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 well-known examples, e.g., short-time behavior of the heat kernel on a manifold and probabilistic proofs of the Gauss-Bonnet-Chern theorem and the Atiyah-Singer 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
• Short-time asymptotics
• Further applications
• Brownian motion and analytic index theorems
• Analysis on path spaces
• Notes and comments
• General notations
• Bibliography
• Index