Analysis for diffusion processes on Riemannian manifolds PDF
Preview Analysis for diffusion processes on Riemannian manifolds
ANALYSIS FOR DIFFUSION PROCESSES ON RIEMANNIAN MANIFOLDS 8737hc_9789814452649_tp.indd 1 14/8/13 2:38 PM ADVANCED SERIES ON STATISTICAL SCIENCE & APPLIED PROBABILITY Editor: Ole E. Barndorff-Nielsen Published Vol. 6 Elementary Stochastic Calculus — With Finance in View by T. Mikosch Vol. 7 Stochastic Methods in Hydrology: Rain, Landforms and Floods eds. O. E. Barndorff-Nielsen et al. Vol. 8 Statistical Experiments and Decisions: Asymptotic Theory by A. N. Shiryaev and V. G. Spokoiny Vol. 9 Non-Gaussian Merton–Black–Scholes Theory by S. I. Boyarchenko and S. Z. Levendorskiĭ Vol. 10 Limit Theorems for Associated Random Fields and Related Systems by A. Bulinski and A. Shashkin Vol. 11 Stochastic Modeling of Electricity and Related Markets . by F. E. Benth, J. Šaltyte Benth and S. Koekebakker Vol. 12 An Elementary Introduction to Stochastic Interest Rate Modeling by N. Privault Vol. 13 Change of Time and Change of Measure by O. E. Barndorff-Nielsen and A. 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Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Advanced Series on Statistical Science and Applied Probability — Vol. 18 ANALYSIS FOR DIFFUSION PROCESSES ON RIEMANNIAN MANIFOLDS Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-4452-64-9 Printed in Singapore August1,2013 18:21 WorldScientificBook-9inx6in ws-book9x6 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 August1,2013 18:21 WorldScientificBook-9inx6in ws-book9x6 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 TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk 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
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