Existence And Uniqueness Of Stationary Solution Of Nonlinear Stochastic Differential Equati

StochasticDifferentialEquationWithMemory.

YuriBakhtin1

1Introduction.

Inthispaperastochasticdifferentialequation(SDE)withinfinitememoryisconsidered.Thedriftcoefficientoftheequationisanonlinearfunctionalofthepasthistoryofthesolution.Sufficientconditionsforexistenceanduniquenessofstationarysolutionaregiven.Thisworkismotivatedbyrecentpapers[1]and[2]wherestochasticallyforcednonlinearequationsofhydrodynamicswereconsideredanditwasshownhowtheinfinite-dimensionalstochasticMarkoviandynamicsrelatedtotheseequationscanbereducedtofinite-dimensionalstochasticdynamics.Thecorrespondingfinite-dimensionalsystemsarehoweveressentiallynon-Markovian.

So,theimportantproblemofexistenceanduniquenessofstationarysolutionsforstochastichydrodynamicalequationsistightlyrelatedtoexistenceanduniquenessofsta-tionarysolutionsofSDEswithinfinitememory.Someresultsintheareawereestablishedin[3].Inthefirstpartofthispapersomenecessarynotionsareintroducedandthemainresultisstated.Aproofofthemainresultisgiveninthesecondpart.Wecombinetheap-proachof[3]withaninterestingmethodforestablishingthedesireduniquenesssuggestedin[1]and[2]fortheproblemsconsideredtherein.

Theequationunderconsiderationis

dX(t)=a(πtX)dt+dW(t).

(1)

HereW(t),t∈Risstandardd−dimensionalWienerprocess(i.e.aGaussianRd-valuedstochasticprocesswithcontinuoustrajectoriesdefinedonthewholereallineRwithindependentandstationaryincrements,W(0)=0,EW(t)=0,EW(t)2=|t|,t∈R),πtisamapfromthespaceCofRd-valuedcontinuosfunctionsdefinedonRtothespaceC−ofcontinuousfunctionsdefinedonR−=(−∞,0]:

πtX(s)=X(s+t),

s∈R−.

Thismapgivesthepasthistoryofacontinuousprocessuptotimet∈R.Fromnowonsupposea(·):C−→Rdtobeacontinuousfunctionalwithrespecttometric

ρ−(f,g)󰀄

∞=

2−n(󰀔f−g󰀔n∧1),

f,g∈C−,

n=1

whichdefinesLU-topologyonthespaceC−.Here󰀔h󰀔n=max−n≤t≤0|h(t)|,and|·|

denotestheEuclideannorm.

ForastochasticprocessXandasetA⊂Rtheσ-algebrageneratedbyr.v.’sX(s),s∈AwillbedenotedbyσA(X)andtheσ-algebrageneratedbyr.v.’sX(s)−X(t),s,t∈AwillbedenotedbyσA(dX).

ConsiderthespaceΩ=C×Cwiththemetricanalogoustothemetricρ−definedabove.

AprobabilitymeasurePonthespaceΩwithBorelσ-algebraBissaidtodefineasolutiontotheequation(1)onRifthefollowingthreeconditionsarefulfilledwithrespecttothemeasureP:

1.TheprojectionW:C×C→C,ω=(ω1,ω2)→ω2,isastandardd-dimensionalWienerprocess.

2.Foranyt∈R

σ(−∞,t](X)∨σ(−∞,t](dW)isindependentof

σ[t,∞)(dW).

(2)

HereandfurtherX:C×C→C,ω=(ω1,ω2)→ω1.

3.Ifs󰀇ta.s.

a(πθX)dθ+W(t)−W(s).X(t)−X(s)=

s

(3)

Ifinadditionthedistributionoftheprocess

(X,dW)≡(X(t),−∞doesnotchangeundertimeshiftsthenthemeasurePissaidtodefineastationarysolution.

Let’sstatethemainresult.

Theorem1.Letthedriftcoefficienta(·)satisfythefollowingconditions:1.ThereexistsuchconstantsK>0,λ>0thattheestimate

󰀇0

eλt|x−(t)−y−(t)|dt(4)|a(x−)−a(y−)|≤K

−∞

isfulfilledwheneverx−,y−∈C−,x−(0)=y−(0)andtheintegralintheright-handsideis

finite.

2.ThereexistsuchconstantsC1≥0andC2>0that

(a(x−),x−(0))≤C1−C2|x−(0)|2,

3.ThereexistsuchaconstantC3>0that

|a(x−)|≤C3|x−(0)|,

x−∈C−.

(6)

x−∈C−.

(5)

ThenthereexistaprobabilisticmeasurePonthespaceC×Cwhichdefinesastationarysolutionoftheequation(1).SuchmeasureisuniqueintheclassofmeasuresforwhichalmosteveryrealizationXpossessesthefollowingproperty:

|X(t)|≤K′eλ|t|,

′

t≤0.(7)

HereK′∈Randλ′∈(0,λ)aresomeconstantsdependingontherealizationX.

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2Proofofthemainresult.

First,letusprovetheexistenceofthestationarysolutionusingtheKrylov–Bogolyubovapproach.

