Some drawbacks of finite modified logarithmic Sobolev inequalities

Modified logarithmic Sobolev inequalities and transportation inequalities

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...

Modified logarithmic Sobolev inequalities in null curvature

We present a logarithmic Sobolev inequality adapted to a log-concave measure. Assume that Φ is a symmetric convex function on R satisfying (1 + ε)Φ(x) 6 xΦ(x) 6 (2 − ε)Φ(x) for x > 0 large enough and with ε ∈]0, 1/2]. We prove that the probability measure on R μΦ(dx) = e /ZΦdx satisfies a modified and adapted logarithmic Sobolev inequality : there exist three constant A,B,D > 0 such that for al...

Logarithmic Sobolev inequalities for some nonlinear PDE’s

The aim of this paper is to study the behavior of solutions of some nonlinear partial differential equations of Mac Kean-Vlasov type. The main tools used are, on one hand, the logarithmic Sobolev inequality and its connections with the concentration of measure and the transportation inequality with quadratic cost; on the other hand, the propagation of chaos for particle systems in mean field in...

Modified logarithmic Sobolev inequalities on R F . Barthe and

We provide a sufficient condition for a measure on the real line to satisfy a modified logarithmic Sobolev inequality, thus extending the criterion of Bobkov and Götze. Under mild assumptions the condition is also necessary. Concentration inequalities are derived. This completes the picture given in recent contributions by Gentil, Guillin and Miclo.