342) and (344)η

=

intH

ge η(

1minus η

2

)

complete

micro10 =

2mminus1prodlowast

j=0

micro j+12m

j2m U

(1j + 1

2m

)minus1

(348)

By (18) we have

(micro 1

20 U

(1

1

2

)minus1)

micro j+12m

j2m

= micro (2j+1)+1

2m+1 2j+1

2m+1lowast

(micro 2j+1

2m+1 2j

2m+1 U

((2j + 1) + 1

2m+12j + 1

2m+1

)minus1)

So

micro j+12m

j2m U

(1j + 1

2m

)minus1

=micro (2j+1)+1

2m+1 2j+1

2m+1 U

(1

(2j + 1) + 1

2m+1

)minus1

lowast

(micro 2j+1

2m+1 2j

2m+1 U

(1

2j + 1

2m+1

)minus1)

=

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

micro10 =

2mminus1prodlowast

j=0

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

=

k=0

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

2〈ξ Rtsξ〉

minusintH

1 + |x|2

)mts(dx) ξ isin H

(350)

intH

1 + |x|2

)mtr(dx)

minusintH

1 + |x|2

)mrs(dx)

minusintH

1 + |x|2

+

intH

i〈ξ U(t r)x〉1 + |x|2

mrs(dx)minusintH

i〈ξ U(t r)x〉1 + |U(t r)x|2

mrs(dx)

+

intHU(t r)x

(1

1 + |x|2

)mrs(dx)

(352)

(355)

minus1

Then

micross = δ0 (357)

(358)

=(microtr lowast

)) U(t0 t)

minus1

=(microtr U(t0 t)

minus1)

=

intH

(359)

)such that for

=

intHotimesn

(361)

E

[exp

(i

nsumj=1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

=nprodj=1

intH

E[exp

〉)]

=

intH

and

E

[exp

(i

nsumj=1

)]=

nprodj=1

E[exp

〉)] (363)

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 17: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

∣∣∣∣le∣∣∣∣int

∣∣∣∣+

intK

lt2ε(1 + finfin)

microts

(A

(εprime

2c

and

micrort

(A

(εprime

2c

2cle εprime

and

δ = min

δtj2

j = 1 2 n

By (18) (342) (343) and (345) we have

=

intH

leintH

intH

ltη

2+η

2= η

=

intH

geintH

intH

ge η(

1minus η

2

)= η minus η2

2

2gt η minus η2

2

=(microts U(tj t)

minus1) U(t0 tj)

then by (18) (342) and (344)η

=

intH

ge η(

1minus η

2

)

complete

micro10 =

2mminus1prodlowast

j=0

micro j+12m

j2m U

(1j + 1

2m

)minus1

(348)

By (18) we have

(micro 1

20 U

(1

1

2

)minus1)

micro j+12m

j2m

= micro (2j+1)+1

2m+1 2j+1

2m+1lowast

(micro 2j+1

2m+1 2j

2m+1 U

((2j + 1) + 1

2m+12j + 1

2m+1

)minus1)

So

micro j+12m

j2m U

(1j + 1

2m

)minus1

=micro (2j+1)+1

2m+1 2j+1

2m+1 U

(1

(2j + 1) + 1

2m+1

)minus1

lowast

(micro 2j+1

2m+1 2j

2m+1 U

(1

2j + 1

2m+1

)minus1)

=

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

micro10 =

2mminus1prodlowast

j=0

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

=

k=0

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

2〈ξ Rtsξ〉

minusintH

1 + |x|2

)mts(dx) ξ isin H

(350)

intH

1 + |x|2

)mtr(dx)

minusintH

1 + |x|2

)mrs(dx)

minusintH

1 + |x|2

+

intH

i〈ξ U(t r)x〉1 + |x|2

mrs(dx)minusintH

i〈ξ U(t r)x〉1 + |U(t r)x|2

mrs(dx)

+

intHU(t r)x

(1

1 + |x|2

)mrs(dx)

(352)

(355)

minus1

Then

micross = δ0 (357)

(358)

=(microtr lowast

)) U(t0 t)

minus1

=(microtr U(t0 t)

minus1)

=

intH

(359)

)such that for

=

intHotimesn

(361)

E

[exp

(i

nsumj=1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

=nprodj=1

intH

E[exp

〉)]

=

intH

and

E

[exp

(i

nsumj=1

)]=

nprodj=1

E[exp

〉)] (363)

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 18: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

and

δ = min

δtj2

j = 1 2 n

By (18) (342) (343) and (345) we have

=

intH

leintH

intH

ltη

2+η

2= η

=

intH

geintH

intH

ge η(

1minus η

2

)= η minus η2

2

2gt η minus η2

2

=(microts U(tj t)

minus1) U(t0 tj)

then by (18) (342) and (344)η

=

intH

ge η(

1minus η

2

)

complete

micro10 =

2mminus1prodlowast

j=0

micro j+12m

j2m U

(1j + 1

2m

)minus1

(348)

