Table Of ContentGrundlehren der
mathematischen Wissenschaften 348
ASeriesofComprehensiveStudiesinMathematics
Serieseditors
M.Berger P.delaHarpe N.J. Hitchin
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
Dominique Bakry (cid:2) Ivan Gentil (cid:2) Michel Ledoux
Analysis and
Geometry of
Markov Diffusion
Operators
DominiqueBakry MichelLedoux
InstitutdeMathématiquesdeToulouse InstitutdeMathématiquesdeToulouse
UniversitédeToulouse UniversitédeToulouse
andInstitutUniversitairedeFrance andInstitutUniversitairedeFrance
Toulouse,France Toulouse,France
IvanGentil
InstitutCamilleJordan
UniversitéClaudeBernardLyon1
Villeurbanne,France
ISSN0072-7830 ISSN2196-9701(electronic)
GrundlehrendermathematischenWissenschaften
ISBN978-3-319-00226-2 ISBN978-3-319-00227-9(eBook)
DOI10.1007/978-3-319-00227-9
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013952461
MathematicsSubjectClassification: 39B62,39B72,47D07,53C21
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HowfarcanyougowiththeCauchy-Schwarz
inequalityandintegrationbyparts?
To Leonard Gross
Preface
SemigroupsofoperatorsonaBanachspaceprovideverygeneralmodelsandtools
in the analysis of time evolution phenomena and dynamical systems. They have a
long history in mathematics and have been studied in a number of settings, from
functionalanalysisandmathematicalphysicstoprobabilitytheory,Riemanniange-
ometry,Liegroups,analysisofalgorithms,andelsewhere.
The part of semigroup theory investigated in this book deals with Markov dif-
fusionsemigroupsandtheirinfinitesimalgenerators,whichnaturallyariseassolu-
tionsofstochasticdifferentialequationsandpartialdifferentialequations.Assuch,
the topic covers a large body of mathematics ranging from probability theory and
partialdifferentialequationstofunctionalanalysisanddifferentialgeometryforop-
eratorsorprocessesonmanifolds.Withintheseframeworks,researchinterestshave
grown over the years, now encompassing a wide variety of questions such as reg-
ularity and smoothing properties of differential operators, Sobolev-type estimates,
heatkernelbounds,non-explosionproperties,convergencetoequilibrium,existence
andregularityofsolutionsofstochasticdifferentialequations,martingaleproblems,
stochasticcalculusofvariationsandsoon.
Thisbookismorepreciselyfocusedontheconcreteinterplaybetweentheana-
lytic,probabilisticandgeometricaspectsofMarkovdiffusionsemigroupsandgen-
eratorsinvolvedinconvergencetoequilibrium,spectralbounds,functionalinequal-
ities and various bounds on solutions of evolution equations linked to geometric
propertiesoftheunderlyingstructure.
Oneprototypicalexampleatthisinterfaceissimplythestandardheatsemigroup
(Pt)t≥0 ontheEuclideanspaceRn whoseGaussiankernel
1
u=u(t,x)=p (x)= e−|x|2/4t, t>0, x∈Rn,
t (4πt)n/2
isafundamentalsolutionoftheheatequation
∂ u=(cid:4)u, u(0,x)=δ ,
t 0
forthestandardLaplaceoperator(cid:4),thuscharacterizedastheinfinitesimalgenerator
ofthesemigroup(Pt)t≥0.
vii
viii Preface
From the probabilistic viewpoint, the family of kernels p (x), t >0, x ∈Rn,
t
representsthetransitionprobabilitiesofastandardBrownianmotion(Bt)t≥0 as
(cid:4)
(cid:2) (cid:3)
E f(x+B ) = f(y)p (x−y)dy=P f(x), t>0, x∈Rn,
2t t t
Rn
forallboundedmeasurablefunctionsf :Rn→R.
Thethirdaspectinvestigatedinthisworkisgeometric,andperhapslessimme-
diately apparent than the analytic and probabilistic aspects. It aims to interpret, in
somesense,thecommutationofderivationandactionofthesemigroupasacurva-
turecondition.ForthestandardEuclideansemigroupexampleabove,thecommu-
tation∇P f =P (∇f)willexpressazerocurvature,althoughthiscorrespondsnot
t t
onlytothecurvatureofEuclideanspaceasaRiemannianmanifoldbutrathertothe
curvatureofEuclideanspaceequippedwiththeLebesguemeasure,invariantunder
theheatflow(Pt)t≥0,andthebilinearoperator(cid:6)(f,g)=∇f ·∇g.
