Stochastic Analysis on Manifolds

By: Elton P. Hsu

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273 Pages

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26.04 x 18.42 x 2.54

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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-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.

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You Can Find This Book In

Non-Fiction Mathematics Geometry Differential & Riemannian Geometry Applied Mathematics Stochastics Topology Algebraic Topology Probability & Statistics

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
Table of Contents provided by Publisher. All Rights Reserved.

Stochastic Analysis on Manifolds

At a Glance

Hardcover

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How to return your order

Defective items

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