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 Poincare, 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 Poincare, 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.

New isoperimetric inequalities and random process techniques have recently appeared at the basis of the modern understanding of Probability in Banach spaces. Based on these tools, the book presents a complete treatment of the main aspects of Probability in Banach spaces (e.g. boundedness and continuity of random processes, integrability and limit theorems for vector valued random variables) and of some of their links to Geometry of Banach spaces. Its purpose is to present some of the main aspects of this theory, from the foundations to the latest developments, treated with the most recent and updated tools. In particular, the most important features are the systematic use of isoperimetry and related concentration of measure phenomena (to study integrability and limit theorems for vector valued random variables), and recent abstract random process techniques (entropy and majorizing measures). Some examples of these probabilistic ideas to classical Banach space theory complete this exposition.

Besides a series of six articles on Levy processes, Volume 38 of the Seminaire de Probabilites contains contributions whose topics range from analysis of semi-groups to free probability, via martingale theory, Wiener space and Brownian motion, Gaussian processes and matrices, diffusions and their applications to PDEs.

As do all previous volumes of this series, it provides an overview on the current state of the art in the research on stochastic processes.

The 37th SA(c)minaire de ProbabilitA(c)s contains A. Lejay's advanced course which is a pedagogical introduction to works by T. Lyons and others on stochastic integrals and SDEs driven by deterministic rough paths. The rest of the volume consists of various articles on topics familiar to regular readers of the SA(c)minaires, including Brownian motion, random environment or scenery, PDEs and SDEs, random matrices and financial random processes.

The 36th Séminaire de Probabilités contains an advanced course on Logarithmic Sobolev Inequalities by A. Guionnet and B. Zegarlinski, as well as two shorter surveys by L. Pastur and N. O'Connell on the theory of random matrices and their links with stochastic processes. The main themes of the other contributions are Logarithmic Sobolev Inequalities, Stochastic Calculus, Martingale Theory and Filtrations. Besides the traditional readership of the Séminaires, this volume will be useful to researchers in statistical mechanics and mathematical finance.

Bakry, Dominique: Hypercontractivity and its Usage in Semigroup Theory.- Ledoux, Michel: Isoperimetry and Gaussian Analysis.- Saloff-Coste, Laurent: Lectures on Finite Markov Chains.

All the papers in the volume are original research papers, discussing fundamental properties of stochastic processes. The topics under study (martingales, filtrations, path properties, etc.) represent an important part of the current research performed in 1996-97 by various groups of probabilists in France and abroad.