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Universitext Wei Liu Michael Röckner Stochastic Partial Differential Equations: An Introduction Universitext Universitext SeriesEditors SheldonAxler SanFranciscoStateUniversity VincenzoCapasso ADAMSS(InterdisciplinaryCentreforAdv) CarlesCasacuberta UniversitatdeBarcelona AngusMacIntyre QueenMaryUniversityofLondon KennethRibet UniversityofCalifornia,Berkeley ClaudeSabbah CNRS,EcolepolytechniqueCentredemathématiques EndreSüli UniversityofOxford WojborA.Woyczynski CaseWesternReserveUniversity,Cleveland,OH Universitext is a series of textbooks that presents material from a wide variety of mathe- maticaldisciplinesatmaster’slevelandbeyond.Thebooks,oftenwellclass-testedbytheir author,mayhaveaninformal,personalevenexperimentalapproachtotheirsubjectmatter. Someofthemostsuccessfulandestablishedbooksintheserieshaveevolvedthroughseveral editions,alwaysfollowingtheevolutionofteachingcurricula,toverypolishedtexts. 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Moreinformationaboutthisseriesathttp://www.springer.com/series/223 Wei Liu (cid:129) Michael RoRckner Stochastic Partial Differential Equations: An Introduction 123 WeiLiu MichaelRoRckner SchoolofMathematicsandStatistics FacultyofMathematics JiangsuNormalUniversity BielefeldUniversity Xuzhou,China Bielefeld,Germany ISSN0172-5939 ISSN2191-6675 (electronic) Universitext ISBN978-3-319-22353-7 ISBN978-3-319-22354-4 (eBook) DOI10.1007/978-3-319-22354-4 LibraryofCongressControlNumber:2015953013 Mathematics Subject Classification (2010): 60-XX, 60H15, 60H10, 60H05, 60J60, 60J25, 35-XX, 35K58, 35K59, 35Q35, 34-XX, 34F05, 34G20, 47-XX, 47J35 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com) Contents 1 Motivation,AimsandExamples ........................................... 1 1.1 MotivationandAims................................................... 1 1.2 GeneralPhilosophyandExamples .................................... 3 2 The Stochastic Integral in General Hilbert Spaces (w.r.t.BrownianMotion).................................................... 9 2.1 Infinite-DimensionalWienerProcesses ............................... 9 2.2 MartingalesinGeneralBanachSpaces................................ 23 2.3 TheDefinitionoftheStochasticIntegral.............................. 27 2.3.1 SchemeoftheConstructionoftheStochasticIntegral....... 27 2.3.2 TheConstructionoftheStochasticIntegralinDetail........ 27 2.4 PropertiesoftheStochasticIntegral................................... 42 2.5 TheStochasticIntegralforCylindricalWienerProcesses ........... 49 2.5.1 CylindricalWienerProcesses.................................. 49 2.5.2 The Definition of the Stochastic Integral forCylindricalWienerProcesses.............................. 52 3 SDEsinFiniteDimensions.................................................. 55 3.1 MainResultandALocalizationLemma.............................. 55 3.2 ProofofExistenceandUniqueness.................................... 62 4 SDEsinInfiniteDimensionsandApplicationstoSPDEs................ 69 4.1 GelfandTriples,ConditionsontheCoefficientsandExamples...... 69 4.2 TheMainResultandAnItôFormula ................................. 89 4.3 MarkovPropertyandInvariantMeasures............................. 109 5 SPDEswithLocallyMonotoneCoefficients............................... 123 5.1 LocalMonotonicity .................................................... 123 5.1.1 MainResult..................................................... 123 5.1.2 ProofoftheMainTheorem.................................... 126 5.1.3 ApplicationtoExamples....................................... 133 v vi Contents 5.2 GeneralizedCoercivity................................................. 145 5.2.1 MainResults ................................................... 145 5.2.2 ProofsoftheMainTheorems.................................. 149 5.2.3 ApplicationtoExamples....................................... 165 6 MildSolutions................................................................ 179 6.1 PrerequisitesforThisChapter ......................................... 179 6.1.1 TheItôFormula................................................ 179 6.1.2 ABurkholder–Davis–GundyTypeInequality................ 180 6.1.3 StochasticFubiniTheorem.................................... 181 6.2 Existence,UniquenessandContinuitywith Respect totheInitialData....................................................... 