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.
Everything the graduate student in probability wants to know about Brownian motion, including the latest research in the field.
Peter Moerters 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 Koenig), 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.
'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-Francois Le Gall, Universite 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. I highly recommend it.' Jeff Steif, Chalmers University of Technology
'I find the style of this book 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 chapters of the book at bedtime, for pure pleasure.' Krzysztof Burdzy, University of Washington, Seattle
'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.' Mathematical Reviews
'The book is, in fact, currently used as a reading course for Ph.D. students in Uppsala. A short informal check tells me that they like it. It is thorough and rigorous, yet intuitive, they enjoy the focus on sample path and geometric properties of Brownian motion ... They also appreciate that it is written with enthusiasm for Brownian motion as a beautiful and fascinating object in its own right (and so do I), yet still highlighting its central role in so many other contexts.' Allan Gurr, Uppsala University