Stochastic Differential Equations in Infinite Dimensions: with Applications to Stochastic Partial Differential Equations

308 Pages·2011·1.53 MB·English

by  Leszek Gawarecki

Probability and Its Applications
PublishedinassociationwiththeAppliedProbabilityTrust
Editors:S. Asmussen,J. Gani,P. Jagers,T.G.Kurtz
Probability and Its Applications
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Leszek Gawarecki (cid:2) Vidyadhar Mandrekar
Stochastic Differential
Equations in Infinite
Dimensions
with Applications to Stochastic
Partial Differential Equations
LeszekGawarecki VidyadharMandrekar
DepartmentofMathematics DepartmentofStatisticsandProbability
KetteringUniversity MichiganStateUniversity
1700UniversityAve A436WellsHall
Flint,MI48504 EastLansing,MI48823
USA USA
[e-mail] [e-mail]
SeriesEditors:
SørenAsmussen PeterJagers
DepartmentofMathematicalSciences MathematicalStatistics
AarhusUniversity ChalmersUniversityofTechnology
NyMunkegade andUniversityofGothenburg
8000AarhusC 41296Göteborg
Denmark Sweden
[e-mail] [e-mail]
JoeGani ThomasG.Kurtz
CentreforMathematicsanditsApplications DepartmentofMathematics
MathematicalSciencesInstitute UniversityofWisconsin-Madison
AustralianNationalUniversity 480LincolnDrive
Canberra,ACT0200 Madison,WI53706-1388
Australia USA
[e-mail] [e-mail]
ISSN1431-7028
ISBN978-3-642-16193-3 e-ISBN978-3-642-16194-0
DOI10.1007/978-3-642-16194-0
SpringerHeidelbergDordrechtLondonNewYork
MathematicsSubjectClassification(2010): 35-XX,60-XX
©Springer-VerlagBerlinHeidelberg2011
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thepreparation ofthisbook

