Brownian Motion: 30 (Cambridge Series in Statistical and Probabilistic Mathematics, Series Number 30)
Cambridge University Press
Edition: Illustrated, 3/25/2010
EAN 9780521760188, ISBN10: 0521760186
Hardcover, 416 pages, 25.4 x 17.8 x 2.3 cm
Language: English
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.
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)
'Brownian Motion by Mörters 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