By (18) we have

(micro 1

20 U

(1

1

2

)minus1)

micro j+12m

j2m

= micro (2j+1)+1

2m+1 2j+1

2m+1lowast

(micro 2j+1

2m+1 2j

2m+1 U

((2j + 1) + 1

2m+12j + 1

2m+1

)minus1)

So

micro j+12m

j2m U

(1j + 1

2m

)minus1

=micro (2j+1)+1

2m+1 2j+1

2m+1 U

(1

(2j + 1) + 1

2m+1

)minus1

lowast

(micro 2j+1

2m+1 2j

2m+1 U

(1

2j + 1

2m+1

)minus1)

=

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

micro10 =

2mminus1prodlowast

j=0

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

=

k=0

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

2〈ξ Rtsξ〉

minusintH

1 + |x|2

)mts(dx) ξ isin H

(350)

intH

1 + |x|2

)mtr(dx)

minusintH

1 + |x|2

)mrs(dx)

minusintH

1 + |x|2

+

intH

i〈ξ U(t r)x〉1 + |x|2

mrs(dx)minusintH

i〈ξ U(t r)x〉1 + |U(t r)x|2

mrs(dx)

+

intHU(t r)x

(1

1 + |x|2

)mrs(dx)

(352)

(355)

minus1

Then

micross = δ0 (357)

(358)

=(microtr lowast

)) U(t0 t)

minus1

=(microtr U(t0 t)

minus1)

=

intH

(359)

)such that for

=

intHotimesn

(361)

E

[exp

(i

nsumj=1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

=nprodj=1

intH

E[exp

〉)]

=

intH

and

E

[exp

(i

nsumj=1

)]=

nprodj=1

E[exp

〉)] (363)

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 19: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

=

intH

geintH

intH

ge η(

1minus η

2

)= η minus η2

2

2gt η minus η2

2

=(microts U(tj t)

minus1) U(t0 tj)

then by (18) (342) and (344)η

=

intH

ge η(

1minus η

2

)

complete

micro10 =

2mminus1prodlowast

j=0

micro j+12m

j2m U

(1j + 1

2m

)minus1

(348)

By (18) we have

(micro 1

20 U

(1

1

2

)minus1)

micro j+12m

j2m

= micro (2j+1)+1

2m+1 2j+1

2m+1lowast

(micro 2j+1

2m+1 2j

2m+1 U

((2j + 1) + 1

2m+12j + 1

2m+1

)minus1)

So

micro j+12m

j2m U

(1j + 1

2m

)minus1

=micro (2j+1)+1

2m+1 2j+1

2m+1 U

(1

(2j + 1) + 1

2m+1

)minus1

lowast

(micro 2j+1

2m+1 2j

2m+1 U

(1

2j + 1

2m+1

)minus1)

=

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

micro10 =

2mminus1prodlowast

j=0

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

=

k=0

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

2〈ξ Rtsξ〉

minusintH

1 + |x|2

)mts(dx) ξ isin H

(350)

intH

1 + |x|2

)mtr(dx)

minusintH

1 + |x|2

)mrs(dx)

minusintH

1 + |x|2

+

intH

i〈ξ U(t r)x〉1 + |x|2

mrs(dx)minusintH

i〈ξ U(t r)x〉1 + |U(t r)x|2

mrs(dx)

+

intHU(t r)x

(1

1 + |x|2

)mrs(dx)

(352)

(355)

minus1

Then

micross = δ0 (357)

(358)

=(microtr lowast

)) U(t0 t)

minus1

=(microtr U(t0 t)

minus1)

=

intH

(359)

)such that for

=

intHotimesn

(361)

E

[exp

(i

nsumj=1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

=nprodj=1

intH

E[exp

〉)]

=

intH

and

E

[exp

(i

nsumj=1

)]=

nprodj=1

E[exp

〉)] (363)

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 20: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

micro10 =

2mminus1prodlowast

j=0

micro j+12m

j2m U

(1j + 1

2m

)minus1

(348)

By (18) we have

(micro 1

20 U

(1

1

2

)minus1)

micro j+12m

j2m

= micro (2j+1)+1

2m+1 2j+1

2m+1lowast

(micro 2j+1

2m+1 2j

2m+1 U

((2j + 1) + 1

2m+12j + 1

2m+1

)minus1)

So

micro j+12m

j2m U

(1j + 1

2m

)minus1

=micro (2j+1)+1

2m+1 2j+1

2m+1 U

(1

(2j + 1) + 1

2m+1

)minus1

lowast

(micro 2j+1

2m+1 2j

2m+1 U

(1

2j + 1

2m+1

)minus1)

