The substitution theorem for semilinear stochastic

THESUBSTITUTIONTHEOREMFORSEMILINEARSTOCHASTICPARTIALDIFFERENTIALEQUATIONSySalah-EldinA.MohammedandTushengZhangAbstract.Inthisarticleweestablishasubstitutiontheoremforsemilinearstochasticevolutionequa-tions(see's)dependingontheinitialconditionasaninnite-dimensionalparameter.Duetotheinnite-dimensionalityoftheinitialconditionsandofthestochasticdynamics,existingnite-dimensionalresultsdonotapply.ThesubstitutiontheoremisprovedusingMalliavincalculustechniquestogetherwithnewestimatesontheunderlyingstochasticsemiow.Applicationsofthetheoremincludedynamicchar-acterizationsofsolutionsofstochasticpartialdierentialequations(spde's)withanticipatinginitialconditionsandnon-ergodicstationarysolutions.Inparticular,ourresultgivesanewexistencetheoremforsolutionsofsemilinearStratonovichspde'swithanticipatinginitialconditions.1.Introduction.Statementofthesubstitutiontheorem.Themainobjectiveofthisarticleistoanswerthefollowingsimple(butbasic)question:Givenanon-anticipatingstochasticpartialdierentialequationwithitsinitialconditionasaninnite-dimensionalparameter,isitjustiedtoreplacetheinitialcondition/parameterbyanarbi-traryrandomvariable?Ananswertothearmativefortheabovequestioniswell-knownforawideclassofnite-dimensionalsde'sviathesubstitutiontheoremsin[Nu.1-2]and[M-S.2]).However,theexistingsubstitutiontheoremsin([Nu.1-2],[M-S.2])donotapplytoinnite-dimensionalsystems.Therearetwoseriousobstructionstothisapproach:Thesubstitutiontheoremsarebasedlargelyonnite-dimensionalselectiontechniquesthatareknowntofailininnite-dimensionalsettings,asindicatedbythefailureofKolmogorov'scontinuitytheoremforinnite-dimensionalrandomelds([Mo.1-2],[M-Z-Z])andthefailureofSobolevinequalitiesininnitedimensions.Theinnite-dimensionalityofthedynamicsrenderstheconditionsofthesubstitutionthe-oremsin[Nu.1-2]inapplicable(cf.Theorem3.2.6[Nu.1],Theorem5.3.4[Nu.2]).BothobstructionsareresolvedusingideasandtechniquesoftheMalliavincalculustogetherwithnewglobalestimatesonthesemiowgeneratedbythespde(Section2)([Ma]).TheuseofMalliavinyJune10,2007.ToappearinJournalofFunctionalAnalysis,2007.TheresearchofthisauthorissupportedinpartbyNSFGrantsDMS-9975462,DMS-0203368andDMS-0705970.TheresearchofthisauthorissupportedinpartbyEPSRCGrantGR/R91144.AMS1991subjectclassications.Primary60H10,60H20;secondary60H25.Keywordsandphrases.Malliavincalculus,stochasticsemiow,Ckcocycle,stochasticevolutionequation(see),anticipatingstochasticpartialdierentialequation(spde).12S.-E.A.MOHAMMEDANDT.S.ZHANGcalculustechniquesinthiscontextseemstobenecessitatedbytheinnite-dimensionalityoftheunderlyingstochasticdynamics.ThedicultyinprovingthesubstitutiontheoremforstochasticsystemswithmemorywaspointedoutbyM.Scheutzowandoneoftheauthorsin([M-S.1],PartII);butnorigorousprooforcounterexamplesareknown.Thepurposeofthediscussionin[M-S.1]istoprovideadynamiccharacterizationofstable/unstablemanifoldsforstochasticsystemswithmemorynearhyerbolicstationarystates.InworkbyGrorud,NualartandSanz-Sole([G-Nu-S])asubstitutiontheoremforStratonovichintegralsinHilbertspaceisdevelopedundertherestrictionthatthesubstitutingrandomvariabletakesvaluesinarelativelycompactsetintheHilbertspace.Thesubstitutionresultin[G-Nu-S]isobtainedwithinthecontextofHilbert-space-valuedstochasticordinarydierentialequations,usingmetricentropytechniques.Cf.also[A-I],wherethesubstitutingrandomvariabletakesvaluesina-compactspace.Inthisarticleweestablishasubstitutiontheoremforsemilinearspde'sforalargeclassofinnite-dimensionalMalliavinsmoothrandomvariables.Westronglybelievethatthetechniquesdevelopedinthisarticlewillyieldasimilarsubstitutiontheoremforsemiowsinducedbysfde's.Weexpecttheresultsinthisarticletobeusefulinestablishingregularityindistributionoftheinvariantmanifoldsforsemilinearspde's.Inordertoformulateourresults,considerthefollowingsemilinearIt^ostochasticevolutionequation(see):du(t;x)=Au(t;x)dt+Fu(t;x)
dt+Bu(t;x)dW(t);t0u(0;x)=x2H)(1:1)inaseparablerealHilbertspaceH.IntheaboveequationA:D(A)H!HisaclosedlinearoperatorontheHilbertspaceH.AssumethatAhasacompleteorthonormalsystemofeigenvectorsfen:n1gwithcorrespondingpositiveeigenvaluesfn;n1g;i.e.,Aen=nen;n1:SupposeAgeneratesastronglycontinuoussemigroupofboundedlinearoperatorsTt:H!H;t0.Furthermore,weletF:H!Hbea(Frechet)C1bnon-linearmap,thatisFhasagloballyboundedcontinuousFrechetderivativeDF:H!L(H).LetEbeaseparableHilbertspaceandW(t);t0;beanE-valuedBrownianmotiondenedonthecanonicallteredWienerspace(;F;(Ft)t0;P)andwithaseparablecovarianceHilbertspaceK.Inparticular,KEisaHilbert-Schmidtembedding.Furthermore,isthespaceofallcontinuouspaths!:R!Esuchthat!(0)=0withthecompactopentopology,FisitsBorel-eld,Ftisthesub--eldofFgeneratedbyallevaluations3!7!!(u)2E;ut,andPisWienermeasureon.TheBrownianmotionisgivenbyW(t;!):=!(t);!2;t2R;THESUBSTITUTIONTHEOREMFORSPDE'S3andmayberepresentedbyW(t)=1Xk=1Wk(t)fk;t2R;(1:2)whereffk:k1gisacompleteorthonormalbasisofK,andWk;k1;arestandardindependentone-dimensionalWienerprocesses([D-Z.1],Chapter4).Notethat,ingeneral,theaboveseriesconvergesabso