Mathematical Surveys and Monographs 2000; 269 pp; softcover Volume: 74 Reprint/Revision History: reprinted 2005 ISBN10: 0821838393 ISBN13: 9780821838396 List Price: US$49 Member Price: US$39.20 Order Code: SURV/74.S
 This book aims to bridge the gap between probability and differential geometry. It gives two constructions of Brownian motion on a Riemannian manifold: an extrinsic one where the manifold is realized as an embedded submanifold of Euclidean space and an intrinsic one based on the "rolling" map. It is then shown how geometric quantities (such as curvature) are reflected by the behavior of Brownian paths and how that behavior can be used to extract information about geometric quantities. Readers should have a strong background in analysis with basic knowledge in stochastic calculus and differential geometry. Professor Stroock is a highlyrespected expert in probability and analysis. The clarity and style of his exposition further enhance the quality of this volume. Readers will find an inviting introduction to the study of paths and Brownian motion on Riemannian manifolds. Readership Graduate students, research mathematicians, and physicists interested in probability theory and stochastic analysis; theoretical physicists; electrical engineers. Table of Contents  Brownian motion in Euclidean space
 Diffusions in Euclidean space
 Some addenda, extensions, and refinements
 Doing it on a manifold, an extrinsic approach
 More about extrinsic Riemannian geometry
 Bochner's identity
 Some intrinsic Riemannian geometry
 The bundle of orthonormal frames
 Local analysis of Brownian motion
 Perturbing Brownian paths
 References
 Index
