Malliavin分析在随机偏微分方程中的应用英文版DavidNualart

《Malliavin分析在随机偏微分方程中的应用英文版DavidNualart》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、ApplicationofMalliavinCalculustoStochasticPartialDifferentialEquationsDavidNualart1IntroductionTheaimofthesenotesistoprovideanintroductiontotheMalliavincalculusanditsapplicationtotheregularityofthesolutionsofaclassofstochasticpartialdifferentialequations.TheMalliavincalculusisadifferentialcalculusonaG

2、aussianspacewhichhasbeendevelopedfromtheprobabilisticproofbyMalliavinofH¨ormander’shypoellipticitytheorem(see[8]).InthenextsectionwepresentanintroductiontotheMalliavincalculus,andwederivethemainpropertiesofthederivativeanddivergenceoperators.Section3isdevotedtoestablishthemaincriteriafortheexistenc

3、eandregularityofdensityforarandomvariableinaGaussianspace.ThelasttwosectionsaredevotedtodiscusstheapplicationsoftheMalli-avincalculustostochasticpartialdifferentialequations.Firstweconsideraone-dimensionalheatequationdrivenbyaspace-timewhitenoiseonthetimeinterval[0,1]withDirichletboundaryconditions,

4、andweshowthatforany(t,x)∈(0,∞)×(0,1),thesolutionu(t,x)hasaninfinitelydifferen-tiabledensityifthecoefficientsaresmoothandthediffusioncoefficientisboundedawayfromzero.Thelastsectiondealswithaclassofstochasticpar-tialdifferentialequationsperturbedbyaGaussiannoiseon[0,∞)×Rdwithhomogeneousspacialcovariance,intr

5、oducedbyDalangin[6].Wesurveytheresultsobtainedinsomerecentpapers[13;16;17]ontheregularityofthedensityforthesolutiontothisclassofequations.2MalliavinCalculusTheMalliavincalculusisaninfinitedimensionalcalculusonaGaussianspace,whichismainlyappliedtoestablishtheregularityofthelawofnonlinearfunctionalsof

6、theunderlyingGaussianprocess.D.KhoshnevisanandF.Rassoul-Agha(eds.)AMinicourseonStochasticPartial73DifferentialEquations.LectureNotesinMathematics1962.cSpringer-VerlagBerlinHeidelberg200974D.NualartSupposethatHisarealseparableHilbertspacewithscalarproductdenotedby·,·H.ConsideraGaussianfamilyofrand

7、omvariablesW={W(h),h∈H}definedinacompleteprobabilityspace(Ω,F,P),withzeromeanandcovarianceE(W(h)W(g))=h,gH.(1)Themappingh→W(h)providesalinearisometryofHontoaclosedsubspaceofHofL2(Ω).1Example2.1.(Brownianmo