**Graduate Studies in Mathematics**

Volume: 38;
2002;
281 pp;
Hardcover

MSC: Primary 58; 60;

Print ISBN: 978-0-8218-0802-3

Product Code: GSM/38

List Price: $53.00

AMS Member Price: $42.40

MAA member Price: $47.70

**Electronic ISBN: 978-1-4704-2090-1
Product Code: GSM/38.E**

List Price: $53.00

AMS Member Price: $42.40

MAA member Price: $47.70

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# Stochastic Analysis on Manifolds

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*Elton P. Hsu*

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 & Endorsements

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

# Table of Contents

## Stochastic Analysis on Manifolds

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents ix10 free
- Preface xiii14 free
- Introduction 116 free
- Chapter 1. Stochastic Differential Equations and Diffusions 520 free
- Chapter 2. Basic Stochastic Differential Geometry 3550
- Chapter 3. Brownian Motion on Manifolds 7186
- Chapter 4. Brownian Motion and Heat Semigroup 101116
- Chapter 5. Short-time Asymptotics 129144
- Chapter 6. Further Applications 157172
- Chapter 7. Brownian Motion and Index Theorems 191206
- Chapter 8. Analysis on Path Spaces 229244
- §8.1. Quasi-invariance of the Wiener measure 230245
- §8.2. Flat path space 234249
- §8.3. Gradient formulas 246261
- §8.4. Integration by parts in path space 250265
- §8.5. Martingale representation theorem 256271
- §8.6. Logarithmic Sobolev inequality and hypercontractivity 258273
- §8.7. Logarithmic Sobolev inequality on path space 263278

- Notes and Comments 267282
- General Notations 271286
- Bibliography 275290
- Index 279294
- Back Cover Back Cover1297