Brownian Motion

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.

• Preface; Frequently used notation; Motivation; 1. Brownian motion as a random function; 2. Brownian motion as a strong Markov process; 3. Harmonic functions, transience and recurrence; 4. Hausdorff dimension: techniques and applications; 5. Brownian motion and random walk; 6. Brownian local time; 7. Stochastic integrals and applications; 8. Potential theory of Brownian motion; 9. Intersections and self-intersections of Brownian paths; 10. Exceptional sets for Brownian motion; Appendix A. Further developments: 11. Stochastic Loewner evolution and its applications to planar Brownian motion; Appendix B. Background and prerequisites; Hints and solutions for selected exercises; References; Index.

'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)