Stochastic analysis on manifolds / Elton P. Hsu.

Format:

Book

Author/Creator:

Hsu, Elton P., 1959- author.

Series:

Graduate studies in mathematics ; v. 38.

Graduate studies in mathematics, 1065-7339 ; volume 38

Language:

English

Subjects (All):

Stochastic differential equations.

Diffusion processes.

Geometry, Differential.

Physical Description:

1 online resource (273 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2002]

Language Note:

English

Summary:

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. Afterpresenting 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 ofthe 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.

Contents:

""Cover""; ""Title""; ""Copyright""; ""Contents""; ""Preface""; ""Introduction""; ""Chapter 1. Stochastic Differential Equations and Diffusions""; ""Â1.1. SDE on euclidean space""; ""Â1.2. SDE on manifolds""; ""Â1.3. Diffusion processes""; ""Chapter 2. Basic Stochastic Differential Geometry""; ""Â2.1. Frame bundle and connection""; ""Â2.2. Tensor fields""; ""Â2.3. Horizontal lift and stochastic development""; ""Â2.4. Stochastic line integrals""; ""Â2.5. Martingales on manifolds""; ""Â2.6. Martingales on submanifolds""; ""Chapter 3. Brownian Motion on Manifolds""

""Â3.1. Laplace-Beltrami operator""""Â3.2. Brownian motion on manifolds""; ""Â3.3. Examples of Brownian motion""; ""Â3.4. Distance function""; ""Â3.5. Radial process""; ""Â3.6. An exit time estimate""; ""Chapter 4. Brownian Motion and Heat Semigroup""; ""Â4.1. Heat kernel as transition density function""; ""Â4.2. Stochastic completeness""; ""Â4.3. C[sub(o)]-property of the heat semigroup""; ""Â4.4. Recurrence and transience""; ""Â4.5. Comparison of heat kernels""; ""Chapter 5. Short-time Asymptotics""; ""Â5.1. Short-time asymptotics: near points""

""Â7.2. Heat equation on differential forms""""Â7.3. Gauss-Bonnet-Chern formula""; ""Â7.4. Clifford algebra and spin group""; ""Â7.5. Spin bundle and the Dirac operator""; ""Â7.6. Atiyah-Singer index theorem""; ""Â7.7. Brownian holonomy""; ""Chapter 8. Analysis on Path Spaces""; ""Â8.1. Quasi-invariance of the Wiener measure""; ""Â8.2. Flat path space""; ""Â8.3. Gradient formulas""; ""Â8.4. Integration by parts in path space""; ""Â8.5. Martingale representation theorem""; ""Â8.6. Logarithmic Sobolev inequality and hypercontractivity""

""Â8.7. Logarithmic Sobolev inequality on path space""""Notes and Comments""; ""General Notations""; ""Bibliography""; ""Index""; ""A""; ""B""; ""C""; ""D""; ""E""; ""F""; ""G""; ""H""; ""I""; ""J""; ""K""; ""L""; ""M""; ""N""; ""O""; ""P""; ""Q""; ""R""; ""S""; ""T""; ""U""; ""V""; ""W""; ""Y""; ""Back Cover""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 275-278) and index.

Description based on print version record.

ISBN:

1-4704-2090-2