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.
Dominique Bakry held his first research position at the CNRS at the University of Strasbourg, and since 1988 has been a professor at the University of Toulouse. Moreover, since 2004 he has been a senior member of the Institut Universitaire de France. He was editor of the journal Potential Analysis. His research interests center on probability, analysis of Markov operators, differential geometry and orthonormal polynomials.
Ivan Gentil held his first position at the University of Paris-Dauphine in 2003 and since 2010 has been a professor at the University of Lyon. His research interests center on analysis, probability, partial differential equations and functional inequalities such as logarithmic Sobolev inequalities.
Michel Ledoux held his first research position at the CNRS, and since 1991 has been a professor at the University of Toulouse. Since 2010 he has been a senior member of the Institut Universitaire de France, having been a junior member from 1997 to 2002. He has been associate editor for various journals including the Annals of Probability and Probability Theory and Related Fields, and is currently chief editor of the Electronic Journal of Probability. His research interests center on probability theory and functional analysis, measure concentration, diffusion operators and functional inequalities, random matrices, probability in Banach spaces.
Part I Markov semigroups, basics and examples: 1.Markov semigroups.- 2.Model examples.- 3.General setting.-
Part II Three model functional inequalities: 4.Poincaré inequalities.- 5.Logarithmic Sobolev inequalities.- 6.Sobolev inequalities.-
Part III Related functional, isoperimetric and transportation inequalities: 7.Generalized functional inequalities.- 8.Capacity and isoperimetry-type inequalities.- 9.Optimal transportation and functional inequalities.-
Part IV Appendices: A.Semigroups of bounded operators on a Banach space.- B.Elements of stochastic calculus.- C.Some basic notions in differential and Riemannian geometry.- Notations and list of symbols.- Bibliography.- Index.
"The book is friendly written and is of a rich content. With simple examples, main ideas of the study are clearly explained and naturally extended to a general framework, so that main progress in the field made for the past decades is presented in a smooth way. The monograph is undoubtedly a significant reference for further development of diffusion semigroups and related topics." (Feng-Yu Wang, zbMATH 1376.60002, 2018)
"It is extremely rich. It is more original and inspirational than a treatise. One can use it and benefit from it in many ways: as a reference book, as an inspiration source, by focusing on a property or on an example. ... From the beginning to the end, this book definitely has a strong personality and a characteristic taste. ... anybody who wants to explore analytic, probabilistic or geometric properties of Markov semigroups to have a look at it first." (Thierry Coulhon, Jahresbericht der Deutschen Math-Vereinigung, Vol. 119, 2017)
"This impressive monograph is about an important and highly active area that straddles the fertile land occupied by both probability and analysis. ... It is written with great clarity and style, and was clearly a labour of love for the authors. I am convinced that it will be a valuable resource for researchers in analysis and probability for many years to come." (David Applebaum, The Mathematical Gazette, Vol. 100 (548), July, 2016)