Quasi-sure Flows Associated with Vector Fields of Low Regularity?

Siyan XU Hua ZHANG

1 Introduction

One of the fundamental problems of the theory of dynamical systems concerns the existence and uniqueness of a global flowUtgenerated by a vector fieldX.To be precise,consider the following integral equation:

whereXis a measurable vector field on some topological vector space equipped with a positive Radon measureμon its Borelσ-algebras.The problem was first treated by Cruzeiro[3–4]and she established the existence of a flow on the Wiener space associated to a general vector field valued in the Cameron-Martin space,under exponential integrability of the vector field as well as its gradient and divergence.Yun[14–15]refined the almost-sure existence to the quasi-sure existence offlows on the Wiener space associated to weakly differentiable vector fields.It was shown in[14]that ifμis the Wiener measure on an abstract Wiener space andXis a vector field taking values in the Cameron-Martin spaceHofμ,belonging to the Sobolev classoverμand satisfying the exponential integrability condition expLp(μ)for allp>1,whereH?nis then-fold tensor product ofH,then there exists a solution{Ut}for all initial values except in an(r,p)-polar set for allr≥2 andp>1.We call this(r,p)-quasi-sure existence for allr≥2 andp>1.

Peters[11]obtained the almost-sure existence under some weaker conditions where only onefold differentiability ofXwas required.Therefore,we expect that this situation can be refined to(1,p)-quasi-sure existence for anyp>1.Unfortunately,the method used by Yun[14]is associated with the Ornstein-Uhlenbeck semigroup which acts on the Banach-valued functional.Since only one-fold differentiability is required,we can not use this method associated with the Ornstein-Uhlenbeck semigroup to give the(1,p)-quasi-sure existence for anyp>1 through the procedure(see[14]).In fact,it is well-known that there is no relation between the derivative operator and the Ornstein-Uhlenbeck operator acting on Banach-valued functionals due to the lack of Meyer inequality;moreover,the fractional order Ornstein-Uhlenbeck operator is not easy to deal with.Consequently,we must take another way to construct a solutionUt(x)associated to a vector field on the Wiener space for all initial values except in a(1,p)-polar set for anyp>1.After the establishment of(1,p)-quasi-sure existence for anyp>1,we can also obtain the(1,p)-quasi-sure flow property for anyp>1 and the equivalence of capacities under the transformations of the Wiener space induced by the solution.

The results in the present paper extend the previous results in[14–15]in the following way.First of all,we will consider the vector whichis the sum of a skew-adjoint operator not necessarily bounded and a non-linear part.This situation is equivalent to considering a nonlinear perturbation of a semigroup of rotations.Secondly,the condition onXonly requires one-fold differentiability.

This paper is organized as follows.In Section 2,we recall some elements in the Malliavin calculus and deal with the capacity theory of the Sobolev spaceIn the last part of Section 2,we recall some results in[11]for later use.Our main work is in Section 3:We prove the(1,p)-quasi-sure existence of the solutions associated with the integral equations for anyp>1,and get the(1,p)-quasi-sure flow property of the solutions for anyp>1.At the end of Section 3,we also prove the equivalence of capacities under the transformations of the Wiener space induced by the solutions,which refines the property of mutual absolute continuity.

2 Preliminaries

2.1 Derivatives and Sobolev spaces

Now let us recall and fix some notations and notions.Let(B,H,μ)be an abstract Wiener space introduced by Gross[6],where

(1)

(2)His a real,separable Hilbert space densely and continuously imbedded inBwith the inner product

(3)μis the standard Gaussian measure,i.e.,the Borel probability measure onBsuch that

whereis a natural pairing ofandB.

