Phi-entropy inequalities for diffusion semigroups

TL;DR

This paper develops new Phi-entropy inequalities for diffusion semigroups, extending classical inequalities like Poincaré and logarithmic Sobolev, and applies them to analyze the asymptotic behavior of linear and nonlinear diffusion equations.

Contribution

It introduces novel Phi-entropy inequalities for diffusion semigroups, broadening the scope of existing inequalities and their applications to diffusion equations.

Findings

Derived new Phi-entropy inequalities for diffusion semigroups.

Extended asymptotic analysis to a wide class of linear Fokker-Planck equations.

Applied inequalities to study nonlinear diffusion equations.

Abstract

We obtain and study new $\Phi$-entropy inequalities for diffusion semigroups, with Poincar\'e or logarithmic Sobolev inequalities as particular cases. From this study we derive the asymptotic behaviour of a large class of linear Fokker-Plank type equations under simple conditions, widely extending previous results. Nonlinear diffusion equations are also studied by means of these inequalities. The $\Gamma_2$ criterion of D. Bakry and M. Emery appears as a main tool in the analysis, in local or integral forms.