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.
- An essential purchase for both pure and applied probabilists
- Material has been class-tested in the USA and Europe
- Features 140 exercises with many solutions and hints also provided
Read more
Reviews & endorsements
'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
'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
See more reviews
Customer reviews

Not yet reviewed
Be the first to review

Log in to review

Review was not posted due to profanity
×

, create a review

(If you're not , sign out)

Please enter the right captcha value

Please enter a star rating.

Your review must be a minimum of 12 words.

How do you rate this item?

Reviews must contain at least 12 words about the product.

×

Product details
- Date Published: April 2010
- format: Hardback
- isbn: 9780521760188
- length: 416 pages
- dimensions: 254 x 178 x 24 mm
- weight: 0.91kg
- contains: 33 b/w illus. 140 exercises
- availability: Available
Table of Contents
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.
Look Inside

General Resources

Resources

Find resources associated with this title

Back to resources home

Type	Name	Unlocked *	Format	Size

Showing of

This title is supported by one or more locked resources. Access to locked resources is granted exclusively by Cambridge University Press to lecturers whose faculty status has been verified. To gain access to locked resources, lecturers should sign in to or register for a Cambridge user account.

Please use locked resources responsibly and exercise your professional discretion when choosing how you share these materials with your students. Other lecturers may wish to use locked resources for assessment purposes and their usefulness is undermined when the source files (for example, solution manuals or test banks) are shared online or via social networks.

Supplementary resources are subject to copyright. Lecturers are permitted to view, print or download these resources for use in their teaching, but may not change them or use them for commercial gain.

If you are having problems accessing these resources please contact lecturers@cambridge.org.

Authors
Peter Mörters, University of Bath
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, Microsoft Research, Redmond, WA
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.