Analysis for diffusion processes on Riemannian manifolds

392 Pages·2014·2.279 MB·English

by  Wang, Feng-Yu

ANALYSIS FOR DIFFUSION
PROCESSES ON RIEMANNIAN
MANIFOLDS
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Vol. 18 Analysis for Diffusion Processes on Riemannian Manifolds
by Feng-Yu Wang
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Vol. 18
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ANALYSIS FOR DIFFUSION
PROCESSES ON RIEMANNIAN
MANIFOLDS
Feng-Yu Wang
Beijing Normal University, China & Swansea University, UK
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Advanced Series on Statistical Science and Applied Probability — Vol. 18
ANALYSIS FOR DIFFUSION PROCESSES ON RIEMANNIAN MANIFOLDS
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Preface
As a cross research field of probability theory and Riemannian geometry,
stochastic analysis on Riemannian manifolds devotes to providing proba-
bilisticsolutionsofproblemsarisingfromdifferentialgeometryanddevelop-
ingacompletetheoryofdiffusionprocessesonRiemannianmanifolds. Since
1980s, many important contributions have been made in this field, which
include, as two typical examples, probabilistic proofs of the H¨ormander
theoremandtheAtiyah-SingerindextheoremmadebyP.MalliavinandJ.
M.Bismutrespectively. Wewouldalsoliketomentionthreepowerfultools
developed in the literature: Malliavian calculus, Bakry-Emery’s semigroup
argument, and coupling method, which have led to numerous results for
diffusion processes and applications to geometry analysis. For instance, as
included in the present book, about twenty equivalent semigroup inequali-
tieshavebeenfoundforthecurvaturelowerboundconditionbyusingthese
tools, and these semigroup inequalities are crucial in the study of various
different topics in the field.
Based on recent progresses made in the last decade, this book aims to
presentaself-containedtheoryconcerning(reflecting)diffusionprocesseson
Riemannian manifolds with or without boundary, and thus complements
some earlier published books in the literature: [Bismut (1984)], [Emery
(1989)], [Elworthy (1982)], [Hsu (2002a)], [Ikeda and Watanabe (1989)],
[Malliavin (1997)], and [Stroock (2000)]. The author did not intend to
include in the book all recent contributions in the field, materials of the
book are selected systematically but mainly according to his own research
interests.
The book consists of five chapters. The first chapter contains neces-
v
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vi Analysis for Diffusion Processes on Riemannian Manifolds
sary preparations for the study, which include a collection of fundamental
results from Riemannian manifold, coupling method and applications, and
a brief theory of functional inequalities. The second chapter is devoted to
the theory of diffusion processes on Riemannian manifolds without bound-
ary, where various equivalent semigroup properties are presented for the
curvature lower bound of the underlying diffusion operator. These equiv-
alent properties have been applied to the study of functional inequalities,
Harnackinequalitiesandapplications,andtransportation-costinequalities.
The third chapter aims to build up a corresponding theory for the reflect-
ing diffusion processes on Riemannian manifold with boundary, for which
equivalent semigroup properties are presented for both the curvature lower
bound and the lower bound of the second fundamental form of the bound-
ary. As applications, functional/Harnack/transportation-cost inequalities
as well as the Robin semigroup are closely investigated. In Chapter 4 we
investigate the stochastic analysis on the path space of the reflecting diffu-
sion process on a Riemannian manifold with boundary. The main content
includes the quasi-invariant flow induced by stochastic differential equa-
tions with reflection, integration by parts formula for the damped gradient
operator, and the log-Sobolev/transpotation-cost inequalities. Finally, in
Chapter 5, functional inequalities and regularity estimates for sub-elliptic
diffusion processes are studied by using Malliavin calculus as well as argu-
ments introduced in the previous chapters.
Most of the book is organized from the author’s recent publications
concerning diffusion processes on manifolds, including joint papers with
colleagueswhoaregratefullyacknowledgedfortheirfruitfulcollaborations.
IwouldliketothankLijuanCheng,XiliangFan,HuaiqianLee,JianWang,
Shaoqin Zhang and Ms. Lai Fun Kwong for reading earlier drafts of the
book and corrections. A main part of Chapter 3 has been presented for
a mini course in the Chinese Academy of Science. I would like to thank
Xiang-Dong Li for the kind invitation and all audience who attended the
mini course. I would also like to thank my colleagues from the probability
groups of Beijing Normal University and Swansea University, in particular
Mu-Fa Chen, Wenming Hong, Niels Jacob, Zenghu Li, Eugene Lytvynov,
Yonghua Mao, Aubrey Truman, Jiang-Lun Wu, Chenggui Yuan and Yuhui
Zhang. Their kind help and constant encouragement have provided an ex-
cellent working environment for me. Finally, financial support from the
NationalNaturalScienceFoundationofChina, SpecializedResearchFoun-
dation for Doctorial Programs, the Fundamental Research Funds for the
August1,2013 18:21 WorldScientificBook-9inx6in ws-book9x6
Preface vii
CentralUniversities,andtheLaboratoryofMathematicsandComplexSys-
tems, are gratefully acknowledged.
Feng-Yu Wang
May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws
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August1,2013 18:21 WorldScientificBook-9inx6in ws-book9x6
Contents
Preface v
1. Preliminaries 1
1.1 Riemannian manifold. . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Differentiable manifold . . . . . . . . . . . . . . . 1
1.1.2 Riemannian manifold . . . . . . . . . . . . . . . . 3
1.1.3 Some formulae and comparison results . . . . . . 9
1.2 Riemannian manifold with boundary . . . . . . . . . . . . 11
1.3 Coupling and applications . . . . . . . . . . . . . . . . . . 15
1.3.1 Transport problem and Wasserstein distance . . . 16
1.3.2 Optimal coupling and optimal map . . . . . . . . 18
1.3.3 Coupling for stochastic processes. . . . . . . . . . 19
1.3.4 Coupling by change of measure. . . . . . . . . . . 22
1.4 Harnack inequalities and applications . . . . . . . . . . . 24
1.4.1 Harnack inequality . . . . . . . . . . . . . . . . . 24
1.4.2 Shift Harnack inequality . . . . . . . . . . . . . . 31
1.5 Harnack inequality and derivative estimate . . . . . . . . 33
1.5.1 Harnack inequality and entropy-gradient estimate 33
1.5.2 Harnack inequality and L2-gradient estimate . . . 36
1.5.3 Harnackinequalitiesandgradient-gradientestimates 37
1.6 Functional inequalities and applications . . . . . . . . . . 39
1.6.1 Poincar´e type inequality and essential spectrum . 39
1.6.2 Exponential decay in the tail norm . . . . . . . . 42
1.6.3 The F-Sobolev inequality . . . . . . . . . . . . . . 42
1.6.4 Weak Poincar´e inequality . . . . . . . . . . . . . . 43
1.6.5 EquivalenceofirreducibilityandweakPoincar´ein-
equality . . . . . . . . . . . . . . . . . . . . . . . . 45
ix