stochastic evolution equations in umd banach spacesnew

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1、STOCHASTICEVOLUTIONEQUATIONSINUMDBANACHSPACESJ.M.A.M.VANNEERVEN,M.C.VERAAR,ANDL.WEISAbstract.Wediscussexistence,uniqueness,andspace-timeH¨olderregular-ityforsolutionsoftheparabolicstochasticevolutionequation(dU(t)=(AU(t)+F(t,U(t)))dt+B(t,U(t))dWH(t),t∈[0,T0],U(0)=u0,whereAgeneratesananal

2、yticC0-semigrouponaUMDBanachspaceEandWHisacylindricalBrownianmotionwithvaluesinaHilbertspaceH.WeprovethatifthemappingsF:[0,T]×E→EandB:[0,T]×E→L(H,E)satisfysuitableLipschitzconditionsandu0isF0-measurableandbounded,thenthisproblemhasauniquemildsolution,whichhastrajectoriesinCλ([0,T];D((−A)

3、θ)providedλ≥0andθ≥0satisfyλ+θ<1.Vari-2ousextensionsofthisresultaregivenandtheresultsareappliedtoparabolicstochasticpartialdifferentialequations.1.IntroductionandstatementoftheresultsInthispaperweproveexistence,uniqueness,andspace-timeregularityresultsfortheabstractsemilinearstochasticCauc

4、hyproblem(dU(t)=(AU(t)+F(t,U(t)))dt+B(t,U(t))dWH(t),t∈[0,T0],(SCP)U(0)=u0.HereAisthegeneratorofananalyticC0-semigroup(S(t))t≥0onaUMDBanachspaceE,HisaseparableHilbertspace,andforsuitableη≥0thefunctionsF:[0,T]×D((−A)η)→EandB:[0,T]×D((−A)η)→L(H,E)enjoysuitableLipschitzcontinuityproperties.T

5、hedrivingprocessWHisanH-cylindricalBrow-nianmotionadaptedtoafiltration(Ft)t≥0.InfactweshallallowconsiderablylessarXiv:0804.0932v1[math.FA]6Apr2008restrictiveassumptionsonFandB;bothfunctionsmaybeunboundedandmaydependontheunderlyingprobabilityspace.AHilbertspacetheoryforstochasticevolutione

6、quationsoftheabovetypehasbeendevelopedsincethe1980sbytheschoolsofDaPratoandZabczyk[10].Muchofthistheoryhasbeenextendedtomartingaletype2-spaces[2,3];seealsotheearlierwork[35].ThisclassofBanachspacescoverstheLp-spacesintherangeDate:April6,2008.2000MathematicsSubjectClassification.Primary:47

7、D06,60H15Secondary:28C20,46B09,60H05.Keywordsandphrases.Parabolicstochasticevolutionequations,UMDBanachspaces,sto-chasticconvolutions,γ-radonifyingoperators,L2γ-Lipschitzfunctions.Thefirstandsecondnamedauthorsaresupportedbya‘VIDIsubsidie’(639.032.201)inthe‘Vernieuwingsimpu