Measure Concentration, Transportation Cost, and Functional Inequalities

— In these lectures, we present a triple description of the concentration of measure phenomenon, geometric (through BrunnMinkoswki inequalities), measure-theoretic (through transportation cost inequalities) and functional (through logarithmic Sobolev inequalities), and investigate the relationships between these various viewpoints. Special emphasis is put on optimal mass transportation and the dual hypercontractive bounds on solutions of Hamilton-Jacobi equations that offer a unified treatment of these various aspects. These notes survey recent developments around the concentration of measure phenomenon through various descriptions, geometric, measure theoretic and functional. These descriptions aim to analyze measure concentration for both product and (strictly) log-concave measures, with a special emphasis on dimension free bounds. Inequalities independent of the number of variables are indeed a key information in the study of a number of models in probability theory and statistical mechanics, with a view towards infinite dimensional analysis. To this task, we review the geometric tool of BrunnMinkowski inequalities, transportation cost inequalities, and functional logarithmic Sobolev inequalities and semigroup methods. Connections are developed on the basis of optimal mass transportation and dual hypercontractive bounds on solutions of Hamilton-Jacobi equations, providing a synthetic view of these recent developments. Results and methods are only outlined in the simplest and basic setting. References to recent PDE extensions are briefly discussed in the last part of the notes. These notes only collect a few basic results on the topics of these lectures, and only aim to give a flavour of the subject. We refer to [Le], [B-G-L], [O-V1], [CE], [CE-G-H], [Vi]... for further material, proofs and detailed references. 1. The concentration of measure phenomenon The concentration of measure phenomenon was put forward in the early seventies by V. Milman [Mi1], [Mi3] in the asymptotic geometry of Banach spaces and the proof of the famous Dvoretzky theorem on spherical sections of convex bodies. Of isoperimetric inspiration, it is of powerful interest in applications, in various areas such as geometry, functional analysis and infinite dimensional integration, discrete mathematics and complexity theory, and probability theory. General references, from various viewpoints, are [Bal], [Grom], [Le], [MD], [Mi2], [Mi4], [M-S], [Sc], [St], [Ta1]... 1.