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1、JournalofDynamicsandDiJferentialEquations,Vol.9,No.2,1997RandomAttractorsHansCrauel,1ArnaudDebussche,2andFrancoFlandoli3ReceivedJune22,1995Inthispaper,wegeneralizethenotionofanattractorforthestochasticdynami-calsystemintroducedin[7].Weprovethatthestochasticatt
2、ractorsatisfiesmostofthepropertiessatisfiedbytheusualattractorinthetheoryofdeterministicdynamicalsystems.WealsoshowthatourresultsapplytothestochasticNavier-Stokesequation,thewhitenoise-drivenBurgersequation,andanonlinearstochasticwaveequation.KEYWORDS:Randomat
3、tractors;stochasticdynamicalsystems;deter-ministicnonautonomoussystems;stochasticNavier-Stokesequation;stochasticBurgersequation;stochasticnonlinearwaveequation.AMSSubjectClassifications:58FI1,58F12,60HI5,34D45,35Q30,35Q35,35L05.O.INTRODUCTIONTheunderstandingo
4、ftheasymptoticbehaviorofdynamicalsystemsisoneofthemostimportantproblemsofmodemmathematicalphysics.Onewaytoattacktheproblemfordissipativedeterministicdynamicalsystemsistoconsideritsglobalattractor.Thisisaninvariantsetthatattractsallthetrajectoriesofthesystem.It
5、sgeometrycanbeverycomplicatedandreflectsthecomplexityofthelong-timedynamicsofthesystems.Oneessentialstepinprovingtheexistenceofaglobalattractoristoderivetheexistenceofaboundedattractingset,i.e.,aboundedsetsuchthatthedistancefromanyorbittothissettendstozerowhen
6、thetimegoestooo.Manypapershavebeendevotedtothissubject(see,e.g.,[15]or[211).iHellwigstraBe17,66121Saarbrficken,Germany.E-mail:crauel(asaar.de2Laboratoired'AnalyseNum6dqueetCNRS,Universit6Paris-Sud,Batiment425,91405OrsayCedex,France.E-mail:Arnaud.DEBUSSCHE@math
7、.u-psud.fr3DipartimentodiMatematicaApplicataU.Dini,ViaBonanno25B,56126Pisa,Italy.E-mail:flandoli@dma.unipi.it307I040-7294,'97/0400.0307512.50/0i"1997PlenumPublishingCorporation308Craud,Debussche,andFlandoliHereweareconcernedwithstochasticdynamicalsystemsthatar
8、eobtainedfromapartialdifferentialequationperturbedbyarandomforcingterm.Itisnotdifficulttoseethattheextensiontothiscaseofthenotionofattractorisnotstraightforward.First,theunderlying