AprobabilisticlawinC×CissaidtodefineasolutionofCauchyproblemfortheequation(1)withinitialdatax−∈C−ifthefollowingconditionsaresatisfied:WisastandardWienerprocess;foreveryt∈Rtherelation(2)istrue;theequality(3)isfulfilledfors=0andeveryt>0;X(t)=x−(t)foranyt<0.ExistencetheoremforsolutionsofCauchyproblemisprovedin[3].

LetP0denotesuchalawfortheinitialdataidenticallyequaltozeroandPsdenotethetimes-shiftofthisdistributioni.e.asolutionoftheCauchyproblemsubjecttozero

initial−1f󰀈datadefinedontheset(−∞,s],s∈R.FormallyPs=P0θwhereθ(f,g)=(f,=f(t−s),g󰀈(t)=g(t−s)−g(−s).

SincethefunctionPs(E)ismeasurablewithrespecttosforallE∈B(see[3]),󰀈g󰀈),(t)ss

forT>0onecandefineapropabilitymeasure

QT(·)=

1

NowletusestimateincrementsoftheprocessX.QT{|X(t2)−X(t1)|>z}

≤QT{|W(t2)−W(t1)|>z/2}+QT

󰀆󰀇

󰀉

a(πθX)dθ>z/2

󰀁󰀇

t2t1

t2

t1

≤16z−4EQT|W(t2)−W(t1)|4+4z−2EQT

a(πθX)dθ

󰀂2

.(9)

ThenextinequalityisaconsequenceoftheFubinitheorem,elementaryinequality|xy|≤(x2+y2)/2,well-knownexpressionformomentsofGaussiandistributionandrelations(6)and(9):

2−2

QT{|X(t2)−X(t1)|>z}≤48z−4|t2−t1|2+4C3zM|t2−t1|2.

(10)

TightnessofthefamilyofprojectionsofmeasuresQTonthefirstcomponent{QTX−1}

andhencethedesiredtightnessofthefamily{QT}isimpliednowby(8)and(10).So,QTn→Q∞whenn→∞forsomesequence(Tn)n∈Nandresultsof[3]implythatQ∞definesastationarysolutionoftheequation(1).

Lemma1.Foranyδ>0thefollowingestimateistrueP−a.s.

t→−∞

Law

lim|Xt|·|t|1/2+δ=0.

Proof.AnestimateformeasureP,analogoustotheestimate(10),imliesthatforanys∈R

󰀆󰀉Pmax|X(t)|>Kz≤P{X(s)>z}+P{X(s+1)>z}+C(z−2+z−4)

t∈[s,s+1]

forsufficientlylargeconstantsK,C>0.UsingthisinequalityandChebyshevinequalityanduniformint∈Rboundednessofthesecond-ordermomentofX(t)oneobtainsthatforallδ0∈(0,δ)theseries

∞󰀄

P

n=0

󰀆

t∈[−n−1,−n]

max

|X(t)|>Kn1/2+δ0

󰀉

isconvergentandthelemmafollowsfromtheBorel–Kantellylemma.

Inparticular,Lemma1impliesthatthetrajectoriesoftheprocessXsatisfythecon-dition(7)P-a.s.

Nowweturntotheproofofuniqueness.ConsideranarbitrarymeasurePwhichdefinesastationarysolutionoftheequation(1).SupposealsothattherealizationsoftheprocessXsatisfycondition(7)P-a.s.IntroduceaspaceC+ofRd-valuedcontinuousfunctionsdefinedonR+=[0,∞).Forx−∈C−wedenotePx−themeasureonΩ+=C+×C+whichdefinesasolutionofCauchyproblemwiththeinitialdatax−.Px−isaconditionaldistributionofthemeasurePconditionedonX−=x−.

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Lemma2.Condition1ofTheorem1impliesthatthereexistsasetA⊂C−suchthatP(π0X∈A)=1andifx−,y−∈Ax−(0)=y−(0)thenthemeasuresPx−andPy−areequivalent.

Proof.Considerx−,y−∈C−suchthateachofthesefunctionsadmitsanexponentialestimatelike(7).ToprovethatPy−isabsolutelycontinuouswithrespecttoPxcondition(see,e.g.,[5,Chapter−,weusetheGirsanovtheoremandverifythNovikov8]).Thesamereasoningwillbevalidforinterchangedx−andy−.

TheNovikovconditioncanbewrittenasfollows:

EPx−exp

󰀆1

SupposetherearetwodifferentergodicmeasuresP(1)P(2)definingstationarysolu-tions.ThereexistsaboundedfunctionalFsuchthat

F2,

andforsomeS>0x(s)=y(s),s∈[0,S]impliesF(x)=F(y).

ThenthereexistsetsB1,B2∈BsuchthatP(i)(Bi)=1and

lim

1

FionBi,i=1,2.

T→∞

Lemmas3and5implythat

P(2)(B1)=

󰀇

P(2)(B1|X(0)=l)P(2)(X(0)∈dl)>0.

Rd

F2.So,P(2)(B1∩B2)>0andB1∩B2=∅,whichcontradictstheassumption

TheauthorisgratefultoProfessorYa.G.Sinaiforstatementoftheproblemandusefuldiscussions.

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KyotoUniv.V.4,1964,p.1–75.[4]BillingsleyP.Convergenceofprobabilitymeasures.N.Y:JohnWiley&Sons,1968.[5]RevuzD.,YorM.ContinuousmartingalesandBrownianmotionI.Berlin–Heidelberg:

Springer-Verlag,1994.

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