We refer to[10](see also[7])for the background in the Malliavin calculus.In the following we fix an orthonormal basis{hi;i≥1}ofHwithhi∈B?for alli≥1.LetGbe a separable Hilbert space.AG-valued functionalFis called a cylindrical if there existsN,M≥1,andsuch that

We denote by Cylin(W,G)the space ofG-valued cylindrical functions.ForF∈Cylin(W,G),we define

where

For anyp>1,we can define the seminorm on Cylin(W,G)as

Now for anyp>1,we defineas the completion of a Cylin(W,G)with respect to the normIfG=,we simply writeand Cylin(W).Furthermore,a Gaussian divergence operatorδ(X)can be defined as the adjoint inL2(B,G)of the gradient alongH:

Letbe an increasing sequence of a finite dimensional subspace ofB?,such that the projectionstrongly.We denote byμnthe Gaussian measure onVnassociated to the restriction of the inner producttoVn.Also denote by theσ-algebra consisting of cylindrical sets based onVn.Obviously,is a filtration of subσ-algebras ofwhereis aσ-algebra generated byThe Ornstein-Uhlenbeck semigroupTεonBis defined by the Mehler formula

This semigroup provides us with smooth approximations of a vector fieldX.

Definition 2.1For X∈L(B,H),the finite dimensional approximations of Xw.r.t.are defined as

whereis a sequence of positive numbers converging to zero.

The following results are due to G.Peters[11].

Proposition 2.1Let1

2.2 Capacity theory on Wiener spaces

We fixp>1.Given an open setOinB,its(1,p)-capacity is defined by

and for any subsetA?B,its(1,p)-capacity is defined by

IfC1,p(A)=0,thenAis called a(1,p)-polar set.If some properties hold except on a(1,p)-polar set,then we say that it holds(1,p)-quasi-everywhere.A subsetAwill be called 1-slim ifC1,p(A)=0 for anyp>1.We also say that it holds 1-quasi-everywhere if some properties hold except on a 1-slim set.

For anyH-valued Wiener functionalf:B→H,iffor anyε>0,there exists an open setOwithC1,p(O)<εsuch thatf:BO→His continuous,then we call thisH-valued Wiener functional(1,p)-quasi-continuous.AnH-valued Wiener functionalfis said to possess a(1,p)-quasi-continuous modificationif among the equivalence classes ofμ-measurable functions off,we can choose a(1,p)-quasi-continuous functionAnH-valued Wiener functionalfis said to be 1-quasi-continuous if it is(1,p)-quasi-continuous for allp>1.

We note that the following property holds for Sobolev spaces on an abstract Wiener space:

By Meyer inequality,we can get the equivalence between Sobolev spacesWp1(B,H)andFp1(B,H),whichis defined through the Ornstein-Uhlenbeck operator.Then it has been proved by Shigekawa[13](see also Denis[5])that anyfadmits a(1,p)-quasi-continuous modification and this is denoted byef,and the following Chebyshev type inequality holds:

Moreover,we can get a capacity version of Kolmogorov’s criterion for path continuity.We refer the readers to Shigekawa[13]for a proof.

Theorem 2.1Let X={X(t),t∈D}be an H-valued process on a domain D ofdandp>1.Suppose that X(t)∈Further,suppose that there exist constants α>0andc>0such that for all(s,t)∈D×D,

Then X(t)admits a(1,p)-quasi-continuous modification(t)for each t∈D,and for(1,p)-quasi-every x∈B,the sample paths of(t)are continuous.

2.3 Anticipating flows on the Wiener space

Now we turn to the integral equation on the Wiener space and recall some results related to this article.We denote by(H,H)the Banach space of a linear continuous operatorL:H→Hequipped with the norm

We also define a nice strongly continuous semigroup onHas follows.

Definition 2.2A strongly continuous semigroup Qtof linear operators on H is said to benice if there exists a measurable normand a constant CTsuch that

Proposition 2.2 summarizes the results of Peters in[11].

Proposition 2.2Letbe a vector field on B,Qtbe a nice stronglycontinuous semigroup of a unitary operator on H,and:B→B denote the measurablelinear extension of Qtto B.Suppose that the vector field X satisfies the following conditions:

Then there exists a solution Ut(x)of the integral equation

for all t∈.

