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1、ErgodicPropertiesofMarkovProcessesLucRey-BelletDepartmentofMathematicsandStatistics,UniversityofMassachusetts,Amherst,MA01003,USAe-mail:lr7q@math.umass.edu1Introduction..................................................12StochasticProcesses...........................................23Marko
2、vProcessesandErgodicTheory...........................43.1Transitionprobabilitiesandgenerators........................43.2StationaryMarkovprocessesandErgodicTheory...............74BrownianMotion..............................................125StochasticDifferentialEquations................
3、................146ControlTheoryandIrreducibility...............................247HypoellipticityandStrong-FellerProperty.......................268LiapunovFunctionsandErgodicProperties......................28References.........................................................391Introducti
4、onInthesenoteswediscussMarkovprocesses,inparticularstochasticdifferentialequa-tions(SDE)anddevelopsometoolstoanalyzetheirlong-timebehavior.Thereareseveralwaystoanalyzesuchproperties,andourpointofviewwillbetousesystem-aticallyLiapunovfunctionswhichallowanicecharacterizationoftheergodicprop
5、-erties.Inthiswefollow,atleastinspirit,theexcellentbookofMeynandTweedie[7].IngeneralaLiapunovfunctionWisapositivefunctionwhichgrowsatin?nityandsatis?esaninequalityinvolvingthegeneratoroftheMarkovprocessL:roughlyspeakingwehavetheimplications(αandβarepositiveconstants)1.LW≤α+βWimpliesexiste
6、nceofsolutionsforalltimes.2.LW≤?αimpliestheexistenceofaninvariantmeasure.3.LW≤α?βWimpliesexponentialconvergencetotheinvariant.measure2LucRey-BelletFor(2)and(3),oneshouldassumeinaddition,forexamplesmoothnessofthetran-sitionprobabilities(i.ethesemigroupetLissmoothing)andirreducibilityofthep
7、rocess(ergodicityofthemotion).ThesmoothingpropertyforgeneratorofSDEsisnaturallylinkedwithhypoellipticityofLandtheirreducibilityisnaturallyex-pressedintermsofcontroltheory.Insuf?cientlysimplesituationsonemightjustguessaLiapunovfunction.Forin-terestingproblems,however
