Brownian Motion

by Peter Mörters, Yuval Peres

Overview

This eagerly awaited textbook covers everything the graduate student in probability wants to know about Brownian motion, as well as the latest research in the area. Starting with the construction of Brownian motion, the book then proceeds to sample path properties like continuity and nowhere differentiability. Notions of fractal dimension are introduced early and are used throughout the book to describe fine properties of Brownian paths. The relation of Brownian motion and random walk is explored from several viewpoints, including a development of the theory of Brownian local times from random walk embeddings. Stochastic integration is introduced as a tool and an accessible treatment of the potential theory of Brownian motion clears the path for an extensive treatment of intersections of Brownian paths. An investigation of exceptional points on the Brownian path and an appendix on SLE processes, by Oded Schramm and Wendelin Werner, lead directly to recent research themes.

Product Details

ISBN-13:	9780521760188
Publisher:	Cambridge University Press
Publication date:	03/25/2010
Series:	Cambridge Series in Statistical and Probabilistic Mathematics , #30
Edition description:	New Edition
Pages:	416
Product dimensions:	7.00(w) x 10.10(h) x 1.30(d)

About the Author

Peter Mörters is Professor of Probability and ESPRC Advanced Research Fellow at the University of Bath. His research on Brownian motion includes identification of the tail behaviour of intersection local times (with König), the multifractal structure of intersections (with Klenke), and the exact packing gauge of double points of three-dimensional Brownian motion (with Shieh).

Yuval Peres is a Principal Researcher at Microsoft Research in Redmond, Washington. He is also an Adjunct Professor at the University of California, Berkeley and at the University of Washington. His research interests include most areas of probability theory, as well as parts of ergodic theory, game theory, and information theory.

