Stochastic Analysis on Manifolds (Graduate Studies in Mathematics)

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. Ch. 1. Stochastic Differential Equations And Diffusions -- 1.1. Sde On Euclidean Space -- 1.2. Sde On Manifolds -- 1.3. Diffusion Processes -- Ch. 2. Basic Stochastic Differential Geometry -- 2.1. Frame Bundle And Connection -- 2.2. Tensor Fields -- 2.3. Horizontal Lift And Stochastic Development -- 2.4. Stochastic Line Integrals -- 2.5. Martingales On Manifolds -- 2.6. Martingales On Submanifolds -- Ch. 3. Brownian Motion On Manifolds -- 3.1. Laplace-beltrami Operator -- 3.2. Brownian Motion On Manifolds -- 3.3. Examples Of Brownian Motion -- 3.4. Distance Function -- 3.5. Radial Process -- 3.6. An Exit Time Estimate -- Ch. 4. Brownian Motion And Heat Semigroup -- 4.1. Heat Kernel As Transition Density Function -- 4.2. Stochastic Completeness -- 4.3. C[subscript 0]-property Of The Heat Semigroup -- 4.4. Recurrence And Transience -- 4.5. Comparison Of Heat Kernels -- Ch. 5. Short-time Asymptotics -- 5.1. Short-time Asymptotics: Near Points -- 5.2. Varadhan's Asymptotic Relation --^ 5.3. Short-time Asymptotics: Distant Points -- 5.4. Brownian Bridge -- 5.5. Derivatives Of The Logarithmic Heat Kernel -- Ch. 6. Further Applications -- 6.1. Dirichlet Problem At Infinity -- 6.2. Constant Upper Bound -- 6.3. Vanishing Upper Bound -- 6.4. Radially Symmetric Manifolds -- 6.5. Coupling Of Brownian Motion -- 6.6. Coupling And Index Form -- 6.7. Eigenvalue Estimates -- Ch. 7. Brownian Motion And Index Theorems -- 7.1. Weitzenbock Formula -- 7.2. Heat Equation On Differential Forms -- 7.3. Gauss-bonnet-chern Formula -- 7.4. Clifford Algebra And Spin Group -- 7.5. Spin Bundle And The Dirac Operator -- 7.6. Atiyah-singer Index Theorem -- 7.7. Brownian Holonomy -- Ch. 8. Analysis On Path Spaces -- 8.1. Quasi-invariance Of The Wiener Measure -- 8.2. Flat Path Space -- 8.3. Gradient Formulas -- 8.4. Integration By Parts In Path Space -- 8.5. Martingale Representation Theorem -- 8.6. Logarithmic Sobolev Inequality And Hypercontractivity --^ 8.7. Logarithmic Sobolev Inequality On Path Space. Elton P. Hsu. Includes Bibliographical References (p. 275-278) And Index. Mainly from the perspective of a probabilist, Hsu shows how stochastic analysis and differential geometry can work together for their mutual benefit. He writes for researchers and advanced graduate students with a firm foundation in basic euclidean stochastic analysis, and differential geometry. He does not include the exercises usual to such texts, but does provide proofs throughout that invite readers to test their understanding. Annotation c. Book News, Inc., Portland, OR (booknews.com) Concerned with probability theory, Elton Hsu's study focuses primarily on the relations between Brownian motion on a manifold and analytical aspects of differential geometry. A key theme is the probabilistic interpretation of the curvature of a manifold A large part of modern analysis is centered around the Laplace operator and its generalizations in various settings.