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On logarithmic Sobolev inequalities. With a preface of Dominique Bakry and Michel Ledoux. (Sur les inégalités de Sobolev logarithmiques.) (French) Zbl 0982.46026
Panoramas et Synthèses. 10. Paris: Société Mathématique de France. xiii, 217 p. (2000).
This book is an overview on logarithmic Sobolev inequalities. These inequalities turned out to be a subject of intense activity during the past years, from analysis and geometry in finite and infinite dimension, to probability theory and statistical mechanics, and many developments are still to be expected.
The book is a pedestrian approach to logarithmic Sobolev inequalities, accessible to a wide audience. It is divided into chapters of independent interest. The fundamental example of two Bernoulli and Gaussian distributions is the starting point to logarithmic Sobolev inequalities as they were defined by Gross in the mid-seventies. Hypercontractivity and tensorisation form two main aspects of these inequalities, that 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 to establish functional inequalities. Another chapter describes a characterization of measure which satisfy logarithmic Sobolev or Poincaré inequalities on the real line, using Hardy’s inequalities.
Interactions with various domains in analytis and probability are developed. A first study deals with the concentration of the measure phenomenon, 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 this inequalities can thus be traced back in the early contributions of Shannon and Stam.
This book focuses on the methods and the characteristics of the treated properties, 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.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60J60 Diffusion processes
58J65 Diffusion processes and stochastic analysis on manifolds
94A17 Measures of information, entropy
47D07 Markov semigroups and applications to diffusion processes
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