Stochastic Analysis on Manifolds (Graduate Studies in Mathematics)

Stochastic Analysis on Manifolds (Graduate Studies in Mathematics)

Hardback

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Description

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.

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.

Product Details

  • ISBN13: 9780821808023
  • Format: Hardback
  • ID: 9780821808023
  • ISBN10: 0821808028

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