Ergodic Propertiesof Markov Processes

《Ergodic Propertiesof Markov Processes》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

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].IngeneralaLiapunovfunctionWisapositivefunctionwhichgrowsatinfinityandsatisfiesaninequalityinvolvingthegeneratoroftheMarkovprocessL: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.InsufficientlysimplesituationsonemightjustguessaLiapunovfunction.Forin-terestingproblems,however