Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation

494 Pages·2012·2.57 MB·English

by  Tomasz Komorowski

Grundlehren der
mathematischen Wissenschaften 345
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Serieseditors
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N.J. Hitchin L.Hörmander A.Kupiainen
G.Lebeau F.-H. Lin S.Mori
B.C.Ngô M.Ratner D.Serre
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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
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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.