Semi Log-Concave Markov Diffusions

TL;DR

This paper develops a comprehensive framework for functional inequalities in diffusion processes with curvature assumptions, extending classical results and providing new bounds and convergence results, especially in the log-concave case.

Contribution

It introduces a novel approach combining coupling and stochastic tools to analyze diffusion processes with curvature, extending existing theories and deriving explicit bounds.

Findings

New results on gradient/semigroup commutation for log-concave diffusions

Explicit bounds for convergence to equilibrium in granular media equations

Extension of curvature-based inequalities to broader diffusion processes

Abstract

In this paper we intend to give a comprehensive approach of functional inequalities for diffusion processes under some "curvature" assumptions. Our notion of curvature coincides with the usual $\Gamma_2$ curvature of Bakry and Emery in the case of a (reversible) drifted Brownian motion, but differs for more general diffusion processes. Our approach using simple coupling arguments together with classical stochastic tools, allows us to obtain new results, to recover and to extend already known results, giving in many situations explicit (though non optimal) bounds. In particular, we show new results for gradient/semigroup commutation in the log concave case. Some new convergence to equilibrium in the granular media equation is also exhibited.