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Conformally Invariant Processes in the Plane Mathematical I Surveys I and I Ponographs I http://dx.doi.org/10.1090/surv/114 Conformally Invariant Processes in the Plane Mathematical I Surveys I and I Ponographs I Volume 114 1 Conformally Invariant Processes in the Plane I Gregory F. Lawler EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber, Chair Michael G. Eastwood Michael P. Loss J. T. Stafford 2000 Mathematics Subject Classification. Primary 30C35, 31A15, 60H30, 60J65, 81T40, 82B27. For additional information and updates on this book, visit www.ams.org/bookpages/surv-114 Library of Congress Cataloging-in-Publication Data Lawler, Gregory F., (Gregory Francis), 1955- Conformally invariant processes in the plane / Gregory F. Lawler. p. cm. — (Mathematical surveys and monographs ; v. 114) Includes bibliographical references and index. ISBN 0-8218-3677-3 (alk. paper) 1. Conformal mapping. 2. Potential theory (Mathematics) 3. Stochastic analysis. 4. Markov processes. I. Title. II. Mathematical surveys and monographs ; no. 114. QA646.L85 2005 515'.9—dc22 2004062341 AMS softcover ISBN 978-0-8218-4624-7 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]. © 2005 by the American Mathematical Society. All rights reserved. Reprinted by the American Mathematical Society, 2008. 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/ 10 9 8 7 6 5 4 3 2 1 13 12 11 10 09 08 Contents Preface ix Some discrete processes 1 0.1. Simple random walk 1 0.2. Loop-erased random walk 3 0.3. Self-avoiding walk 5 0.4. Infinitely growing self-avoiding walk 7 0.5. Percolation exploration process 7 Chapter 1. Stochastic calculus 11 1.1. Definition 11 1.2. Integration with respect to Brownian motion 12 1.3. Ito's formula 17 1.4. Several Brownian motions 18 1.5. Integration with respect to semimartingales 19 1.6. Ito's formula for semimartingales 20 1.7. Time changes of martingales 22 1.8. Examples 22 1.9. Girsanov's transformation 23 1.10. Bessel processes 25 1.11. Diffusions on an interval 30 1.12. A Feynman-Kac formula 39 1.13. Modulus of continuity 39 Chapter 2. Complex Brownian motion 43 2.1. Review of complex analysis 43 2.2. Conformal invariance of Brownian motion 45 2.3. Harmonic functions 46 2.4. Green's function 52 Chapter 3. Conformal mappings 57 3.1. Simply connected domains 57 3.2. Univalent functions 60 3.3. Capacity 66 3.4. Half-plane capacity 69 3.5. Transformations on D 76 3.6. Caratheodory convergence 78 3.7. Extremal distance 80 3.8. Beurling estimate and applications 84 3.9. Conformal annuli 88 vi CONTENTS Chapter 4. Loewner differential equation 91 4.1. Chordal Loewner equation 91 4.2. Radial Loewner equation 97 4.3. Whole-plane Loewner equation 100 4.4. Chains generated by curves 104 4.5. Distance to the curve 108 4.6. Perturbation by conformal maps 109 4.7. Convergence of Loewner chains 114 Chapter 5. Brownian measures on paths 119 5.1. Measures on spaces of curves 119 5.2. Brownian measures on /C 123 5.3. H-excursions 130 5.4. One-dimensional excursion measure 135 5.5. Boundary bubbles 137 5.6. Loop measure 141 5.7. Brownian loop soup 144 Chapter 6. Schramm-Loewner evolution 147 6.1. Chordal SLE 147 6.2. Phases 150 6.3. The locality property for K — 6 152 6.4. The restriction property for K = 8/3 153 6.5. Radial SLE 156 6.6. Whole-plane SLEK 162 6.7. Cardy's formula 163 6.8. SLEQ in an equilateral triangle 167 6.9. Derivative estimates 169 6.10. Crossing exponent for SLEQ 171 6.11. Derivative estimates, radial case 174 Chapter 7. More results about SLE 111 7.1. Introduction 177 7.2. The existence of the path 181 7.3. Holder continuity 182 7.4. Dimension of the path 183 Chapter 8. Brownian intersection exponent 187 8.1. Dimension of exceptional sets 187 8.2. Subadditivity 190 8.3. Half-plane or rectangle exponent 191 8.4. Whole-plane or annulus exponent 200 Chapter 9. Restriction measures 205 9.1. Unbounded hulls in M 205 9.2. Right-restriction measures 209 9.3. The boundary of restriction hulls 211 9.4. Constructing restriction measures 213 Appendix A. Hausdorff dimension 217 CONTENTS vii A.l. Definition 217 A.2. Dimension of Brownian paths 220 A.3. Dimension of random "Cantor sets" in [0,1] 221 Appendix B. Hypergeometric functions 229 B.l. The case a = 2/3, /? = 1/3,7 = 4/3 230 B.2. Confluent hypergeometric functions 230 B.3. Another equation 231 Appendix C. Reflecting Brownian motion 233 