181 6.3 SmoothingProperty of the Semigroup:Pathwise ContinuityoftheMildSolution ....................................... 201 A TheBochnerIntegral........................................................ 209 A.1 DefinitionoftheBochnerIntegral..................................... 209 A.2 PropertiesoftheBochnerIntegral..................................... 211 B NuclearandHilbert–SchmidtOperators ................................. 215 C ThePseudoInverseofLinearOperators.................................. 221 D SomeToolsfromRealMartingaleTheory ................................ 225 E WeakandStrongSolutions:TheYamada–WatanabeTheorem........ 227 F ContinuousDependenceofImplicitFunctionsonaParameter ........ 241 G Strong,MildandWeakSolutions.......................................... 243 H SomeInterpolationInequalities............................................ 247 I Girsanov’sTheoreminInfiniteDimensionswithRespect toaCylindricalWienerProcess............................................ 251 References......................................................................... 261 Index............................................................................... 265 Chapter 1 Motivation, Aims and Examples 1.1 Motivation andAims In this course we will concentrate on (nonlinear) stochastic partial differential equations (SPDEs) of evolutionary type. All kinds of dynamics with stochastic influence in nature or man-made complex systems can be modeled by such equations. As we shall see from the examples at the end of this section, the state spaces of their solutions are necessarily infinite dimensional, such as spaces of (generalized)functions.Inthiscoursethestatespaces,denotedbyH,willbemostly separableHilbertspaces,sometimesseparableBanachspaces. There is also enormousresearch activity on SPDEs where the state spaces are notlinear,butratherspacesofmeasures(particlesystems,dynamicsinpopulation genetics) or infinite-dimensionalmanifolds(path or loop spaces over Riemannian manifolds). There are basically three approaches to analyzing SPDEs: the “martingale (or martingale measure) approach” (cf. [80]), the “semigroup (or mild solution) approach” (cf. [26, 27]) and the “variational approach” (cf. [75]). There is an enormously rich literature on all three approaches which cannot be listed here. We refer instead to the above and the following other monographs on SPDEs: [6,13,16,19,20,26–28,37,46,48,50,53,66]andthereferencestherein. The purpose of this course is to give an introduction to the “variational approach”,asself-containedaspossible,includingthe“localcase”,i.e.where,e.g. thestandard(weak)monotonicityconditionsonlyholdlocally.Inthe“globalcase” thisapproachwasinitiatedinthepioneeringworkofE.Pardoux[64,65]andfurther developed by N. Krylov and B. Rozovski˘ı in [54] for continuous martingales as integratorsin the noise term and later by I. Gyöngyand N. Krylovin [40–42] for notnecessarilycontinuousmartingales. The predecessor [67] of this monograph grew out of a two-semester graduate coursegivenbythesecond-namedauthoratPurdueUniversityin2005/2006.This extendededitionof[67]istheoutcomeofatwosemestergraduatecourseheldatthe ©SpringerInternationalPublishingSwitzerland2015 1 W.Liu,M.Röckner,StochasticPartialDifferentialEquations:AnIntroduction, Universitext,DOI10.1007/978-3-319-22354-4_1 2 1 Motivation,AimsandExamples UniversityofBielefeldin2012/2013.Prerequisiteswouldbeanadvancedcoursein probabilitytheory,coveringstandardmartingaletheory,stochasticprocessesinRd and maybe basic stochastic integration, thoughthe latter is not formally required. Sincegraduatestudentsinprobabilitytheoryareusuallynotfamiliarwiththetheory of Hilbert spaces or basic linear operator theory, all required material from these areasis includedin the text,mostof it in the appendices.Forthe same reason we minimizethegeneraltheoryofmartingalesonHilbertspaces,paying,however,the pricethatsomeproofsconcerningstochasticintegrationonHilbertspaceare a bit lengthy,sincetheyhavetobedone“withbarehands”. For simplicity we specialize to the case where the integrator in the noise term is just a cylindricalWiener process, but everythingis spelt outin such a way that itgeneralizesdirectlytocontinuouslocalmartingales.Inparticular,integrandsare alwaysassumedtobepredictableratherthanjustadaptedandproductmeasurable. The