Optimal heat kernel bounds under logarithmic Sobolev inequalities

Modified logarithmic Sobolev inequalities and transportation inequalities

Weak logarithmic Sobolev inequalities and entropic convergence

In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities...). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincaré inequality can not be used for d...

Modified Logarithmic Sobolev Inequalities in Discrete Settings

Motivated by the rate at which the entropy of an ergodic Markov chain relative to its stationary distribution decays to zero, we study modified versions of logarithmic Sobolev inequalities in the discrete setting of finite Markov chains and graphs. These inequalities turn out to be weaker than the standard log-Sobolev inequality, but stronger than the Poincare’ (spectral gap) inequality. We sho...

Isoperimetry and Symmetrization for Logarithmic Sobolev Inequalities

Using isoperimetry and symmetrization we provide a unified framework to study the classical and logarithmic Sobolev inequalities. In particular, we obtain new Gaussian symmetrization inequalities and connect them with logarithmic Sobolev inequalities. Our methods are very general and can be easily adapted to more general contexts.

Logarithmic Sobolev Inequalities and the Information Theory

In this paper we present an overview on logarithmic Sobolev inequalities. These inequalities have become a subject of intense research activity during the past years, from analysis and geometry in finite and infinite dimension, to probability and statistical mechanics, and of course many others developments and applications are expected. We have divided this paper into three parts. The first pa...