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Random Walk Intersections Large Deviations and Related Topics Mathematical Surveys and Monographs Volume 157 Random Walk Intersections Large Deviations and Related Topics Xia Chen American Mathematical Society http://dx.doi.org/10.1090/surv/157 Random Walk Intersections Large Deviations and Related Topics Mathematical Surveys and Monographs Volume 157 Random Walk Intersections Large Deviations and Related Topics Xia Chen American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Jerry L. Bona Michael G. Eastwood Ralph L. Cohen, Chair J. T. Stafford Benjamin Sudakov 2000 Mathematics Subject Classification. Primary 60F05, 60F10, 60F15, 60F25, 60G17, 60G50, 60J65, 81T17, 82B41, 82C41. This work was supported in part by NSF Grant DMS-0704024 For additional information and updates on this book, visit www.ams.org/bookpages/surv-157 Library of Congress Cataloging-in-Publication Data Chen, Xia, 1956– Random walk intersections : large deviations and related topics / Xia Chen. p. cm.— (Mathematical surveys and monographs ; v. 157) Includes bibliographical references and index. ISBN 978-0-8218-4820-3 (alk. paper) 1. Random walks (Mathematics) 2. Large deviations. I. Title. QA274.73.C44 2009 519.282–dc22 2009026903 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2010 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 151413121110 To the memory of my great grandmother Ding, Louyi Contents Preface ix Chapter 1. Basics on large deviations 1 1.1. G¨artner-Ellis theorem 1 1.2. LDP for non-negative random variables 8 1.3. LDP by sub-additivity 19 1.4. Notes and comments 22 Chapter 2. Brownian intersection local times 25 2.1. Introduction 25 2.2. Mutual intersection local time 27 2.3. Self-intersection local time 42 2.4. Renormalization 48 2.5. Notes and comments 53 Chapter 3. Mutual intersection: large deviations 59 3.1. High moment asymptotics 59 ×···× 3.2. High moment of α([0,τ 1] [0,τp]) 67 3.3. Large deviation for α [0, 1]p 77 3.4. Notes and comments 84 Chapter 4. Self-intersection: large deviations 91 4.1. Feynman-Kac formula 91 4.2. One-dimensional case 102 4.3. Two-dimensional case 111 4.4. Applications to LIL 121 4.5. Notes and comments 126 Chapter 5. Intersections on lattices: weak convergence 133 5.1. Preliminary on random walks 133 5.2. Intersection in 1-dimension 139 5.3. Mutual intersection in sub-critical dimensions 145 5.4. Self-intersection in dimension two 160 5.5. Intersection in high dimensions 164 5.6. Notes and comments 171 Chapter 6. Inequalities and integrabilities 177 6.1. Multinomial inequalities 177 6.2. Integrability of In and Jn 187 6.3. Integrability of Qn and Rn in low dimensions 191 6.4. Integrability of Qn and Rn in high dimensions 198 vii viii CONTENTS 6.5. Notes and comments 204 Chapter 7. Independent random walks: large deviations 207 7.1. Feynman-Kac minorations 207 7.2. Moderate deviations in sub-critical dimensions 222 7.3. Laws of the iterated logarithm 226 7.4. What do we expect in critical dimensions? 230 7.5. Large deviations in super-critical dimensions 231 7.6. Notes and comments 247 Chapter 8. Single random walk: large deviations 253 8.1. Self-intersection in one dimension 253 8.2. Self-intersection in d = 2 257 8.3. LDP of Gaussian tail in d = 3 264 8.4. LDP of non-Gaussian tail in d = 3 270 8.5. LDP for renormalized range in d =2, 3 278 8.6. Laws of the iterated logarithm 287 8.7. What do we expect in d ≥ 4? 289 8.8. Notes and comments 291 Appendix 297 A. Green’s function 297 B. Fourier transformation 299 C. Constant κ(d, p) and related variations 303 D. Regularity of stochastic processes 309 E. Self-adjoint operators 313 Bibliography 321 List of General Notations 329 Index 331 Preface This book aims to provide a systematic account for some recent progress on the large deviations arising from the area of sample path intersections, including calculation of the tail probabilities of the intersection local times, the ranges and the intersections of the ranges of random walks and Brownian motions. The phrase “related topics” appearing in the title of the book mainly refers to the weak law and the law of the iterated logarithm for these models. The former is the reason for certain forms of large deviations known as moderate deviations; while the latter appears as an application of the moderate deviations. Quantities measuring the amount