# Sur Les Inégalités de Sobolev Logarithmiques

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*S. Blanchere; D. Chafai; P. Fougeres; I. Gentil; F. Malrieu; C. Roberto; G. Scheffer*

A publication of the Société Mathématique de France

This book is an overview of logarithmic Sobolev inequalities. These
inequalities have been the subject of intense activity in recent years, from
analysis and geometry in finite and infinite dimensions to probability theory
and statistical mechanics. And many developments are still to come.

The book is a “pedestrian approach” to logarithmic Sobolev
inequalities, accessible to a wide audience. It is divided into several
chapters of independent interest. The fundamental example of the Bernoulli and
Gaussian distributions is the starting point for logarithmic Sobolev
inequalities, as they were defined by Gross in the mid-seventies.
Hypercontractivity and tensorisation form two main aspects of these
inequalities, which are actually part of the larger family of classical Sobolev
inequalities in functional analysis.

A chapter is devoted to the curvature-dimension criterion, which is an
efficient tool for establishing functional inequalities. Another chapter
describes a characterization of measures which satisfy logarithmic Sobolev or
Poincaré inequalities on the real line, using Hardy's inequalities.

Interactions with various domains in analysis and probability are developed.
A first study deals with the concentration of measure phenomenon, which is
useful in statistics as well as geometry. The relationships between logarithmic
Sobolev inequalities and the transportation of measures are considered, in
particular through their approach to concentration. A control of the speed of
convergence to equilibrium of finite state Markov chains is described in terms
of the spectral gap and the logarithmic Sobolev constants. The last part is a
modern reading of the notion of entropy in information theory and of the
several links between information theory and the Euclidean form of the Gaussian
logarithmic Sobolev inequality. The genesis of these inequalities can be traced
back to the early contributions of Shannon and Stam.

This book focuses on the specific methods and the characteristics of
particular topics, rather than the most general fields of study. Chapters are
mostly self-contained. The bibliography, without being encyclopedic, tries to
give a rather complete state of the art on the topic, including some very
recent references.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

#### Readership

Graduate students and research mathematicians.