Also,the image of Gaussian measureμunder Uthas the Radon-Nikodym density

and for T>0,there existandsuch that

The solution Utenjoys the crude flow property,i.e.,for every s∈there exists a set Es?B such thatμ(Es)=1and

Remark 2.1The measurable linear extensionalways exists and preserves the Gaussian measureμ(see[8]).As in[11,Section 4],under the condition thatQtis nice,can be considered as a(possibly unbounded)linear operator onBwith a domain inoffull measure,wheredoes not depend ont.Sinceis a linear operator and isH-differentiable,we obtainx∈B,for eacht∈.

The strategy for proving this proposition is to find a solution to approximate integral equations

whereare approximations ofXwhich are defined as in Definition 2.1,and then to show that the limitexistsμ-a.e.inx∈Band prove the theorem.More details of the proof can be found in[11].

We give the following result from[11]which will be used later.

Proposition 2.3For T>0,we can choose pT>1and constants C(pT,T),that areindependent of n,such that the finite dimensional flowis a Radon-Nikodym derivative thatsatisfies

Also,for each s∈there exists a set Essuch thatμ(Es)=1and

3 1-Quasi-sure Analysis of Integral Equations on the Wiener Space

We now give the main result concerning the 1-quasi-sure flows associated with a vector field of low regularity.

Theorem 3.1Letbe a vector field on B,Qtbe a nice strongly continuoussemigroup of a unitary operators on H,anddenote the measurable linear extensionof Qtto B.Further,let the vector field X fulfill the following conditions:

Then we can choose a1-quasi-continuous modificationof X(x)defined everywhere on B,and we can construct Ut(x),t∈Rand x∈B,satisfying the following integral equation:

for all t∈.

Moreover,the solution Uthas the1-quasi-sure flow property,i.e.,for all s∈,

for all t∈.

Finally,the mapping x→Ut(x)preserves the class of1-slims set for all t∈.

Remark 3.1Thanks to the following lemma which shows that bounded continuous operators fromBtoHare of the Hilbert-Schmidt class when restricted toH,our results also hold under exponential integrability assumptions expfor allp>1.Hence our results refine the almost-sure existence in[11]to 1-quasi-sure existence.

Lemma 3.1(see[2,Theorem 3.5.10])Let(B,H,μ)be an abstract Wiener space,and then one can find an orthonormal basis{en}in H such that

We also need the following lemma(see[1]),and for the convenience of the readers,we include the proof.

Lemma 3.2Let DX(x)∈(H,H)be a linear continuous operator,and then we have

with C depending only on B andμ.

ProofBy the above lemma,we can find a complete orthonormal systemofHsuch that

The rest of this article is devoted to proving Theorem 3.1.For convenience,we fixedT>0.If we obtain the desired results,then sinceT>0 is arbitrary,the results can also be extended to the case whent∈.

3.1 Existence

We divide the proof of existence into four steps.As before,we defineXnasXn=PVn?whereis a sequence of positive numbers converging to zero.

Step 1First we note that ifVt(x)solves the integral equation:

thensolves the original equation(3.1).By the results in[11],we can deduce thatVt(x)∈H.Therefore,we can use the theory of the Malliavin calculus and the capacity version of Kolmogorov’s criterion to investigate the 1-quasi-sure property.

Step 2We introduce the following process parameterized by[0,1]×[0,T]:

wheresolves the following approximating integral equation:

We have the following proposition.

Proposition 3.1Z(s,t)has a1-quasi-continuous modificationfor each(s,t)∈[0,1]×[0,T].Moreover,the sample paths ofare continuous for1-quasi-every x∈B.

To prove this proposition,we need some lemmas.The following lemma can be found in[11].

Lemma 3.3For all p>1,we have

We also need the following simple lemma.

Lemma 3.4For all p>1,we have

ProofSince

we have

Therefore,

By Gronwall’s lemma,

Thus using Proposition 2.3,we have

wherepTis as in Proposition 2.3 andBy the assumptions of Theorem 3.1 we can conclude that

The following two lemmas will play a crucial role.

