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1、SomelinearSPDEsdrivenbyafractionalnoisewithHurstindexgreaterthan1/2RalucaM.Balan∗†February18,2011AbstractInthisarticle,weidentifythenecessaryandsufficientconditionsfortheexistenceofarandomfieldsolutionforsomelinears.p.d.e.’sofparabolicandhyperbolictype.T
2、heseequationsrelyonaspatialop-eratorLgivenbytheL2-generatorofad-dimensionalL´evyprocessX=(Xt)t≥0,andaredrivenbyaspatially-homogeneousGaussiannoise,whichisfractionalintimewithHurstindexH>1/2.Asanapplication,weconsiderthecasewhenXisaβ-stableprocess,with
3、β∈(0,2].Intheparaboliccase,wedevelopaconnectionwiththepo-tentialtheoryoftheMarkovprocessX¯(definedasthesymmetrizationofX),andweshowthattheexistenceofthesolutionisrelatedtotheexistenceofa“weighted”intersectionlocaltimeoftwoindependentcopiesofX¯.MSC2000s
4、ubjectclassification:Primary60H15;secondary60H05,60G51arXiv:1102.3992v1[math.PR]19Feb2011Keywordsandphrases:stochasticpartialdifferentialequations,fractionalBrownianmotion,spatiallyhomogeneousGaussiannoise,L´evyprocesses∗DepartmentofMathematicsandStatis
5、tics,UniversityofOttawa,585KingEdwardAvenue,Ottawa,ON,K1N6N5,Canada.E-mailaddress:rbalan@uottawa.ca†ResearchsupportedbyagrantfromtheNaturalSciencesandEngineeringResearchCouncilofCanada.11IntroductionIn1944,inhisseminalarticle[19],Itˆointroducedthestoc
6、hasticintegralwithrespecttotheBrownianmotion,whichturnedouttobeoneofthemostfruitfulideasinmathematicsinthe20thcentury.Thisleadtothetheoryofdiffusions(whoseoriginscanbetracedbackto[20]),andthedevelopmentofthestochasticcalculuswithrespecttomartingales(in
7、itiatedin[25]).Theseideashavegrownintoasolidbranchofprobabilitytheorycalledstochasticanalysis,whichincludesthestudyofstochasticpartialdifferentialequations(s.p.d.e.’s)Traditionally,therehavebeenseveralapproachesforthestudyofs.p.d.e.’s.Themostimportanta
8、re:theWalshapproachwhichreliesonstochasticin-tegralswithrespecttomartingale-measures(see[33]),theDaPratoandZabczykapproachwhichusesstochasticintegralswithrespecttoHilbert-space-valuedWienerprocesses(see[10]),andtheKrylovapproachwhichusesthecon
