Uniform Poincaré and logarithmic Sobolev inequalities for mean field particle systems

Logarithmic Sobolev and Poincaré inequalities for the circular Cauchy distribution ∗

In this paper, we consider the circular Cauchy distribution μx on the unit circle S with index 0 ≤ |x| < 1 and we study the spectral gap and the optimal logarithmic Sobolev constant for μx, denoted respectively by λ1(μx) and CLS(μx). We prove that 1 1+|x| ≤ λ1(μx) ≤ 1 while CLS(μx) behaves like log(1 + 1 1−|x| ) as |x| → 1.

From Poincaré to Logarithmic Sobolev Inequalities: A Gradient Flow Approach

We use the distances introduced in a previous joint paper to exhibit the gradient flow structure of some drift-diffusion equations for a wide class of entropy functionals. Functional inequalities obtained by the comparison of the entropy with the entropy production functional reflect the contraction properties of the flow. Our approach provides a unified framework for the study of the Kolmogoro...

SOBOLEV - POINCARÉ INEQUALITIES FOR p < 1

If Ω is a John domain (or certain more general domains), and |∇u| satisfies a certain mild condition, we show that u ∈ W 1,1 loc (Ω) satisfies a Sobolev-Poincaré inequality`R Ω |u − a| q ´ 1/q ≤ C `R Ω |∇u| p ´ 1/p for all 0 < p < 1, and appropriate q > 0. Our conclusion is new even when Ω is a ball.

Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential

We consider a non-interacting unbounded spin system with conservation of the mean spin. We derive a uniform logarithmic Sobolev inequality (LSI) provided the single-site potential is a bounded perturbation of a strictly convex function. The scaling of the LSI constant is optimal in the system size. The argument adapts the two-scale approach of Grunewald, Otto, West-dickenberg, and Villani from ...

Modified logarithmic Sobolev inequalities and transportation inequalities