of self-intersection of a random walk, or of mutual intersection of several independent random walks have been studied inten- sively for more than twenty years; see e.g. [57], [59], [124], [125], [116], [22], [131], [86], [135][136], [17], [90], [11], [10], [114]. This research is often moti- vated by the role that these quantities play in renormalization group methods for quantum field theory (see e.g. [78], [51], [52], [64]); our understanding of polymer models (see e.g. [134], [19],[96], [98][162], [165], [166],[167],[63], [106], [21], [94], [93]); or the analysis of stochastic processes in random environments (see e.g. [107], [111],[43], [44][82], [95], [4], [42][79], [83]). Sample path intersection is also an important subject within the probability field. It has been known ([48], [138], [50]) that sample path intersections have a deep link to the problems of cover times and thick points through tree-encoding techniques. In addition, it is impossible to write a book on sample path intersec- tion without mentioning the influential work led by Lawler, Schramm and Werner ([118], [119], [120], [117]) on the famous intersection exponent problem and on other Brownian sample path properties in connection to the Stochastic Loewner Evolution, which counts as one of the most exciting developments made in the fields of probability in recent years. Contrary to the behavior patterns investigated by Lawler, Schramm and Werner, where the sample paths avoid each other and are loop-free, most of this book is concerned with the probability that the random walks and Brownian motions in- tersect each other or themselves with extreme intensity. When these probabilities decay at exponential rates, the problem falls into the category of large deviations. In recent years, there has been some substantial input about the new tools and new ideas for this subject. The list includes the method of high moment asymp- totics, sub-additivity created by moment inequality, and the probability in Banach space combined with the Feynman-Kac formula. Correspondent to the progress in methodology, established theorems have been accumulated into a rather complete ix xPREFACE picture of this field. These developments make it desirable to write a monograph on this subject which has not been adequately exposed in a systematic way. This book was developed from the lecture notes of a year-long graduate course at the University of Tennessee. Making it accessible to non-experts with only basic knowledge of stochastic processes and functional analysis has been one of my guidelines in writing it. To make it reasonably self-contained, I added Chapter 1 for the general theory of large deviations. Most of the theorems listed in this chapter are not always easy to find in literature. In addition, a few exercises are included in the “Notes and comments” section in each chapter, an effort to promote active reading. Some of them appear as extensions of, or alternative solutions to the main theorems addressed in the chapter. Others are not very closely related to the main results on the topic, such as the exercises concerning small ball probabilities, but are linked to our context by sharing similar ideas and treatments. The challenging exercises are marked with the word “hard”. The mainspring of the book does not logically depend on the results claimed in the exercises. Consequently, skipping any exercise does not compromise understanding the book. The topics and results included in the book do reflect my taste and my involve- ment on the subject. The “Notes and comments” section at the end of each chapter is part of the effort to counterbalance the resulted partiality. Some relevant works not included in the other sections may appear here. In spite of that, I would like to apologize in advance for any possible inaccuracy of historic perspective appearing in the book. In the process of investigating the subject and writing the book, I benefitted from the help of several people. It is my great pleasure to acknowledge the contributions, which appear throughout the whole book, made by my collaborators R. Bass, W. Li,P.M¨orters and J. Rosen in the course of several year’s collaboration. I would like to express my special thanks to D. Khoshnevisan, from whom I learned for the first time the story about intersection local times.
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