Inordertocarryouttheinvestigationalongtheselines,theexpositionemphasizes
thebasicstructureofaMarkovTriple1 (E,μ,(cid:6))consistingofa(measurable)state
spaceE,acarréduchampoperator(cid:6)andameasureμinvariantunderthedynamics
inducedby(cid:6).Thenotionofacarréduchampoperator(cid:6)associatedwithaMarkov
semigroup(Pt)t≥0 withinfinitesimalgeneratorLgiven(onasuitablealgebraAof
functionsonE)by
(cid:5) (cid:6)
1
(cid:6)(f,g)= L(fg)−fLg−gLf ,
2
willbeacentraltoolofinvestigation,theassociated (cid:6)-calculusproviding,atleast
at a formal level, a kind of algebraic framework encircling the relevant properties
andresults.
Theseanalytic,stochasticandgeometricfeaturesformthebasisoftheinvestiga-
tionundertakeninthisbook,describingMarkovsemigroupsviatheirinfinitesimal
generatorsassolutionsofsecondorderdifferentialoperatorsandtheirprobabilistic
representationsasMarkovprocesses,andanalyzingthemwithrespecttocurvature
properties. The investigation is limited to symmetric (reversible in the Markovian
terminology) semigroups, although various ideas and techniques go beyond this
framework. We also restrict our attention to the diffusion setting, that is when the
carréduchampoperatorisaderivationoperatorinitstwoarguments,eveninthose
caseswheretheresultcouldbeextendedtoamoregeneralsetting.Theserestrictions
ruleoutmanyinterestingfieldsofapplications(discreteMarkovchains,modelsof
statisticalmechanics,mostoftheanalysisofalgorithmsofinterestinoptimization
theory or approximations of partial differential equations, for example), but allow
ustoconcentrateoncentralfeaturesintheanalysisofsemigroups,inthesameway
thatordinarydifferentialequationsareingeneraleasiertohandlethandiscretese-
quences.Evenwithinthefieldofsymmetricdiffusionsemigroups,wehavenottried
1Theterminology“Markovtriple”shouldnot,ofcourse,beconfusedwithsolutionsoftheMarkov
Diophantineequationx2+y2+z2=3xyz!
Preface ix
tocoverallthepossibleinterestingcases.Inordertokeepthemonographwithina
reasonablesize,wehavehadtoomit,amongotherthings,thespecificanalysisre-
lated to hypoelliptic diffusions, special features of diffusions on Lie groups, and
manyinterestingdevelopmentsarisingfrominfiniteinteractingparticlesystems.
Inaddition,althoughwehavelargelybeenmotivatedbytheanalysisofthebe-
haviorofdiffusionprocesses(thatis,solutionsoftimehomogeneousstochasticdif-
ferential equations), rather than concentrating on the probabilistic aspects of the
subject,suchasalmostsureconvergenceoffunctionalsofthetrajectoriesoftheun-
derlyingMarkovprocesses,recurrenceortransience,weinsteadchosetotranslate
most of the features of interest into functional analytic properties of the Markov
structure(E,μ,(cid:6))underinvestigation.
Heatkernelbounds,functionalinequalitiesandtheirapplicationstoconvergence
to equilibrium and geometric features of Markov operators are among the main
topicsofinterestdevelopedinthismonograph.Aparticularemphasisisplacedon
familiesofinequalitiesrelating,onaMarkovTriple(E,μ,(cid:6)),functionalsoffunc-
tionsf :E→Rtotheenergyinducedbytheinvariantmeasureμandthecarrédu
champoperator(cid:6),
(cid:4)
E(f,f)= (cid:6)(f,f)dμ.
E
Typicalfunctionalsarethevariance,entropyorLp-normsleadingtothemainfunc-
tionalinequalitiesofinterest,thePoincaréorspectralgapinequality,thelogarithmic
SobolevinequalityandtheSobolevinequality.Aparticulargoalistoestablishsuch
familiesofinequalitiesundersuitablecurvatureconditionswhichmaybedescribed
bythecarréduchampoperator(cid:6)anditsiterated(cid:6) operator.