Table of Contents

Preface viii

Frequently used notation x

Motivation 1

1 Brownian motion as a random function 7

1.1 Paul Lévy's construction of Brownian motion 7

1.2 Continuity properties of Brownian motion 14

1.3 Nondifferentiability of Brownian motion 18

1.4 The Cameron-Martin theorem 24

Exercises 30

Notes and comments 33

2 Brownian motion as a strong Markov process 36

2.1 The Markov property and Blumenthal's 0-1 law 36

2.2 The strong Markov property and the reflection principle 40

2.3 Markov processes derived from Brownian motion 48

2.4 The martingale property of Brownian motion 53

Exercises 59

Notes and comments 63

3 Harmonic functions, transience and recurrence 65

3.1 Harmonic functions and the Dirichlet problem 65

3.2 Recurrence and transience of Brownian motion 71

3.3 Occupation measures and Green's functions 76

3.4 The harmonic measure 84

Exercises 91

Notes and comments 94

4 Hausdorff dimension: Techniques and applications 96

4.1 Minkowski and Hausdorff dimension 96

4.2 The mass distribution principle 105

4.3 The energy method 108

4.4 Frostman's lemma and capacity 111

Exercises 115

Notes and comments 116

5 Brownian motion and random walk 118

5.1 The law of the iterated logarithm 118

5.2 Points of increase for random walk and Brownian motion 123

5.3 Skorokhod embedding and Donsker's invariance principle 127

5.4 The arcsine laws for random walk and Brownian motion 135

5.5 Pitman's 2M - B theorem 140

Exercises 146

Notes and comments 149

6 Brownian local time 153

6.1 The local time at zero 153

6.2 A random walk approach to the local time process 165

6.3 The Ray-Knight theorem 170

6.4 Brownian local time as a Hausdorff measure 178

Exercises 186

Notes and comments 187

7 Stochastic integrals and applications 190

7.1 Stochastic integrals with respect to Brownian motion 190

7.2 Conformal invariance and winding numbers 201

7.3 Tanaka's formula and Brownian local time 209

7.4 Feynman-Kac formulas and applications 213

Exercises 220

Notes and comments 222

8 Potential theory of Brownian motion 224

8.1 The Dirichlet problem revisited 224

8.2 The equilibrium measure 227

8.3 Polar sets and capacities 234

8.4 Wiener's test of regularity 248

Exercises 251

Notes and comments 253

9 Intersections and self-intersections of Brownian paths 255

9.1 Intersection of paths: Existence and Hausdorff dimension 255

9.2 Intersection equivalence of Brownian motion and percolation limit sets 263

9.3 Multiple points of Brownian paths 272

9.4 Kaufman's dimension doubling theorem 279

Exercises 285

Notes and comments 287

10 Exceptional sets for Brownian motion 290

10.1 The fast times of Brownian motion 290

10.2 Packing dimension and limsup fractals 298

10.3 Slow times of Brownian motion 307

10.4 Cone points of planar Brownian motion 312

Exercises 322

Notes and comments 324

Appendix A Further developments

11 Stochastic Loewner evolution and planar Brownian motion Oded Schramm Wendelin Werner 327

11.1 Some subsets of planar Brownian paths 327

11.2 Paths of stochastic Loewner evolution 331

11.3 Special properties of SLE(6) 339

11.4 Exponents of stochastic Loewner evolution 340

Notes and comments 344

Appendix B Background and prerequisites 346

12.1 Convergence of distributions 346

12.2 Gaussian random variables 349

12.3 Martingales in discrete time 351

12.4 Trees and flows on trees 358

Hints and solutions for selected exercises 361

Selected open problems 383

Bibliography 386

Index 400

Editorial Reviews

"This splendid account of the modern theory of Brownian motion puts special emphasis on sample path properties and connections with harmonic functions and potential theory, without omitting such important topics as stochastic integration, local times or relations with random walk. The most significant properties of Brownian motion are derived via powerful and elegant methods. This book, which fills a gap in the existing literature, will be of interest both to the beginner, for the clarity of exposition and the judicious choice of topics, and to the specialist, who will find neat approaches to many classical results and to some more recent ones. This beautiful book will soon become a must for anybody who is interested in Brownian motion and its applications."
Jean-François Le Gall, Université Paris 11 (Paris-Sud, Orsay)

"Brownian Motion" by Moerters and Peres, a modern and attractive account of one of the central topics of probability theory, will serve both as an accessible introduction at the level of a Masters course and as a work of reference for fine properties of Brownian paths. The unique focus of the book on Brownian motion gives it a satisfying concreteness and allows a rapid approach to some deep results. The introductory chapters, besides providing a careful account of the theory, offer some helpful points of orientation towards an intuitive and mature grasp of the subject matter. The authors have made many contributions to our understanding of path properties, fractal dimensions and potential theory for Brownian motion, and this expertise is evident in the later chapters of the book. I particularly liked the marking of the ‘leaves' of the theory by stars, not only because this offers a chance to skip on, but also because these are often the high points of our present knowledge."
James Norris, University of Cambridge

"This excellent book does a beautiful job of covering a good deal of the theory of Brownian motion in a very user-friendly fashion. The approach is hands-on which makes it an attractive book for a first course on the subject. It also contains topics not usually covered, such as the "intersection-equivalence" approach to multiple points as well as the study of slow and fast points. Other highlights include detailed connections with random fractals and a short overview of the connections with SLE.
Jeff Steif, Chalmers University of Technology

"I find the style of this book is extremely user-friendly. I am sure that it will be considered a very gentle introduction to stochastic analysis by many graduate students, and I guess that many established researchers will read some of chapters of this book at bedtime, for pure pleasure."
Krzysztof Burdzy for SIAM Review

"This is a well-written book guiding the interested reader from the humble beginnings to the cutting edge of current research in Brownian motion. It excels in its careful selection of topics and very clear presentation and, though quite advanced material is presented, never gives the reader the impression of being fraught with technicalities."
Rene L. Schilling, Mathematical Reviews

"written with enthusiasm for Brownian motion as a beautiful and fascinating object in its own right, yet, still highlighting its central role in so many other contexts."
Allan GUT, Journal of the American Statistical Association

From the Publisher