Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation PDF
Preview Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation
Grundlehren der mathematischen Wissenschaften 345 ASeriesofComprehensiveStudiesinMathematics Serieseditors M.Berger P.delaHarpe F.Hirzebruch N.J. Hitchin L.Hörmander A.Kupiainen G.Lebeau F.-H. Lin S.Mori B.C.Ngô M.Ratner D.Serre N.J.A. Sloane A.M. Vershik M.Waldschmidt Editor-in-Chief A.Chenciner J.Coates S.R.S.Varadhan Forfurthervolumes: www.springer.com/series/138 Tomasz Komorowski (cid:2) Claudio Landim (cid:2) Stefano Olla Fluctuations in Markov Processes Time Symmetry and Martingale Approximation TomaszKomorowski ClaudioLandim InstituteofMathematics InstitutodeMatemática(IMPA) MariaCurie-SkłodowskaUniversity RiodeJaneiro,Brazil Lublin,Poland and and CNRSUMR6085 InstituteofMathematics UniversitédeRouen PolishAcademyofSciences Saint-Étienne-du-Rouvray,France Warsaw,Poland StefanoOlla CEREMADE,CNRSUMR7534 UniversitéParis-Dauphine Paris,France ISSN0072-7830 GrundlehrendermathematischenWissenschaften ISBN978-3-642-29879-0 ISBN978-3-642-29880-6(eBook) DOI10.1007/978-3-642-29880-6 SpringerHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2012943039 MathematicsSubjectClassification: 60F05,60J25,60K35,60K37,60H25,60G60,35B27,76M50 ©Springer-VerlagBerlinHeidelberg2012 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To Raghu Varadhan, a continuous source of inspiration Preface Instatisticalmechanicsasinmanyotherfields,diffusionphenomenamayarisefrom Markovian stochastic modeling. This has motivated, since the early investigations ofKolmogorovandDoeblin’sseminalpaper(Doeblin,1938),anincreasinginterest in central limit theorems for Markov processes. Doeblin made a close connection betweenthevalidityofsuchtheoremsandthetimemixingpropertiesoftheMarkov process.Suchmixingconditions,nowadayscalledDoeblin’sconditions,aresome- times difficult to verify. In particular many problems in statistical mechanics are intrinsically infinite dimensional, and in that case the Doeblin mixing assumption typicallydoesnothold. Inthe1960sGordin(1969)introducedanothermoreanalyticalapproach,based on martingale approximation of additive functionals of Markov processes. Con- sideraMarkovprocess{X ,t ≥0}withanergodicstationaryprobabilitymeasure t π(dx),andletV(x)beafunctiononthestatespaceoftheprocesswithmeanzero with respect to π. Let L be the generator of the corresponding Markovian semi- group Ptu(x)=Ex[u(X(cid:2)t)]. Then if u is a function in the domain of L, the pro- cess u(X )−u(X )− tLu(X )ds is a martingale. Solving the Poisson equation t 0 0 s (cid:2) Lu=V withanicefunctionuinthedomainofL,onecanexpress tV(X )dsasa 0 s martingaleplusaboundaryterm,andreducetheproblemtoprovingacentrallimit theorem for martingales. As we will explain in Chaps. 1 and 2, central limit theo- remsformartingalesrequiresonlyergodicityoftheprocessandaglobalcontrolof thecorrespondingquadraticvariation. Of course the possibility of inverting the generator L is related to the mixing properties of the process, but it permits to tailor the conditions to the particular function V we are studying. If L possesses a spectral gap, then the corresponding PoissonequationcanbesolvedforanyfunctionV withmeanzero. Howeverinmanyapplicationsinstatisticalmechanics,thegeneratorLdoesnot haveaspectralgap.Thetypicalinfinitedimensionalsituationdealswithadynamics thathasconservationlawsandthereisanentirefamilyofstationaryergodicmea- sures. The classical example is given by the problem of the macroscopic diffusive behaviorofataggedparticleinasystemofinteractingparticles.Thedynamicsof vii viii Preface theinteractingparticlesmayconservethedensityandeventuallysomeotherquan- tities. The problem of the tagged particle in exclusion processes (studied in detail in Part II of this book) motivated Kipnis and Varadhan to develop a general central limittheoremthatexploitsthetimesymmetriesoftheprocess.Theideainthearticle (Kipnisan(cid:2)dVaradhan,1986)canbesummarizedasfollows:theasymptoticvariance fort−1/2 tV(X )ds isgivenby 0 s (cid:3) ∞ (cid:4) (cid:5) σ2(V)=2 E V(X )V(X ) dt, π t 0 0 whereE istheexpectationwithrespecttothepathmeasureoftheprocessstarting π fromthestationarymeasureπ.Ifthemeasureistime-reversible,i.e.thegeneratorL