=

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

micro10 =

2mminus1prodlowast

j=0

2j+1prodlowast

k=2j

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

=

k=0

micro k+1

2m+1 k

2m+1 U

(1k + 1

2m+1

)minus1

2〈ξ Rtsξ〉

minusintH

1 + |x|2

)mts(dx) ξ isin H

(350)

intH

1 + |x|2

)mtr(dx)

minusintH

1 + |x|2

)mrs(dx)

minusintH

1 + |x|2

+

intH

i〈ξ U(t r)x〉1 + |x|2

mrs(dx)minusintH

i〈ξ U(t r)x〉1 + |U(t r)x|2

mrs(dx)

+

intHU(t r)x

(1

1 + |x|2

)mrs(dx)

(352)

(355)

minus1

Then

micross = δ0 (357)

(358)

=(microtr lowast

)) U(t0 t)

minus1

=(microtr U(t0 t)

minus1)

=

intH

(359)

)such that for

=

intHotimesn

(361)

E

[exp

(i

nsumj=1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

=nprodj=1

intH

E[exp

〉)]

=

intH

and

E

[exp

(i

nsumj=1

)]=

nprodj=1

E[exp

〉)] (363)

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 21: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

2〈ξ Rtsξ〉

minusintH

1 + |x|2

)mts(dx) ξ isin H

(350)

intH

1 + |x|2

)mtr(dx)

minusintH

1 + |x|2

)mrs(dx)

minusintH

1 + |x|2

+

intH

i〈ξ U(t r)x〉1 + |x|2

mrs(dx)minusintH

i〈ξ U(t r)x〉1 + |U(t r)x|2

mrs(dx)

+

intHU(t r)x

(1

1 + |x|2

)mrs(dx)

(352)

(355)

minus1

Then

micross = δ0 (357)

(358)

=(microtr lowast

)) U(t0 t)

minus1

=(microtr U(t0 t)

minus1)

=

intH

(359)

)such that for

=

intHotimesn

(361)

E

[exp

(i

nsumj=1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

=nprodj=1

intH

E[exp

〉)]

=

intH

and

E

[exp

(i

nsumj=1

)]=

nprodj=1

E[exp

〉)] (363)

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 22: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

(355)

minus1

Then

micross = δ0 (357)

(358)

=(microtr lowast

)) U(t0 t)

minus1

=(microtr U(t0 t)

minus1)

=

intH

(359)

)such that for

=

intHotimesn

(361)

E

[exp

(i

nsumj=1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

=nprodj=1

intH

E[exp

〉)]

=

intH

and

E

[exp

(i

nsumj=1

)]=

nprodj=1

E[exp

〉)] (363)

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 23: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

=

intH

(359)

)such that for

=

intHotimesn

(361)

E

[exp

(i

nsumj=1

rang)]

=

intHotimesn

exp

(i

nsumj=1

〈zj yj〉

=nprodj=1

intH

E[exp

〉)]

=

intH

and

E

[exp

(i

nsumj=1

)]=

nprodj=1

E[exp

〉)] (363)

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 24: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

(ΩF ) defined by

=

intH

timesintH

1Bn

(nminus1sumj=1

U(sn sj)yj + yn

(365)

E[f(Ys Yt)] =

intHtimesH

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 25: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

=

intH

)

limεdarr0

=

intH

limεdarr0

=

Xts =

int t

s

U(t σ) dZσ t ge s

Xt0s minusXt0t =

int t0

s

t

int t

s

U(t σ) dZσ

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 26: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

s

)= microts(U(t0 t)

Yt minus U(t s)Ys =

int t

s0

int s

s0

U(s σ) dZσ

=

int t

s0

U(t σ) dZσ

=

int t

s

U(t σ) dZσ

(368)

limεrarr0

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 27: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

limεdarr0

limεrarr0ηrarr0

Hpstf(x) νs(dx) =

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 28: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

41 Some properties

)= νt (42)

=

intH

=

intH

=

)= νt

)= microtr lowast

)

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 29: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

This proves (18)

=

intH

1B1(y1)νt1(dy1)

intH

timesintH

1Bn

(nminus1sumj=1

U(tn tj)yj + yn

(46)

=

intH

Xts =

int t

s

U(t σ) dZσ t ge s

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 30: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

Xtminusinfin =

int t

(2)t )tisinR

ν(2)t = ν

and

ν(2)t (ξ) = ν

(1)s

(U(t s)lowastξ

)σt(ξ)

= microts(ξ)ν(1)s

(U(t s)lowastξ

(2)s

(U(t s)lowastξ

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 31: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