Lemma 3.5For all p>1,we have

ProofSince

and

we put

We have

For fixedt,0≤t≤T,we have

Then by Gronwall’s lemma,

By Jensen’s inequality

Thus

Note that by the assumptions of Theorem 3.1,we have

whereis as in Proposition 2.2 and=1,and

Thus it remains to show that

Since

for the second term in(3.3)we have

whereis as in Proposition 2.2 andThe first term in(3.3)is handled as follows.First by[11,Section 7],taking a subsequence if necessary we obtain that there exists a subsetA∈Bsuch thatμ(A)=1,andx∈A.Thus from the smoothness ofDXn,we know that

Using Egoroff’s theorem,for everyε>0,there exists a measurable subsetKεsuch thatconverges touniformly onKε.Thus we have

Sinceconverges touniformly onKε,the first part in(3.4)converges to zero asntends to infinity.The second part is dealt with as follows:

wherepTis as in Proposition 2.3 and=1,andis as in Proposition 2.2 and=1.Thus we have

Lemma 3.6For any t,s∈[0,T]and all p>1,we have

ProofSince

then from the assumptions of Theorem 3.1,Proposition 2.3 and Lemma 3.4,we deduce that

Proof of Proposition 3.1First by Lemma 3.3,we have

Therefore,extracting a subsequence still denoted by{n},we have

On the other hand,together with the assumptions of Theorem 3.1 and Proposition 2.3,we have

wherepTis as in Proposition 2.3 andThen from inequalities(3.5)–(3.6),taking further subsequence if necessary,for allp>1,we have

Hence we can takeplarge enough such that

for someε>0 and(s1,s2)∈[0,1]×[0,1],(t1,t2)∈[0,T]×[0,T].

It remains to show that for somep,we have

for someε>0 and(s1,s2)∈[0,1]×[0,1],(t1,t2)∈[0,T]×[0,T].

By Lemmas 3.5–3.6,taking further subsequence if necessary,we obtain

Then by the same procedure we can get that

for someε>0 and(s1,s2)∈[0,1]×[0,1],(t1,t2)∈[0,T]×[0,T].

Therefore,the conclusion follows from Proposition 2.1.

We deduce from this proposition immediately the following proposition.

Proposition 3.2For each t∈[0,T],andhave a1-quasi-continuous modifica-tionand,respectively.

Proposition 3.3There exists a1-slim set A such thatfor alland all t∈[0,T].

Step 3Sincewe can take a 1-quasi-continuous modificationofXby

Then for fixeds,there exists a subsequence still denoted by{n}such thatconverges to1-quasi-everyx∈Band the limit is 1-quasi-continuous.Repeating the same argument as in Lemma 3.3 and Lemma 3.5,we can deduce the following lemma.

Lemma 3.7For all p>1,

We denote

and

By Lemma 3.7,for fixedtwe have

Then we can deduce thatis a Cauchy sequence inand therefore,we can take a subsequencesuch that for anyε>0,there exists a closed setAwithconverges toFt(x)uniformly inx∈A.Thus we can deduce thatis 1-quasi-continuous.Therefore for fixedt∈[0,T],we have

and the limit is 1-quasi-continuous.

However,this is not our purpose because for differents∈[0,T]we have different 1-slim sets.Thus we still need to show that there exists a common setAwithC1,p(A)=0 for anyp>1,such that for allFor this purpose,we need the following lemma.

Lemma 3.8For any s,t∈[0,T]and all p>1,taking further subsequence if necessary,we have

ProofThe proof of the first formula can be seen in[11].The proof of the other formulas is just a repetition of Lemma 3.5 and Lemma 3.6.

Hence using the same skills,we have the following proposition.

Proposition 3.4For each t∈[0,T],Ft(x)has a1-quasi-continuous modification

Proposition 3.5There exist a1-slim set A such thatfor alland t∈[0,T].

Though we obtainasntends to infinity for 1-quasi-everyx∈B,we don’t know the expression ofeFt(x).For this,we proceed as follows.By(3.7),we know that for eacht∈[0,T],there exists a 1-slim setAtsuch that for allHowever,bothandhave continuous sample paths,and hence there exists a common 1-slim setAsuch that for alldsfor allt∈.

Therefore,we conclude that there exists a subsequence still denoted by{n}and a 1-slim setA,such that

for allx∈Acand allt∈[0,T].