isself-adjointinL2(π),thenE [V(X )V(X )]≥0forall t≥0andσ2(V)willbe π t 0 finite if this time correlation decays sufficiently fast. The condition σ2(V)<+∞ is much weaker than asking that V belongs to the range of L, and it translates to anintegrabilityconditiononthecorrespondingspectralmeasure(supportedonthe realline)aroundzerothatisrelativelyeasytoverify.Asimpleargumentusingthis spectral measure shows that the martingale approximation can be done using the resolventsolutionu =(λ−L)−1V,evenwhenV isoutsidetherangeofL.Inthe λ introductoryChap.1wewillexplainthisapproachindetailforasimpleexampleof areversible(discretetime)Markovchain. Some of the ideas discussed above were already present in the literature deal- ingwiththeproblemofhomogenizationofdiffusionsinstationaryergodicrandom environments (cf. Kozlov 1979; Papanicolaou and Varadhan 1981), but it is in the aforementioned work of Kipnis and Varadhan (1986) that the connection with re- versibility has been exploited fully and the finiteness of the variance σ2(V) has beenformulatedasasufficientconditionforthecentrallimittheoremsforreversible Markovprocesses. Lateronthetheoryhasbeenextendedtocertainclassesofnon-reversibleMarkov processes,i.e.Markovprocesseswithastationaryergodic(butnon-reversible)prob- ability measure π. When this measure is explicitly known, the generator can be decomposed as the sum of a symmetric and an anti-symmetric operator in L2(π), L=S+A.IfforasufficientlylargesetC,thatisacommoncoreofLandS,there existsafiniteconstantK suchthat (cid:6)(cid:3) (cid:7) (cid:3) (cid:3) 2 fLgdπ ≤K f(−S)f dπ g(−S)gdπ, f,g∈C then we say that L satisfies a sector condition. The name illustrates the fact that thespectrumofL,ingeneralasubsetofthecomplexplane,iscontainednowina conearoundthenegativerealstouchingtheimaginaryaxisonlyat0,wherethecon- stantK isthetangentofthecorrespondingsemiangleofthecone.Therearemany interesting examples of Markov processes that satisfy the sector condition: asym- metricexclusionprocesseswithnullaveragejumprate(studiedinSect.5.3),cyclic randomwalksinrandomenvironment(Sect.3.3),doublystochasticrandomwalks Preface ix inonedimension(Sect.3.6),diffusionswiththerandomgeneratorinadivergence form(Chap.9). Another class of processes where the theory can be extended is provided by Markov processes with normal generators L and their bounded perturbations (Sect. 2.7.5). This condition allows to deal with examples like random walks, or diffusionsintimedependentrandomenvironment(Sect.9.9). AfurtherextensionconcernsprocesseswherethegeneratorLsatisfiesagraded sector condition: this is the case where the spac(cid:8)e L2(π) can be decomposed into adirectsumof orthogonalsubspaces, L2(π)= A , and L satisfies asector n≥0 n condition on each subspace with a constant K eventually growing to infinity not n too fast (see Sect. 2.7.4 for the precise formulation of this condition). Examples includeasymmetricexclusionprocessesindimensiond≥3(Sect.5.5,thisexample motivatedtheextension),anddiffusionswithGaussiandrifts(Chap.12). Wealsopresentsomeexamplesthatgobeyondsectortypeconditions: • diffusions in divergence free fields with the stream matrix that is square inte- grable (i.e. finite Péclet number), can be dealt with by an approximation proce- dure(Chap.11); • doublystochasticrandomwalksinspace-mixingenvironmentindimension3or higher,whereanapproximationprocedurecanbeimplemented(Sect.3.5); • Ornstein–Uhlenbeck process in a random potential (position-velocity Langevin diffusion).Thisisaverydegeneratediffusion(noiseactingonlyonthevelocity), but time symmetry of the Gibbs measure can be exploited by changing sign of thevelocityinthereversedprocess(seeChap.13). Alltheseexamplesofthecentrallimittheoremforprocesseswiththegenerator notsatisfyinganysectorconditionexploitparticularfeaturesofthedynamics.What ismissingisageneraltheoremfornon-reversibleMarkovprocesses.Weexpected the validity of the central limit theorem for any Markov process with stationary, ergodic measure π and a function V such that (−S)−1/2V belongs to L2(π). We arenotawareofanycounterexamplestothisstatement. Description of the Content of This Book Part I concerns the general theory, and could be used as the base for a graduate course. Only some basic