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1、Probab.TheoryRelat.Fields100,365-393(1994)ProbabilityTheoryRelatedFields9Springer-Verlag1994AttractorsforrandomdynamicalsystemsHansCrauel,~FrancoFlandoli21FachbereichMathematik,Universit~itdesSaarlandes,D-66041Saarbrticken,Germany2ScuolaNormaleSuperiore,Piazza
2、deiCavalieri7,1-56126Pisa,ItalyReceived:17November1992/Inrevisedform:6April1994Summary.AcriterionforexistenceofglobalrandomattractorsforRDSisestablished.ExistenceofinvariantMarkovmeasuressupportedbytheran-domattractorisproved.ForSPDEthisyieldsinvariantmeasuresf
3、ortheassociatedMarkovsemigroup.TheresultsareappliedtoreactiondiffusionequationswithadditivewhitenoiseandtoNavier-Stokesequationswithmultiplicativeandwithadditivewhitenoise.MathematicsSubjectClassification:58Fll,58F12,34D45,35Q301IntroductionOneofthemostimportan
4、tdiscoveriesinmathematicalphysicsduringthepast20yearsisthatoffinite-dimensionalattractorsinmathematicalmodelsforfluiddynamics.However,alltheanalysisbreaksdownassoonasonewantstotakerandominfluencesonthesystemunderinvestigationinaccount.Inparticular,whensubjectin
5、gthesystemtoadditivewhitenoise,thereisnochancethatboundedsubsetsofthestatespaceremaininvariant.Whitenoisepushesthesystemoutofeveryboundedsetwithprobabilityone.Thepresentpaperisanattempttoovercomethisproblem.Weinfactdodiscovercompactinvariantsets-however,theyare
6、notfixed,buttheydependonchance,andtheymovewithtime(inacoherent,"stationary"manner).Typicallytheyleaveeveryboundeddeterministicset.Stilltheyarecompact-infact,oftenevenfinite-dimensional-andtheyattractallboundedsubsetsofthestatespace.Theapproachweuseisthatofrando
7、mdynamicalsystems(RDS).Bytakinganabstract,ergodictheoreticalpointofview,RDScoversomeofthemostcommonclassesofsystemsinvolvingrandomnessandtimeevolution.Amongstthemarestochasticflows,randomflows,andproductsofstationaryrandommaps.SeeArnold[1];forasurveyonRDSseeArn
8、oldandCrauel[3].366H.Crauel,F.FlandoliSincewewanttoapplytheresultstoinfinite-dimensionalsystems,wewillhavetotakecarethattherespectivesystemdoesgenerateastochasticflow.Infini
