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.

First book to give systematic account of the rich interplay between analytic, probabilistic and geometric aspects of Markov diffusion operators

Authors are leading players in the field

Covers large body of results and techniques from the early developments to the current achievements

Includes supplementary material: [...]

Introduction.- 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.