Appendix. Bibliography 237 Index 240 Index of symbols 242 Preface A number of two-dimensional lattice models in statistical physics have contin­ uum limits that are conformally invariant. For example, the limit of simple random walk is Brownian motion. This book will discuss the nature of conformally invari­ ant limits. Most of the processes discussed in this book are derived in one way or another from Brownian motion. The exciting new development in this area is the Schramm-Loewner evolution (SLE), which can be considered as a Brownian motion on the space of conformal maps. These notes arise from graduate courses at Cornell University given in 2002- 2003 on the mathematics behind conformally invariant processes. This may be considered equal doses of probability and conformal mapping. It is assumed that the reader knows the equivalent of first-year graduate courses in real analysis, complex analysis, and probability. Here is an outline of the book. We start with a quick introduction to some discrete processes which have scaling limits that are conformally invariant. We only present enough here to whet the appetite of the reader, and we will not use this section later in the book. We will not prove any of the important results concerning convergence of discrete processes. A good survey of some of these results is [83]. Chapter 1 gives the necessary facts about one-dimensional Brownian motion and stochastic calculus. We have given an essentially self-contained treatment; in order to do so, we only integrate with respect to continuous semimartingales derived from Brownian motion and we only integrate adapted processes that are continuous (or piecewise continuous). The latter assumption is more restrictive than one generally wants for other applications of stochastic calculus, but it suffices for our needs and avoids having to discuss certain technical aspects of stochastic calculus. More detailed treatments can be found in many books, e.g., [5, 32, 72, 73]. Sections 1.10 and 1.11 discuss some particular stochastic differential equations that arise in the analysis of SLE. The reader may wish to skip these sections until Chapter 6 where these equations appear; however, since they discuss properties of one-dimensional equations it logically makes sense to include them in the first chapter. The next chapter introduces the basics of two-dimensional (i.e., one complex dimension) Brownian motion. It starts with the basic fact (dating back to Levy [65] and implicit in earlier work on harmonic functions) that complex Brownian motion is conformally invariant. Here we collect a number of standard facts about harmonic functions and Green's function for complex Brownian motion. Because much of this material is standard, a number of facts are labeled as exercises. Conformal mapping is the topic of Chapter 3. The purpose is to present the material about conformal mapping that is needed for SLE, especially material that would not appear in a first course in complex variables. References for much of this ix X PREFACE chapter are [2, 23, 30, 71]. However, our treatment here differs in some ways, most significantly in that it freely uses Brownian motion. We start with simple connectedness and a proof of the Riemann mapping theorem. Although this is really a first-year topic, it is so important to our discussion that it is included here. The next section on univalent functions follows closely the treatment in [30]; since the Riemann mapping theorem gives a correspondence between simply connected domains and univalent functions on the disk it is natural to study the latter. We then discuss two kinds of capacity, logarithmic capacity in the plane which is classical and a "half-plane" capacity that is not as well known but similar in spirit. Important uniform estimates about certain conformal transformations are collected here; these are the basis for Loewner differential equations. Extremal distance (extremal length) is an important conformally invariant quantity and is discussed in Section 3.7. The next section discusses the Beurling estimate, which is a corollary of a stronger result, the Beurling projection theorem. This is used to derive a number of estimates about conformal maps of simply connected domains near the boundary; what makes this work is the fact that the boundary of a simply connected domain is connected.
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