Preface
Stochastic differential equations are playing an increasingly important role in
applications to finance, numerical analysis, physics, and biology. In the finite-
dimensional case, there are two definitive books: one by Gikhman and Sko-
rokhod[25],whichstudiestheexistenceanduniquenessproblemalongwithproba-
bilisticpropertiesofsolutions,andanotherbyKhasminskii[39],whichstudiesthe
asymptoticbehaviorofthesolutionsusingtheLyapunovfunctionmethod.Ourob-
ject in this book is to study these topics in the infinite-dimensional case. The two
main problems one faces are the invalidity of the Peano theorem in the infinite-
dimensionalcaseandtheappearanceofunboundedoperatorsifonewantstoapply
finite-dimensionaltechniquestostochasticpartialdifferentialequations(SPDEs).
Motivated by these difficulties, we discuss the theory in the deterministic case
fromtwopointsofview.Thefirstmethod(seePazy[63]andButzerandBerens[6])
involvessemigroupsgeneratedbyunboundedoperatorsandresultsinconstructing
mildsolutions.Theothersets uptheequationinaGelfandtriplet V (cid:2)→H (cid:2)→V∗
∗
ofHilbertspaceswiththespaceV asthedomainoftheunboundedoperatorandV
itscontinuousdual.Inthiscasevariationalsolutionsareproduced.Thisapproachis
studiedbyAgmon[1]andLions[48],whoassumethateithertheinjectionV (cid:2)→H
iscompactandtheunboundedoperatoriscoerciveorthattheunboundedoperator
iscoerciveandmonotone.
The systematic study of the first approach to SPDEs was first undertaken by
Ichikawa [32, 33] and is explained in the timely monographs of Da Prato and
Zabczyk[11,12].TheapproachofJ.P.LionswasfirstusedbyViot[75](seealso
Metivier [56] and Metivier and Viot [58]). Working under the assumption that the
embedding V (cid:2)→H is compact and that the coefficients of the equation are coer-
cive,theexistenceofaweaksolutionwasproven.Theseresultsweregeneralizedby
Pardoux[62],whoassumedcoercivityandmonotonicity,andusedthecrucialdeter-
ministicresultofLions[48]toproducethestrongsolution.Later,Krylov,andRo-
zovskii[42]establishedtheabove-mentionedresultofLionsinthestochasticcase
andalsoproducedstrongsolutions.TheinitialpresentationwasgivenbyRozovskii
in[66].However,ratherrigorousandcompleteexpositioninaslightlygeneralform
isprovidedbyPrévôtandRöckner[64].
vii
viii Preface
InadditiontopresentingtheseresultsonSPDEs,wediscusstheworkofLehaand
Ritter[46,47]onSDEsinR∞ withapplicationstointeractingparticlesystemsand
therelatedworkofAlbeverioetal.[2,3],andalsoofGawareckiandMandrekar[20,
21]ontheequationsinthefieldofGlauberdynamicsforquantumlatticesystems.
InbothcasestheauthorsstudyinfinitesystemsofSDEs.
WedonotpresentheretheapproachusedinKalliapurandXiong[37],asitre-
quiresintroducingadditionalterminologyfornuclearspaces.Forthistypeofprob-
lem (referred to as “type 2” equations by K. Itô in [35]), we refer the reader to
[22,23],and[24],aswellasto[37].
Athirdapproach,whichinvolvessolutionsbeingHidadistributionispresented
byHoldenetal.inthemonograph[31].
The book is divided into two parts. We begin Part I with a discussion of the
semigroup and variational methods for solving PDEs. We simultaneously develop
stochasticcalculuswithrespecttoaQ-WienerprocessandacylindricalWienerpro-
cess,relyingontheclassicapproachpresentedin[49].Thesefoundationsallowusto
developthetheoryofsemilinearpartialdifferentialequations.Weaddressthecase
of Lipschitz coefficients first and produce unique mild solutions as in [11]; how-
ever,wethenextendourresearchtothecasewheretheequationcoefficientsdepend
ontheentire“past”ofthesolution,invokingthetechniquesofGikhmanandSko-
rokhod [25]. We also prove Markov and Feller properties for mild solutions, their
dependenceontheinitialcondition,andtheKolmogorovbackwardequationforthe
relatedtransitionsemigroup.HerewehaveadaptedtheworkofB.Øksendal[61],
S.Cerrai[8],andDaPratoandZabczyk [11].
TogobeyondtheLipschitzcase,wehaveadaptedthemethodofapproximating
continuousfunctionsbyLipschitzfunctionsf :[0,T]×Rn→Rn fromGikhman
andSkorokhod[25]tothecaseofcontinuousfunctionsf :[0,T]×H →H [22].
This technique enabled us to study the existence of weak solutions for SDEs with
continuouscoefficients,withthesolutionidentifiedinalargerHilbertspace,where
theoriginalHilbertspaceiscompactlyembedded.Thisarrangementisused,aswe
havealreadymentioned,duetotheinvalidityofthePeanotheorem.Inaddition,we
studymartingalesolutionstosemilinearSDEsinthecaseofacompactsemigroup
andforcoefficientsdependingontheentirepastofthesolution.
ThevariationalmethodisaddressedinChap.4,wherewestudyboththeweak
andstrongsolutions.Theproblemoftheexistenceofweakvariationalsolutionsis
not well addressed in the existent literature, and our original results are obtained
withthehelpoftheideaspresentedinKallianpuretal.[36].Wehavefollowedthe
approachofPrévôtandRöcknerinourpresentationoftheproblemoftheexistence
anduniquenessofstrongsolutions.
WeconcludePartIwithaninterestingproblemofaninfinitesystemofSDEsthat
doesnotarisefromastochasticpartialdifferentialequationandservesasamodelof
aninteractingparticlesystemandinGlauberdynamicsforquantumlatticesystems.
InPartIIofthebook,wepresenttheasymptoticbehaviorsofsolutionstoinfinite-
dimensionalstochasticdifferentialequations.Thestudyofthistopicwasundertaken
forspecificcasesbyIchikawa[32,33]andDaPratoandZabczyk[12]inthecase
ofmildsolutions.A generalLyapunovfunctionapproachfor strongsolutionsina
Preface ix
GelfandtripletsettingforexponentialstabilitywasoriginatedintheworkofKhas-
minskii and Mandrekar [40] (see also [55]). A generalization of this approach for
mild and strong solutions involving exponential boundedness was put forward by
R.LiuandMandrekar[52,53].Thisworkallowsreaderstostudytheexistenceof
invariant measure [52] and weak recurrence of the solutions to compact sets [51].
Some of these results were presented by K. Liu in a slightly more general form
in[50].
Althoughwehavestudiedtheexistenceanduniquenessofnon-Markoviansolu-
tions, we do not investigate the ergodic properties of these processes, as the tech-
niquesinthisfieldarestillindevelopment[28].
EastLansing LeszekGawarecki
October,2010 VidyadharMandrekar