existence and uniqueness proof (cf. Sect.4.2) is our personal version of the proofin [54,75] and is largelytaken from [69] presented there in a more general framework.Theresultsoninvariantmeasures(cf.Sect.4.3)wecouldnotfindinthe literature for the “variational approach”.They are, however,quite straightforward modificationsofthoseinthe“semigroupapproach”in[27]. Tokeepthiscoursereasonablyself-containedwealsoincludeacompleteproof ofthefinite-dimensionalcasein Chap.3,whichisbasedonthe veryfocussedand beautiful exposition in [52], which uses the Euler approximation. Among other complementing topics such as Chap.6, which contains a concise introduction to the “semigroup (or mild solution) approach”, the appendices contain a detailed accountoftheYamada–WatanabeTheoremontherelationbetweenweakandstrong solutions(cf.AppendixE),anda detailedproofofGirsanov’sTheoremininfinite dimensions(cf.AppendixI). The structureofthis monographis, we hope,obviousfromthe list ofcontents. Here,weonlymentionthatasubstantialpartconsistsofaverydetailedintroduction to stochastic integration on Hilbert spaces (see Chap.2), major parts of which (as well as Appendices A–C) are taken from the Diploma thesis of Claudia Prévôt (née Knoche) and Katja Frieler (see [34]), which in turn was based on [26] and supervisedbythesecondnamedauthorofthismonograph.Wewouldliketothank bothofthematthispointfortheirpermissiontodothis.Wethankallcoauthorsof thosejointpaperswhichformanothercomponentforthebasisofthismonograph.It wasreallyapleasureworkingwiththeminthisexcitingareaofprobabilitytheory. We would also like to thank Nelli Schmelzer, Matthias Stephan, Sven Wiesinger and Lukas Wresch for the excellent type job, as well as the participants of the graduatecoursesatPurdueandBielefeldUniversityforcheckinglargepartsofthe textcarefully.Special thanksgo to MichaelScheutzow and Byron Schmulandfor spotting a number of misprints and small errors in [67]. We also thank Claudia Prévôtforgivingpermissionforthisextensionof[67].Furthermore,thefirstnamed authoracknowledgesthefinancialsupportfromNSFC(No.11201234,11571147) and a project funded by PAPD of Jiangsu Higher Education Institutions. The last named author would like to thank the German Science Foundation (DFG) for its financial support through SFB 701 and also Jose Luis da Silva for his hospitality 1.2 GeneralPhilosophyandExamples 3 duringaverypleasantstayattheUniversityofMadeirawherethefinalproofreading ofthismonographwasdone. 1.2 General PhilosophyandExamples Before starting with the main body of this course, let us briefly recall the general philosophy of describing stochastic dynamics by stochastic differential equations (SDE) in a more heuristic and intuitive way. These usually take values in a space H of(generalized)functions,e.g.onadomainƒ (cid:2) Rd,oradifferentialmanifold, afractalorevenmerelyinanarbitrarymeasurablespace.ƒ;B/.Abstractly,H isa separableHilbertorBanachspace.ThengivenamapF WŒ0;T(cid:2)(cid:3)H(cid:3)U !H,where T 2(cid:2)0;1Œ andU isanotherseparableHilbertspace,oneconsidersthe differential equations dX.t/ DF.t;X.t/;WP.t// (1.1) dt on H. Here WP.t/;t 2 Œ0;T(cid:2), is a U-valued white noise in time, more precisely, thegeneralisedtimederivativeofaU-valuedcylindricalBrownianmotionW.t/ D .Wk.t//k2N;t 2Œ0;T(cid:2),onsomeprobabilityspace.(cid:3);F;P/.HenceWP.t/;t 2Œ0;T(cid:2), areindependentcentredGaussianvariableswithinfinitevariance,henceinregardto (1.1)the“worst”(Gaussian)randomperturbationthatcanoccurin(1.1).Employing aTaylorexpansionforFaround02Uand,neglectingtermsoforder2andhigher, turns(1.1)into dX.t/ DF.t;X.t/;0/CD3F.t;X.t/;0/WP.t/; (1.2) dt whereD3FdenotesthederivativeofFwithrespecttoitsthirdcoordinate.Setting A.t;x/WDF.t;x;0/; B.t;x/WDD3F.t;x;0/ andtakingintoaccountthenon-differentiabilityofW.t/int,(1.2)turnsinto dX.t/DA.t;X.t//dtCB.t;X.t//dW.t/; (1.3) to be rigorously understoodin integral form. We mention here that the stochastic termin(1.3)isoftencalledorinterpretedasa“stochasticforce”,thoughtheequa- tion is first order. This can, however, be justified by the Kramers–Smoluchowski approximation(see [14, 15, 33, 51, 76]). Furthermore, A is called the drift of the equation. We briefly recall here that the linear pendant of (1.3) is given by the associ- ated Fokker–Planck–Kolmogorovequationsobtained as a linearization of (1.3) as