Analysis for diffusion processes on Riemannian manifolds / Feng-Yu Wang, Beijing Normal University, China & Swansea University, UK.

Format:

Book

Author/Creator:

Wang, Feng-Yu, author.

Series:

Advanced series on statistical science & applied probability ; v. 18.

Advanced Series on Statistical Science & Applied Probability ; Volume 18

Language:

English

Subjects (All):

Riemannian manifolds.

Diffusion processes.

Physical Description:

1 online resource (xii, 379 pages) : illustrations.

Place of Publication:

New Jersey : World Scientific, [2014]

Language Note:

English

Summary:

Stochastic analysis on Riemannian manifolds without boundary has been well established. However, the analysis for reflecting diffusion processes and sub-elliptic diffusion processes is far from complete. This book contains recent advances in this direction along with new ideas and efficient arguments, which are crucial for further developments. Many results contained here (for example, the formula of the curvature using derivatives of the semigroup) are new among existing monographs even in the case without boundary.

Contents:

Preface; Contents; 1. Preliminaries; 1.1 Riemannian manifold; 1.1.1 Differentiable manifold; 1.1.2 Riemannian manifold; 1.1.3 Some formulae and comparison results; 1.2 Riemannian manifold with boundary; 1.3 Coupling and applications; 1.3.1 Transport problem and Wasserstein distance; 1.3.2 Optimal coupling and optimal map; 1.3.3 Coupling for stochastic processes; 1.3.4 Coupling by change of measure; 1.4 Harnack inequalities and applications; 1.4.1 Harnack inequality; 1.4.2 Shift Harnack inequality; 1.5 Harnack inequality and derivative estimate

1.5.1 Harnack inequality and entropy-gradient estimate1.5.2 Harnack inequality and L2-gradient estimate; 1.5.3 Harnack inequalities and gradient-gradient estimates; 1.6 Functional inequalities and applications; 1.6.1 Poincar e type inequality and essential spectrum; 1.6.2 Exponential decay in the tail norm; 1.6.3 The F-Sobolev inequality; 1.6.4 Weak Poincare inequality; 1.6.5 Equivalence of irreducibility and weak Poincare inequality; 2. Diffusion Processes on Riemannian Manifolds without Boundary; 2.1 Brownian motion with drift; 2.2 Formulae for Pt and RicZ

2.3 Equivalent semigroup inequalities for curvature lower bound2.4 Applications of equivalent semigroup inequalities; 2.5 Transportation-cost inequality; 2.5.1 From super Poincare to weighted log-Sobolev inequalities; 2.5.2 From log-Sobolev to transportation-cost inequalities; 2.5.3 From super Poincare to transportation-cost inequalities; 2.5.4 Super Poincare inequality by perturbations; 2.6 Log-Sobolev inequality: Different roles of Ric and Hess; 2.6.1 Exponential estimate and concentration of; 2.6.2 Harnack inequality and the log-Sobolev inequality

2.6.3 Hypercontractivity and ultracontractivity2.7 Curvature-dimension condition and applications; 2.7.1 Gradient and Harnack inequalities; 2.7.2 HWI inequalities; 2.8 Intrinsic ultracontractivity on non-compact manifolds; 2.8.1 The intrinsic super Poincare inequality; 2.8.2 Curvature conditions for intrinsic ultracontractivity; 2.8.3 Some examples; 3. Reflecting Diffusion Processes on Manifolds with Boundary; 3.1 Kolmogorov equations and the Neumann problem; 3.2 Formulae for Pt, RicZ and I; 3.2.1 Formula for Pt; 3.2.2 Formulae for RicZ and I; 3.2.3 Gradient estimates

3.3 Equivalent semigroup inequalities for curvature conditionand lower bound of I3.3.1 Equivalent statements for lower bounds of RicZ and I; 3.3.2 Equivalent inequalities for curvature-dimension condition and lower bound of I; 3.4 Harnack inequalities for SDEs on Rd and extension to nonconvex manifolds; 3.4.1 Construction of the coupling; 3.4.2 Harnack inequality on Rd; 3.4.3 Extension to manifolds with convex boundary; 3.4.4 Neumann semigroup on non-convex manifolds; 3.5 Functional inequalities; 3.5.1 Estimates for inequality constants on compact manifolds

3.5.2 A counterexample for Bakry-Emery criterion

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on online resource; title from PDF title page (ebrary, viewed March 31, 2014).

ISBN:

9789814452656

9814452653

OCLC:

860388715