Analysis for diffusion processes on Riemannian manifolds (Q2842218)

This book is devoted to the stochastic analysis of diffusion processes on Riemannian manifolds from the point of view of functional inequalities. In particular, Sobolev and Harnack inequalities and transportation-cost inequalities are stated and proved in various settings, in finite dimensions and on manifolds with or without boundary, as well as in infinite-dimensional settings, for global functionals on the path space over a manifold. The setting of manifolds with boundary naturally gives rise to reflecting diffusion processes which are considered in Chapters~3 and 4.NEWLINENEWLINEChapter~1 starts from the notion of Riemannian manifold, and then moves on to the various functional inequalities available on measure spaces. It also presents the coupling technique which will play a major role in the derivation of functional inequalities, along with Bakry-Emery semigroup type arguments.NEWLINENEWLINEChapter~2 is based on Brownian motion and diffusion processes on manifolds without boundary. Here, functional inequalities are proved together with the derivation of their mutual relationships and their equivalent forms. In particular, logarithmic Sobolev inequalities are obtained when the curvature is bounded below, and Harnack inequalities are derived via the coupling technique.NEWLINENEWLINEIn Chapter~3, functional inequalities are obtained for reflecting processes on concave and non-convex manifolds. This includes Harnack inequalities for the solutions of stochastic differential equations on \({\mathbb R}^d\) and their extension to manifolds with convex boundary, and log-Sobolev inequalities on locally concave and non-convex manifolds.NEWLINENEWLINEChapter~4 deals with global stochastic analysis on the path space of reflecting diffusions on manifolds with boundary, which is the occasion to present and apply the main elements of the Malliavin calculus on Riemannian manifolds, such as integration by parts formulas for the damped gradient operator.NEWLINENEWLINEChapter~5 is devoted to hypoelliptic diffusion processes on Riemannian manifolds. Poincaré inequalities, Nash and log-Sobolev inequalities, and Harnack inequalities are derived under generalized curvature-dimension conditions. Bismut type identities are also derived using the Malliavin calculus and coupling arguments.NEWLINENEWLINEThe book is largely arranged around the author's own research interests, and covers work done over the past 20 years on diffusion processes on manifolds, including joint publications.