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Hardcover ISBN:  9780821808023 
Product Code:  GSM/38 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
eBook ISBN:  9781470420901 
Product Code:  GSM/38.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9780821808023 
eBook ISBN:  9781470420901 
Product Code:  GSM/38.B 
List Price:  $184.00 $141.50 
MAA Member Price:  $165.60 $127.35 
AMS Member Price:  $147.20 $113.20 

Book DetailsGraduate Studies in MathematicsVolume: 38; 2002; 281 ppMSC: Primary 58; 60
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 wellknown examples, e.g., shorttime behavior of the heat kernel on a manifold and probabilistic proofs of the GaussBonnetChern theorem and the AtiyahSinger index theorem for Dirac operators. The book concludes with an introduction to stochastic analysis on the path space over a Riemannian manifold.
ReadershipAdvanced graduate students, research mathematicians, probabilists and geometers interested in stochastic analysis or differential geometry; mathematical physicists interested in global analysis.

Table of Contents

Chapters

Introduction

Chapter 1. Stochastic differential equations and diffusions

Chapter 2. Basic stochastic differential geometry

Chapter 3. Brownian motion on manifolds

Chapter 4. Brownian motion and heat semigroup

Chapter 5. Shorttime asymptotics

Chapter 6. Further applications

Chapter 7. Brownian motion and index theorems

Chapter 8. Analysis on path spaces

Notes and comments


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


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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 wellknown examples, e.g., shorttime behavior of the heat kernel on a manifold and probabilistic proofs of the GaussBonnetChern theorem and the AtiyahSinger index theorem for Dirac operators. The book concludes with an introduction to stochastic analysis on the path space over a Riemannian manifold.
Advanced graduate students, research mathematicians, probabilists and geometers interested in stochastic analysis or differential geometry; mathematical physicists interested in global analysis.

Chapters

Introduction

Chapter 1. Stochastic differential equations and diffusions

Chapter 2. Basic stochastic differential geometry

Chapter 3. Brownian motion on manifolds

Chapter 4. Brownian motion and heat semigroup

Chapter 5. Shorttime asymptotics

Chapter 6. Further applications

Chapter 7. Brownian motion and index theorems

Chapter 8. Analysis on path spaces

Notes and comments

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