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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.

Markov operators. --- Markov processes. --- Analysis, Markov --- Chains, Markov --- Markoff processes --- Markov analysis --- Markov chains --- Markov models --- Models, Markov --- Processes, Markov --- Operators, Markov --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Functional analysis. --- Partial differential equations. --- Differential geometry. --- Probabilities. --- Analysis. --- Probability Theory and Stochastic Processes. --- Differential Geometry. --- Partial Differential Equations. --- Functional Analysis. --- Ergodic theory --- Linear operators --- Markov processes --- Stochastic processes --- Global analysis (Mathematics). --- Distribution (Probability theory. --- Global differential geometry. --- Differential equations, partial. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Partial differential equations --- Geometry, Differential --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Differential geometry --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- 517.1 Mathematical analysis --- Mathematical analysis

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Stochastic processes --- Probabilities --- 519.21 --- Probability theory. Stochastic processes --- 519.21 Probability theory. Stochastic processes --- Congresses --- Probabilities - Congresses