A dynamical approximation for stochastic partial d

arXiv:math/0607050v2[math.DS]7Oct2007ADYNAMICALAPPROXIMATIONFORSTOCHASTICPARTIALDIFFERENTIALEQUATIONSWEIWANG&JINQIAODUANAbstract.Randominvariantmanifoldsoftenprovidegeometricstruc-turesforunderstandingstochasticdynamics.Inthispaper,adynamicalapproximationestimateisderivedforaclassofstochasticpartialdiffer-entialequations,byshowingthattherandominvariantmanifoldisalmostsurelyasymptoticallycomplete.Theasymptoticdynamicalbehavioristhusdescribedbyastochasticordinarydifferentialsystemontherandominvari-antmanifold,undersuitableconditions.Asanapplication,stationarystates(invariantmeasures)isconsideredforoneexampleofstochasticpartialdif-ferentialequations.Date:July16,2007(revisedversion);May4,2007(originalversion).2000MathematicsSubjectClassification.Primary37L55,35R60;Secondary60H15,37H20,34D35.Keywordsandphrases.Stochasticpartialdifferentialequations(SPDEs),dynamicalim-pactofnoise,randominvariantmanifoldreduction,dynamicalapproximation,stationarypatterns.TheauthorswouldliketothankDirkBl¨omker,TomasCaraballoandPeterE.Kloedenforhelpfulcomments.ThisworkwaspartlysupportedbytheNSFGrantsDMS-0209326,DMS-0542450,NSFCGrantNo-10626052andtheOutstandingOverseasChineseScholarsFundoftheChineseAcademyofSciences.12W.WANG&J.DUAN1.IntroductionStochasticpartialdifferentialequations(SPDEsorstochasticPDEs)ariseasmacroscopicmathematicalmodelsofcomplexsystemsunderrandominflu-ences.Therehavebeenrapidprogressesinthisarea[12,22,9,26,2,14].Morerecently,SPDEshavebeeninvestigatedinthecontextofrandomdynamicalsystems(RDS)[1];seeforexample,[3,4,5,7,8,23,10,11],amongothers.Invariantmanifoldsarespecialinvariantsetsrepresentedbygraphsinstatespaces(functionspaces)wheresolutionprocessesofSPDEslive.Arandomin-variantmanifoldprovidesageometricstructuretoreducestochasticdynamics.Stochasticbifurcation,inasense,isaboutthechangesininvariantstructuresforrandomdynamicalsystems.Thisincludesqualitativechangesofinvariantmanifolds,randomattractors,andinvariantmeasuresorstationarystates.Duanetal.[10,11]haverecentlyprovedresultsonexistenceandsmooth-nessofrandominvariantmanifoldsforaclassofstochasticpartialdifferentialequations.Inthispaper,wefurtherderiveadynamicalapproximationesti-matebetweenthesolutionsofstochasticpartialdifferentialequationsandtheorbitsontherandominvariantmanifolds.Thisisachievedbyshowingthattherandominvariantmanifoldisalmostsurelyasymptoticallycomplete(seeDefinition4.1).Theasymptoticdynamicalbehaviorthuscanbedescribedbyastochasticordinarydifferentialsystemontherandominvariantmanifold,undersuitableconditions.InthisapproachonekeyassumptionisthattheglobalLipschitzconstantofnonlineartermissmallenough.Iftheinvariantmanifoldisalmostsurelyasymptoticallycompletewecanapproximatetheinfinitedimensionalsystembyasystemrestrictedontherandominvariantmanifoldwhichisinfactfinitedimensional.ThatistheINVARIANTMANIFOLDREDUCTIONANDDYNAMICALAPPROXIMATION3infinitedimensionalsystemisreducedtoafinitedimensionalsystem,whichisusefulforunderstandingasymptoticbehavioroftheoriginalstochasticsystem[19].Asaapplication,in§5,westudytheexistenceofstationarysolutionsofahyperbolicSPDE.Specifically,wewillconsiderthefollowingstochastichy-perbolicpartialdifferentialequation,in§5,withlargediffusivityandhighlydampedterm,onthespace-timedomain[0,2π]×(0,+∞)utt(t,x)+αut(t,x)=νΔu(t,x)+f(u(t,x),x)+u(t,x)◦˙W(t),(1.1)withu(0,x)=u0,ut(0,x)=u1,u(t,0)=u(t,2π)=0,t0whereνandαarebothpositive.Andf∈C2(R,R)isaboundedgloballyLipschitznonlinearity.NotethatthestochasticSine-Gordonequation(f=sinu)isanexample.Whenthedampingislargeenoughtheexistenceofthestationarysolutionsfor(1.1)isobtainedbyconsideringthestochasticsystemontherandominvariantmanifold.Invariantmanifoldsareoftenusedasatooltostudythestructureofat-tractors[6,25].InthispaperwefirstconsiderinvariantmanifoldsforaclassinfinitedimensionalRDSdefinedbySPDEs,thenreducetherandomdynamicstotheinvariantmanifolds.Whentheinvariantmanifoldsareshowntobeal-mostsurelyasymptoticallycomplete(seeDefinition4.1),weobtaindynamicalapproximationsofthesolutionsofstochasticPDEsbyorbitsontheinvariantmanifolds.Almostsureconeinvarianceconcept(seeDefinition4.2)isusedto4W.WANG&J.DUANprovealmostsureasymptoticcompletenesspropertyoftherandominvariantmanifolds.Thispaperisorganizedasfollows.WestatethemainresultondynamicalapproximationforaclassofSPDEsinsection2.Thenwerecallbackgroundmaterialsinrandomdynamicalsystemsandtheexistenceresultofinvariantmanifolds[10]insection3.Themainresultisprovedinsection4,andappli-cationsindetectingstationarystatesarediscussedinthefinalsection5.2.MainresultWeconsiderthestochasticevolutionarysystemdu(t)=(Au(t)+F(u(t)))dt+u(t)◦dW(t)(2.1)whereAisthegeneratorofaC0-semigroup{etA}t≥0onrealvaluedseparableHilbertspace(H,|·|)withinnerproducth,·,i;F:R→RisacontinuousnonlinearfunctionwithF(0)=0andLipschitzconstantLFisassumedtobesmall;andW(t)isastandardrealvaluedWienerprocess.Moreover,◦denotesthestochasticdifferentialinthesenseofStratonovich.Supposethatσ(A),thespectrumofoperatorA,splitsasσ(A)={λk,k∈N}=σc∪σs,σc,σs6=∅(2.2)withσc⊂{z∈C:Rez≥0},σs⊂{z∈C:Rez0}whereCdenotesthecomplexnumbersset.σcisa