Description
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.
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.
