Graduate Studies in Mathematics 2002; 281 pp; hardcover Volume: 38 ISBN10: 0821808028 ISBN13: 9780821808023 List Price: US$50 Member Price: US$40 Order Code: GSM/38
 Probability theory has become a convenient language and a useful tool in many areas of modern analysis. The main purpose of this book is to explore part of this connection concerning the relations between Brownian motion on a manifold and analytical aspects of differential geometry. A dominant theme of the book is the probabilistic interpretation of the curvature of a manifold. The book begins with a brief review of stochastic differential equations on Euclidean space. After presenting the basics of stochastic analysis on manifolds, the author introduces Brownian motion on a Riemannian manifold and studies the effect of curvature on its behavior. He then applies Brownian motion to geometric problems and vice versa, using many wellknown examples, e.g., shorttime behavior of the heat kernel on a manifold and probabilistic proofs of the GaussBonnetChern theorem and the AtiyahSinger index theorem for Dirac operators. The book concludes with an introduction to stochastic analysis on the path space over a Riemannian manifold. Readership Advanced graduate students, research mathematicians, probabilists and geometers interested in stochastic analysis or differential geometry; mathematical physicists interested in global analysis. Reviews "The purpose of this fine book is to explore connections between Brownian motion and analysis in the area of differential geometry, from a probabilist's point of view."  Zentralblatt MATH Table of Contents  Introduction
 Stochastic differential equations and diffusions
 Basic stochastic differential geometry
 Brownian motion on manifolds
 Brownian motion and heat kernel
 Shorttime asymptotics
 Further applications
 Brownian motion and analytic index theorems
 Analysis on path spaces
 Notes and comments
 General notations
 Bibliography
 Index
