An introduction to the analysis of paths on a Riemannian manifold / Daniel W. Stroock.

Format:

Book

Author/Creator:

Stroock, Daniel W., author.

Series:

Mathematical surveys and monographs ; v. 74.

Mathematical surveys and monographs, 0076-5376 ; volume 74

Language:

English

Subjects (All):

Riemannian manifolds.

Brownian motion processes.

Physical Description:

1 online resource (289 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2000]

Summary:

This book aims to bridge the gap between probability and differential geometry. It gives two constructions of Brownian motion on a Riemannian manifold: an extrinsic one where the manifold is realized as an embedded submanifold of Euclidean space and an intrinsic one based on the ``rolling'' map. It is then shown how geometric quantities (such as curvature) are reflected by the behavior of Brownian paths and how that behavior can be used to extract information about geometricquantities. Readers should have a strong background in analysis with basic knowledge in stochastic calculus and differential geometry. Professor Stroock is a highly-respected expert in probability and analysis. The clarity and style of his exposition further enhance the quality of this volume. Readers willfind an inviting introduction to the study of paths and Brownian motion on Riemannian manifolds.

Contents:

""Contents""; ""Preface""; ""Chapter 1 Brownian Motion in Euclidean Space""; ""1.1. Wiener Measure""; ""1.1.1. Deconstructing Brownian Paths""; ""1.1.2. Levy's Construction""; ""1.1.3. Modulus of Continuity""; ""1.1.4. Multi-dimensional Brownian Motion""; ""1.2. The Infinite Dimensional Sphere and Related Matters""; ""1.2.1. Square Variation of Brownian Paths""; ""1.2.2. Paley�Wiener Integrals""; ""1.2.3. Fourier Characterization""; ""1.2.4. Extension to Higher Dimensions""; ""1.2.5. The Cameron�Martin Formula""; ""1.2.6. Integration by Parts""

""1.3. Feynman's Picture of Wiener Measure""""1.3.1. Rescaling Feynman's Picture""; ""1.4. Wiener Measure, the Laplacian, and Martingales""; ""1.4.1. A Preliminary Manipulation""; ""1.4.2. Reinterpretation""; ""1.4.3. A Heuristic Interpretation""; ""Chapter 2 Diffusions in Euclidean Space""; ""2.1. Martingale Problems for Operators in Hormander Form""; ""2.2. The Abelian Case""; ""2.2.1. A Single Vector Field""; ""2.2.2. A Single Vector Field Squared""; ""2.2.3. Several Commuting Vector Fields""; ""2.3. The Non-Abelian Case""; ""2.3.1. The Scheme for Smooth Paths""

""2.3.2. The Scheme in the Stochastic Case""""2.3.3. Basic Size Estimates""; ""2.3.4. A Continuity Estimate""; ""2.4. Derivatives""; ""2.4.1. Burkholder's Inequality""; ""2.4.2. Estimating Derivatives""; ""2.4.3. A Little Bit of Sobolev""; ""2.4.4. Existence of a Smooth Choice""; ""2.4.5. Loosening Things Up""; ""2.5. The Flow Property""; ""2.5.1. Renewal at Stopping Times""; ""2.5.2. The Heat Flow Semigroup for L and Uniqueness""; ""Chapter 3 Some Addenda, Extensions, and Refinements""; ""3.1. Explosion and Non-explosion""; ""3.1.1. An Example""; ""3.2. Localization""

""3.2.1. Random Paths which may Explode""""3.2.2. Splicing""; ""3.2.3. Localizing the Martingale Problem""; ""3.2.4. A Non-explosion Criterion""; ""3.2.5. Well-posed Martingale Problems""; ""3.3. A Polygonal Approximation Scheme""; ""3.3.1. The Bounded Case""; ""3.3.2. The General Case""; ""3.4. Subordination""; ""3.4.1. Time Dependent Vector Fields""; ""3.4.2. Subordination for Diffusions""; ""3.5. Semigroups of Diffeomorphisms""; ""3.5.1. Flowing Backwards""; ""3.5.2. Existence of a Continuous Version""; ""3.5.3. Non-Degenerate Jacobian""; ""3.5.4. In General""

""3.6. Invariant and Symmetric Measures""""3.6.1. Criterion for Invariance""; ""3.6.2. Symmetric Measures""; ""3.6.3. An Application to the Explosion Problem""; ""Chapter 4 Doing it on a Manifold, An Extrinsic Approach""; ""4.1. Diffusions on a Submanifold of R[sup(N)]""; ""4.1.1. The Martingale Problem""; ""4.1.2. Invariant and Symmetric Measures""; ""4.1.3. Non-Explosion Criterion""; ""4.2. Brownian Motion on a Submanifold""; ""4.2.1. Extrinsic Expressions""; ""4.2.2. Extrinsic Brownian Motion""; ""4.2.3. Brownian Motion Normal to a Submanifold""

""4.2.4. An Extrinsic Non-Explosion Criterion for Brownian Motion""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 265-266) and index.

Description based on print version record.

ISBN:

1-4704-1301-9