Concentration of measure and logarithmic Sobolev inequalities. (English) Zbl 0957.60016

Ledoux, Michel

Azéma, Jacques (ed.) et al., Séminaire de probabilités XXXIII. Berlin: Springer. Lect. Notes Math. 1709, 120-216 (1999).

From the author’s summary: The first part of this set of notes includes an introduction to isoperimetry and concentration for Gaussian and Boltzmann measures. The second part then presents spectral gap and logarithmic Sobolev inequalities, and describes Herbst’s basic Laplace transform argument. In the third part, we investigate by this method deviation and concentration inequalities for product measures. While concentration inequalities do not necessarily tensorize, we show that they actually follow from stronger logarithmic Sobolev inequalities. We thus recover most of M. Talagrand’s recent results on isoperimetric and concentration inequalities in product spaces. We briefly mention there the information theoretic inequalities by K. Marton which provide an alternate approach to concentration also based on entropy, and which seems to be well suited to dependent structures. We then develop the subject of modified logarithmic Sobolev inequalities investigated in joint works with S. Bobkov. We examine in this way concentration properties for the product measure of the exponential distribution, as well as, more generally, of measures satisfying a Poincaré inequality. In the next section, the analogous questions for discrete gradients are addressed, with particular emphasis on Bernoulli and Poisson measures. We then present some applications to large deviation upper bounds and to tail estimates for Brownian motion on a manifold. In the final part, we discuss some recent results on the logarithmic Sobolev constant in Riemannian manifolds with non-negative Ricci curvature. The last section is an addition of L. Saloff-Coste on the logarithmic Sobolev constant and the diameter of Markov chains.
For the entire collection see [Zbl 0924.00016].

Reviewer: Walter Farkas (Regensburg)

MSC:

60E15	Inequalities; stochastic orderings
60J35	Transition functions, generators and resolvents
60J65	Brownian motion
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

Sobolev inequality; isoperimetric inequality; Markov semigroup; Dirichlet form; Brownian motion on a manifold; Gaussian measures; Boltzmann measures; Herbst’s basic Laplace transform; concentration inequalities; Bernoulli measures; Poisson measures; Riemannian manifolds with non-negative Ricci curvature