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 highly-respected 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