2
Similar inequalities are investigated at the level of the underlying semigroup
(Pt)t≥0 for the heat kernel measures, comparing Pt(ϕ(f)) (for some ϕ :R→R)
to P ((cid:6)(f,f)) or (cid:6)(P f,P f), which give rise to heat kernel bounds. With this
t t t
taskinmind,wewilldevelopthemainpowerfultoolofheatflowmonotonicity,or
semigroupinterpolation,withnumerousillustrativeapplicationsandstrongintuitive
content.Toillustratetheprinciple,asawinktowardswhatistocome,letusbriefly
presenthereaheatflowproofoftheclassicalHölderinequalitywhichisverymuch
in the spirit of this book. In particular, the reduction to a quadratic bound is typi-
caloftheargumentsdevelopedinthiswork.Let f,g besuitable(strictly)positive
functionsonRn andθ ∈(0,1).Forfixedt>0,consider,atanypoint(omitted),the
interpolation
(cid:2) (cid:3)
(cid:9)(s)=P eθlogPt−sf+(1−θ)logPt−sg , s∈[0,t],
s
where (Pt)t≥0 is the standard heat semigroup on Rn as recalled above. Together
withtheheatequation∂ P =(cid:4)P =P (cid:4),thederivativeins of(cid:9)isgivenby
s s s s
(cid:2) (cid:2) (cid:3) (cid:5) (cid:2) (cid:3) (cid:2) (cid:3)(cid:6)(cid:3)
(cid:9)(cid:6)(s)=P (cid:4) eH −eH θe−F(cid:4) eF +(1−θ)e−G(cid:4) eG
s
x Preface
where F =logPt−sf, G=logPt−sg and H =θF +(1−θ)G.Nowbystandard
calculus,
(cid:2) (cid:3) (cid:5) (cid:2) (cid:3) (cid:2) (cid:3)(cid:6)
e−H(cid:4) eH − θe−F(cid:4) eF +(1−θ)e−G(cid:4) eG
=|∇H|2−θ|∇F|2−(1−θ)|∇G|2
which is negative by convexity of the square function. Hence (cid:9)(s), s ∈[0,t], is
decreasing,andthus
(cid:2) (cid:3)
(cid:9)(t)=P fθg1−θ ≤(P f)θ(P g)1−θ =(cid:9)(0).
t t t
Normalizingbytn/2 andlettingt tendtoinfinityyieldsHölder’sinequalityforthe
Lebesgue measure. Actually, the same argument may be performed at the level of
a Markov semigroup with invariant finite discrete measure, thus yielding Hölder’s
inequalityforarbitrarymeasures.
Whilefunctionalinequalitiesandtheirrelatedapplicationsareanimportantfocal
point, they also give us the opportunity to discuss a number of issues related to
examples and properties of Markov semigroups and operators. One objective of
thisworkisthusalsotopresentthebasictoolsandideasrevolvingaroundMarkov
semigroupsandtoillustratetheirusefulnessindifferentcontexts.
Themonographcomprisesthreemainparts.
The first part, covering Chaps. 1 to 3, presents some of the main features,
properties and examples of Markov diffusion semigroups and operators as con-
sidered in this work. In a somewhat informal but intuitive way, Chap. 1 intro-
duces Markov semigroups, their infinitesimal generators and associated Markov
processes, stochastic differential equations and diffusion semigroups. It also de-
scribes a few of the standard operations and techniques while working with semi-
groups. Chapter 2 develops in detail a number of central geometric models which
will serve as references for later developments, namely the heat semigroups and
LaplaciansontheflatEuclideanspace,thesphereandthehyperbolicspace.Sturm-
Liouvilleoperatorsontheline,andsomeofthemostrelevantexamples(Ornstein-
Uhlenbeck,LaguerreandJacobi),arealsopresentedtherein.Onthebasisofthese
preliminary observations and examples, Chap. 3 then tries to describe a general
framework of investigation. While it would not be appropriate to try to cover in a
uniqueformalmouldallthecasesofinterest,itisneverthelessusefultoemphasize
thebasicpropertiesandtoolsinordertoeasilyandsuitablydevelopthe(cid:6)-calculus.
In particular,it is necessaryto describewithsomecare thevarious classes andal-
gebras of functions that we shall be dealing with and to show their relevance in
theclassicalsmoothsettings.Notethatwhileinfinite-dimensionalmodelswouldre-
quirefurthercareinthisabstractformalism,themethodsandprinciplesemphasized
throughoutthisworkaresimilarlyrelevantforthem.Takingthemoreclassicalpic-
tureasgranted,Chap.3maybeskippedatfirstreading(orlimitedtothesummary
Sect.3.4).
PartII,formingthecoreofthetext,includesChaps.4to6andcoversthethree
mainfunctionalinequalitiesofinterest,Poincaréorspectralgapinequalities,loga-
rithmicSobolevinequalitiesandSobolevinequalities.Foreachfamily,somebasic