probability is required, in particular ergodic theorems and martingales. In the first chapter we exposethecentrallimittheoremforacountablestatespace,discretetime,reversible Markovchains.Thisisthemostelementary,non-trivialset-upwherewecanillus- trate the basic ideas without spending much time on technicalities. In Chap. 2 we developthetheoryforgeneralcontinuoustimeMarkovprocesses,andintroducethe varioussectorconditions.InChap.3weapplythetheorytorandomwalksinran- dom environment on Zd, that are nice and relatively simple, although at the same timequitenon-trivialexamplesofthegeneraltheory.Weconcludethispartofthe bookwithChap.4thatcontainsestimatesandvariationalformulasfortheasymp- totic variance σ2(V), which turn out to be quite useful in applications. In fact in some examples the central limit theorem follows by the applicationof the general theory,whileprovingstrictpositivityofthevariancerequiressomeextraeffort. x Preface PartIIiscompletelydedicatedtocentrallimittheoremsforexclusionprocesses, theproblemwhichmotivatedthedevelopmentofthetheorypresentedinthisbook. InChap.5,weintroducethesimpleexclusionprocess,provethemainproperties ofitsgenerator,andexaminecentrallimittheoremsforadditivefunctionals.Inthe case where the jump rates of the particles are symmetric, this result is a straight- forwardconsequenceofthefirstpartofthebooksincethegeneratorisself-adjoint. Ifthejumprateshavemeanzero,thegeneratoroftheexclusionprocesssatisfiesa sectorconditionandwemaystillapplythegeneraltheory. In the asymmetric case, the picture is different. The duality introduces an or- thogonaldecompositionoftheL2 spacewhichpermitstoconsiderthecentrallimit theoremfromtheperspectiveofthegradedsectorcondition.However,oneneedsto assumethatthedimensionislargerthanorequaltothreetoprovethattheasymmet- ricpartofthegeneratorwhichchangesthedegreeofafunctionsatisfieshypothesis (2.45)assumedinthegeneraltheory.Moreover,theasymmetricpartofthegener- atorwhichkeepsthedegreesofthefunctionsdoesnotsatisfycondition(2.50).To overcomethisobstacleamethod,knownastheremovalofthehardcoreinteraction, hasbeendevelopedandispresentedinTheorem5.19. Toproceedstepbystep,increasingprogressivelythelevelofdifficulty,wefirst present a central limit theorem for additive functionals of asymmetric exclusion processesindimensiond≥3whenthedensityofparticlesisequalto1/2,inwhich case the asymmetricpart of thegenerator whichkeeps the degreeof the functions does not appear. In the following section we examine the full asymmetric case in d≥3.InthelastsectionofthischapterwepresentsomeresultsontransientMarkov chainsneededinthechapterandwhichhaveintrinsicinterest. Inthefollowingtwochapters,weextendthecentrallimittheoremtothecaseofa taggedparticle,derivingtheself-diffusioncoefficientofexclusionprocesses,andto thecaseofasecondclassparticlewhichisconnectedtotheequilibriumfluctuations fromthehydrodynamiclimitoftheempiricalmeasure,KipnisandLandim(1999). In thelast chapterof thissecondpartof thebookweprovethattheasymptotic variance depends smoothly on the density of particles. In particular, we show that theself-diffusionandthebulkdiffusioncoefficientsaresmoothfunctions,aproperty whichhasimportantconsequencesinthetheoryofhydrodynamiclimits. PartIIIdealsentirelywithdiffusionsinrandomenvironments.Chapter9contains themainapplicationsofthegeneraltheory.Itstartswiththeperiodicenvironments, where a spectral gap is present in the dynamics. In the quasi-periodic case we il- lustrate the loss of compactness and of the spectral gap property. This motivates thestudyofgeneralergodicrandomenvironments.Chapter10containsvariational principlesforthehomogenizeddiffusionmatrix,whilethefollowingchapterscon- tainsomeotherapplicationsandextensions:divergencefreedrifts(Chap.11,where the homogenized diffusion is enhanced by the microscopic convection), Gaussian drifts(seeChap.12),wherewehaveincludedalsothediscussiononsuperdiffusion effectduetolargeconvection.Chapter13dealswiththeOrnstein–Uhlenbeckpro- cess in a random potential. The final Chap. 14 is dedicated to the relation of this probabilistic approach with the classical analytic homogenization theory and the notionsofG-convergenceofoperatorsandΓ-convergenceofquadraticforms.
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