Analysis and Geometry of Markov Diffusion Operators (Grundlehren der mathematischen Wissenschaften Book 348)

The present volume is an extensive monograph on the analytic and geometric aspects of Markov diffusion operators. It focuses on the geometric curvature properties of the underlying structure in order to study convergence to equilibrium, spectral bounds, functional inequalities such as Poincaré, Sobolev or logarithmic Sobolev inequalities, and various bounds on solutions of evolution equations. At the same time, it covers a large class of evolution and partial differential equations. The book is intended to serve as an introduction to the subject and to be accessible for beginning and advanced scientists and non-specialists. Simultaneously, it covers a wide range of results and techniques from the early developments in the mid-eighties to the latest achievements. As such, students and researchers interested in the modern aspects of Markov diffusion operators and semigroups and their connections to analytic functional inequalities, probabilistic convergence to equilibrium and geometric curvature will find it especially useful. Selected chapters can also be used for advanced courses on the topic. "The present volume is an extensive monograph on the analytic and geometric aspects of Markov diffusion operators. It focuses on the geometric curvature properties of the underlying structure in order to study convergence to equilibrium, spectral bounds, functional inequalities such as Poincaré, Sobolev or logarithmic Sobolev inequalities, and various bounds on solutions of evolution equations. At the same time, it covers a large class of evolution and partial differential equations. The book is intended to serve as an introduction to the subject and to be accessible for beginning and advanced scientists and non-specialists. Simultaneously, it covers a wide range of results and techniques from the early developments in the mid-eighties to the latest achievements. As such, students and researchers interested in the modern aspects of Markov diffusion operators and semigroups and their connections to analytic functional inequalities, probabilistic convergence to equilibrium and geometric curvature will find it especially useful. Selected chapters can also be used for advanced courses on the topic"--Back cover Front Matter....Pages I-XX Front Matter....Pages 1-1 Markov Semigroups....Pages 3-75 Model Examples....Pages 77-118 Symmetric Markov Diffusion Operators....Pages 119-174 Front Matter....Pages 175-175 Poincaré Inequalities....Pages 177-233 Logarithmic Sobolev Inequalities....Pages 235-275 Sobolev Inequalities....Pages 277-343 Front Matter....Pages 345-345 Generalized Functional Inequalities....Pages 347-389 Capacity and Isoperimetric-Type Inequalities....Pages 391-431 Optimal Transportation and Functional Inequalities....Pages 433-469 Back Matter....Pages 471-552