Weak approximation of stochastic partial different

arXiv:0804.1304v1[math.NA]8Apr2008WEAKAPPROXIMATIONOFSTOCHASTICPARTIALDIFFERENTIALEQUATIONS:THENONLINEARCASEARNAUDDEBUSSCHEAbstract.WestudytheerroroftheEulerschemeappliedtoastochasticpartialdifferentialequation.Weprovethatasitisoftenthecase,theweakorderofconvergenceistwicethestrongorder.AkeyingredientinourproofisMalliavincalculuswhichenablesustogetridoftheirregulartermsoftheerror.Weapplyourmethodtothecaseasemilinearstochasticheatequationdrivenbyaspace-timewhitenoise.April9,20081.IntroductionWhenoneconsidersanumericalschemeforastochasticequation,twotypesoferrorscanbeconsidered.Thestrongerrormeasuresthepathwiseapproximationofthetruesolutionbyanumericalone.Thisproblemhasbeenextensivelystudiedinfinitedimensionforstochasticdifferentialequations(seeforinstance[20],[26],[27],[32])andalsomorerecentlyininfinitedimensionforvarioustypesofstochasticpartialdifferentialequations(SPDEs)(seeamongothers[1],[4],[6],[10],[11],[12],[13],[14],[15],[16],[17],[18],[22],[23],[29],[30],[34],[35],[36]).Anotherwaytomeasuretheerroristheso-calledweakorderofconvergenceofanumericalschemewhichisconcernedwiththeapproximationofthelawofthesolutionatafixedtime.Inmanyapplications,thiserrorismorerelevant.PioneeringworkbyMilstein([24],[25])andTalay([33])havebeenfollowedbymanyarticles(seereferencesinthebookscitedabove).VeryfewworksexistintheliteraturefortheweakapproximationofsolutionofSPDEs.Adelayedstochasticdifferentialequationhasbeenstudiedin[3].WeakorderforaSPDEhasbeenstudiedonlyrecentlyin[7],[8],[19].Inordertoexplainthenoveltyofthepresentarticle,letusfocusonaspecificexample.WeconsiderastochasticnonlinearheatequationinaboundedintervalI=(a,b)⊂RwithDirichletboundaryconditionsanddrivenbyaspace-timewhitenoise:(1.1)∂X∂t=Xξξ+f(X)+σ(X)˙η,ξ∈I,t0,X(a,t)=X(b,t)=0,t0,X(ξ,0)=x(ξ),ξ∈I.1991MathematicsSubjectClassification.35A40,60H15,60H35.Keywordsandphrases.Weakorder,stochasticheatequation,Eulerscheme.ENSdeCachan,AntennedeBretagne,CampusdeKerLann,Av.R.Schuman,35170BRUZ,FRANCE([e-mail]).Acknowledgments:PartofthisworkwasdonewhiletheauthorvisitedtheInstitutMittag-Leffler(Djursholm,Sweden)duringthesemester”StochasticPartialDifferentialEquations”.12A.DEBUSSCHEWherefandσaresmoothLipschitzfunctionsfromRtoR.Weintroducetheclassicalabstractframeworkextensivelyusedinthebook[5].WesetH=L2(I),A=∂ξξ,D(A)=H2(I)∩H10(I),WisacylindricalWienerprocesssothatthespace-timewhitenoiseismathematicallyrepresentedasthetimederivativeofW.Wesetf(x)(ξ)=f(x(ξ)),x∈Handdefineσ:H→L(H)byσ(x)h(ξ)=σ(x(ξ))h(ξ),x,h∈H.Wethenrewrite(1.1)as(1.2)dX=(AX+f(X))dt+σ(X)dW,X(0)=x.Itiswellknownthatthisequationhasauniquesolution.WeinvestigatetheerrorcommittedwhenapproximatingthissolutionbythesolutionoftheEulerscheme(1.3)Xk+1−Xk=Δt(AXk+1+f(Xk))+σ(Xk)(W((k+1)Δt)−W(kΔt)),X0=x,whereΔt=T/N,N∈N,T0.Thestudyoftheweakerroraimstoproveboundsofthetype:|E(ϕ(X(nΔt)))−E(ϕ(Xn)|≤cΔtδ,withaconstantcwhichmaydependonϕ,x,Nandonthevariousparameterintheequation.AlsoϕisassumedtobeasmoothfunctiononH.Ifsuchaboundistrue,wesaythattheschemehasweakorderδ.Incomparison,thestrongerrorisgivenbyE(|(X(nΔt))−Xn|)orE(supn=0,...,N|(X(nΔt))−Xn|).Clearly,iftheschemehasstrongorder˜δthenithasweakorderδ≥˜δ.Indeed,thetestfunctionsϕareLipschitz.Ingeneral,itisexpectedthattheweakorderislargerthanthethestrongorder.InthecaseoftheEulerschemeappliedtoastochasticdifferentialequation,itiswellknownthatthestrongorderis1/2whereastheweakorderis1(see[32]).TheclassicalproofofthisusestheKolmogorovequationassociatedtothestochasticequation.Themaindifficultytogeneralizethisprooftotheinfinitedimensionalequation(1.2)isthatthisKolmogorovequationisthenapartialdifferentialequationwithaninfinitenumberofvariablesandinvolvingunboundedoperators(see(3.6)below).Thedelayedstochasticdifferentialequationstudiedin[3]isaninfinitedimensionalproblembutsincetheequationdoesnotcontaindifferentialoperatorstheKolmogorovequationissimplertostudy.In[19],aSPDEsimilarto(1.2)isconsideredbutveryparticulartestfunctionsϕareused.Theyareallowedtodependonlyonfinitedimensionalprojectionsoftheunknownandtheboundoftheweakerrorinvolvesaconstantwhichstronglydependsonthedimension.In[7],[8],theKolmogorovequationisnotuseddirectly.Achangeofvariableisusedinordertosimplifyit.In[7],thestochasticnonlinearSchr¨odingerequationisconsideredandthefactthatthelinearSchr¨odingerequationgeneratesaninvertiblegroupisusedinanessentialway.Thisisobviouslywrongfortheheatequationconsideredhere.Thesamechangeofunknownworksinthecaseofalinearequationwithadditivenoiseasshownin[8]butthereitisusedthatthesolutioncanbewrittendownexplicitly.Wehavenotbeenabletogeneralizethisideatothenonlinearequationconsideredhere.WeuseinfacttheoriginalmethoddevelopedbyTalayinthefinitedimensionalcase.TheweakerrorisdecomposedthankstotheKolmogorovequationsoneachtimestep.EachtermrepresentstheerrorbetweenthesolutionoftheKolmogorovequationononetimestepandtheWEAKAPPROXIMATIONOFSPDES3approximationgivenbythenumericalsolution.Duetothepresenceofunboundedoperators,thisapparentlyrequiresalotofsmoothnessonthenumericalsolution.Themainideahereistoobserv