Step 4We first note that by Peters[11]there exists a solutionVn(x)satisfying the following integral equation:

whereμ(A0)=1.Thus by Proposition 3.2,there exists a 1-quasi-continuous modificationand

SinceQt:H→Hleaves the subspaceVninvariant,from(3.9),we knowThe fact that in finite dimensions the embeddingforp>ngenerates the implication:IfthenAis empty.This implies thatis a continuous modification of

For anyy∈B,there exists a sequenceconverging toy.Sinceis continuous,converges toashconverges to infinity.By Proposition 2.1,we know thatareC∞cylindrical functionals based on the subspaceVn.Then together with the factifx∈A0,we have

Therefore we have

and this implies thatexists for allx∈Band satisfies

As in Step 3,we see that there is a subsetA1withC1,p(A1)=0 for anyp>1,such that for all

Proposition 3.3 implies that there exists a subsetA2withC1,p(A2)=0 for anyp>1,such that for alland allThus combining this with(3.10),for allwithfor anyp>1,we have

Hence(x)satisfies the integral equation(3.2)for 1-quasi-everyx∈Band for allt∈.Thensatisfies the following integral equation:

and for allt∈,and the 1-quasi-sure existence is established.

3.2 Quasi-sure flow property

Now we show that the solutionhas the 1-quasi-sure flow property,i.e.,for eachs∈it satisfies

for 1-quasi-everyx∈Band for allt∈.

First we need the following lemma.

Lemma 3.9For every s∈R,we have that for all p>1,

ProofSincewe have

Also we have

Thus by Lemma 3.5 we have

wherepTis as in Proposition 2.3 andand the lemma established.

Proposition 3.6The solutionconstructed in Subsection3.1enjoys the1-quasi-sureflow properties,i.e.,for every s∈,it satisfies

and for all t∈.

ProofWe denote byUt(x)the solution constructed by Peters[11].First note that by Proposition 3.2is 1-quasi-continuous.Then,is also 1-quasi-continuous andUt(x)=for almost-everyx∈B.By the almost-everywhere flow property ofUt(x),we havefor almost-everyx∈B.Butis 1-quasi-continuous and hence,if we can show thatis 1-quasi-continuous,we havefor 1-quasi-everyx∈B.

By Lemma 3.9,is a Cauchy sequence inand therefore,we can take a subsequencesuch that for anyε>0,there exists a closed setAwithandconverges touniformly inx∈A.Hence from the smoothness of the solutionand 1-quasi-continuous ofwe can deduce thatis 1-quasicontinuous.

3.3 Equivalence of capacities

Letbe the solution constructed in Subsection 3.1.

Lemma 3.10For any1

for all

ProofCombining Proposition 2.3 with Lemma 3.4,we obtain

wherepTis as in Proposition 2.3,Thus we have

for some constantC.Therefore,the proofis completed if we prove thatconverges to

Since

we can prove that

by the same method as in the proof of Lemma 3.5.

To show the equivalence,we need another lemma which has been proved by Yun[15].

Lemma 3.11There exists an increasing sequenceof compact sets such that forall p>1,

andthe restriction ofto Fn,is a homeomorphism.

With the above preparations,now we can show the equivalence of capacities between a setAinBandUt(A).

Proposition 3.7For10and C2>0such that

ProofBy Lemma 3.11,there exists an increasing sequenceof compact sets such thatis a homeomorphism.LetObe an open set inB.ThenO∩Fnis open inFnandis open inThus there exists an open setO′inBsuch that

We can show that

andis an open set.Then we have

By Lemma 3.10,we have

Therefore,for an open setOinB,for some constantC1.Then for an arbitrary setA?B,it is easy to show

We note that for an arbitrary setA,

Thus we can easily get the first inequality of(3.11).

Corollary 3.1The flowconstructed in Theorem3.1preserves the class of1-slimsets,that is,if A?B is a1-slim set,thenis also a1-slim set for every t∈.

AcknowledgementThe authors would like to thank the referees for their careful reading of this manuscript.

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