Brownian Motion

Front Cover
Cambridge University Press, Mar 25, 2010 - Mathematics
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.
 

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Contents

Brownian motion as a random function
7
Brownian motion as a strong Markov process
36
Harmonic functions transience and recurrence
65
Techniques and applications
96
Brownian motion and random walk
118
Brownian local time
153
Stochastic integrals and applications
190
Potential theory of Brownian motion
224
Intersections and selfintersections of Brownian paths
255
Exceptional sets for Brownian motion
290
Further developments
327
Hints and solutions for selected exercises
361
Selected open problems
383
Index
400
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About the author (2010)

Peter Mrters 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 Knig), 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.

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