Displacement convexity of generalized relative entropies

TL;DR

This paper explores the convexity properties of a generalized relative entropy on weighted manifolds, linking it to curvature conditions and deriving inequalities and equations relevant to diffusion processes.

Contribution

It establishes the equivalence between displacement convexity of the $m$-relative entropy and curvature plus weight function conditions, extending known inequalities and connecting to diffusion equations.

Findings

Displacement $K$-convexity of $m$-relative entropy is characterized by curvature and weight function conditions.

Derived variants of Talagrand, HWI, and logarithmic Sobolev inequalities.

Gradient flow of the $m$-relative entropy solves porous medium and fast diffusion equations.

Abstract

We investigate the $m$-relative entropy, which stems from the Bregman divergence, on weighted Riemannian and Finsler manifolds. We prove that the displacement $K$-convexity of the $m$-relative entropy is equivalent to the combination of the nonnegativity of the weighted Ricci curvature and the $K$-convexity of the weight function. We use this to show appropriate variants of the Talagrand, HWI and the logarithmic Sobolev inequalities, as well as the concentration of measures. We also prove that the gradient flow of the $m$-relative entropy produces a solution to the porous medium equation or the fast diffusion equation.