Isoperimetry and Gaussian analysis. (English) Zbl 0874.60005

Ledoux, Michel

Dobrushin, R. (ed.) et al., Lectures on probability theory and statistics. Ecole d’été de probabilités de Saint-Flour XXIV – 1994. Lectures given at the summer school in Saint-Flour, France, July 7–23, 1994. Berlin: Springer. Lect. Notes Math. 1648, 165-294 (1996).

These notes present the course given by the author at the summer school in St-Flour in 1994. In a concise and selfcontained form the fundamental results on Gaussian processes and measures based on the isoperimetric tool are presented. Eight chapters are devoted to this. Chapter 1 is a short introduction to isoperimetry. Classical isoperimetric inequalities on \(R^d\) and Gaussian isoperimetric inequalities are introduced. In Chapter 2 the concentration of measure phenomenon from a functional analysis point of view based on semigroup theory is developed. Chapter 3 presents several isoperimetric and concentration inequalities for product measures due to M. Talagrand. In Chapter 4 the integrability properties of functionals of a Gaussian measure and large deviations statements are investigated using concentration inequalities. A large deviations statement for the Ornstein-Uhlenbeck process is presented, too. Chapter 5 deals with the corresponding questions for Wiener chaos and is based on the results of C. Borell. In Chapter 6, a complete treatment of boundedness and continuity of Gaussian processes based on the results of R. M. Dudley, X. Fernique, V. N. Sudakov and M. Talagrand is provided. The author aimed to demonstrate the actual simplicity of majorizing measure techniques that usually are considered as difficult and obscure. This aim is achieved successfully. Chapter 7 is devoted to the study of Gaussian small ball probabilities. Some correlation and conditional inequalities for norms of Gaussian measures are investigated, too. In the final Chapter 8 the tight relationship between isoperimetry and semigroup techniques is further investigated. The Gaussian isoperimetric inequality based on hypercontractivity is presented.
For the entire collection see [Zbl 0855.00026].

Reviewer: A.J.Rachkauskas (Vilnius)

MSC:

60B05	Probability measures on topological spaces
60-02	Research exposition (monographs, survey articles) pertaining to probability theory
60G15	Gaussian processes
60B11	Probability theory on linear topological spaces
60E15	Inequalities; stochastic orderings
60F10	Large deviations

Keywords:

Isoperimetric inequalities; concentration inequalities; Gaussian measures; large deviations; Wiener chaos