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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