Stochastic Analysis on Manifolds, Hardback Book

Stochastic Analysis on Manifolds Hardback

Part of the Graduate Studies in Mathematics series

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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-Chem 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.

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