)

mts(A) ge mtr(A)

supslet

trRts ltinfin (410)

〈Rtsx y〉 (411)

trRtr le trRts

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 32: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

limsrarrminusinfin

Rts ltinfin

intH

)= microtr (412)

)= νt

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 33: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

(418) hold

(νs U(t s)minus1

)

(419)

supnisinN

and

supnisinN

(trRtminusn +

int|x|lt1

|x|2 mtminusn(dx)

)ltinfin (421)

trRts ltinfin

supslet

intH

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 34: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

νt =(δats lowast

(νs U(t s)minus1

(νs

) (422)

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 35: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

Because

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 36: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

For each t ge s set

)(f)

=

)(pjstf) = pgst(p

jstf)(x)

with domain

(pstf(x))α le exp

2(αminus 1)

)pstf

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 37: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

(pgstf(x))α le exp

2(αminus 1)

)pgstf

(pstf(x)

)α=(pgst(p

jstf)(x)

)αle exp

2(αminus 1)

)(pgst(p

jstf)α

)(y)

le exp

2(αminus 1)

)(pgst(p

jstf

α))(y)

= exp

2(αminus 1)

)(pstf

α)(y)

dN(0 Rts)(z)

= exp

minus12ts z

rangminus 1

2|Rminus12

(54)

intH

(1

2|R12

ts h|2) (55)

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 38: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

pgstf(x)

=

prime)

=

= exp

(minus1

2|Γts(xminus y)|2

minus12ts z

rang)microgts(dz

prime)

(56)

minus12ts z

rang)microgts(dz

prime)

le(int

prime)

)1α

middot(int

Hexp

(α

)microgts(dz

prime)

)(αminus1)α

le(pgstf

α(y))1α middot

(exp

(1

2

α2

=(pgstf

α(y))1α

exp

(α

)

(57)

pgstf(x)

le exp

(minus1

2|Γts(xminus y)|2

)(pgstf

α(y))1α

exp

(α

)= exp

(1

)(pgstf

α(y))1α

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 39: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

exp

(minusn2π2

int t

s

a(r) dr

Ren = λnen

Xs = x(58)

rts =infinsumn=0

λn

int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du ltinfin

WU(t s) =

int t

s

U(t r) dWr

Rts =

int t

s

pstf(x) =

Rtsen = λn

[int t

s

exp

(minus2n2π2

int t

s

a(r) dr

)du

]en

We have

exp(minusn2π2

int tsa(r) dr

)[λnint ts

exp(minus2n2π2

int tsa(r) dr

)du]12

en

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 40: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

=infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)λnint ts

exp(minus2n2π2

int tsa(r) dr

)du〈x en〉2

(pstf(x))α le exp

α

2(αminus 1)

infinsumn=1

exp(minus2n2π2

int tsa(r) dr

)〈xminus y en〉2

λnint ts

exp(minus2n2π2

int tsa(r) dr

)du

pstfα(y)

12ts (H) (510)

(pstf(x))α le exp

2(αminus 1)

)pstf

pstf(x) =

xn =1

nx0 isin H n = 1 2

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 41: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

le(

minpstf

(513)

pgsth2(z)minus

(pgsth(z)

)2

lepgstpjstf

2(z)minus(pgstp

jstf(z)

)2= pstf

pgsth(x) =

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 42: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

Therefore we obtain

=

(intH

)2

leintH

intH

]2microgts(dz)

=

gts(dz)minus 1

)middot(int

)=(

Cst(α ε) =

intH

[intH

exp

2(αminus 1)

)νs(dy)

]minus(1+ε)

νs(dx)

(pstf(x))α exp

2(αminus 1)

]le pstf

α(y) x y isin H

(pst|f |)α(x)

intH

exp

2(αminus 1)

)νs(dy) le

Hence

Hexp

2(αminus 1)

)νs(dy)

]minus(1+ε)

fα(1+ε)Lα(Hνt)

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 43: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

z(s) = x(61)

z(t s x) = U(t s)x+

int t

s

Πtsx =

int t

s

= inf

int t

s

(63)

int t

s

U(t r)C(r)u(r) dr

lowastts

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 44: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

U(t s)(H) sub Π12ts (H) (65)

r dr(int tsξr dr

)2 x isin H (66)

sh(r)minus1 dr

x isin H (67)

REFERENCES

1 Introduction
      - References

Page 45: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

1 Introduction
      - References

Page 46: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

1 Introduction
      - References

Page 47: bibos.math.uni-bielefeld.de · TIME INHOMOGENEOUS GENERALIZED MEHLER SEMIGROUPS AND SKEW CONVOLUTION EQUATIONS SHUN-XIANG OUYANG AND MICHAEL ROCKNER¨ ABSTRACT.A time …

1 Introduction
      - References