Displacement convexity of generalized relative entropies.II

TL;DR

This paper introduces a new class of generalized relative entropies on Wasserstein spaces over weighted manifolds, linking their convexity to curvature conditions and deriving related inequalities and measure concentration results.

Contribution

It establishes the equivalence between entropy convexity and curvature conditions, extending the curvature-dimension framework to generalized relative entropies.

Findings

Convexity of generalized entropies is equivalent to nonnegative weighted Ricci curvature.

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

Analyzed gradient flows of the generalized relative entropy.

Abstract

We introduce a class of generalized relative entropies (inspired by the Bregman divergence in information theory) on the Wasserstein space over a weighted Riemannian or Finsler manifold. We prove that the convexity of all the entropies in this class is equivalent to the combination of the nonnegative weighted Ricci curvature and the convexity of another weight function used in the definition of the generalized relative entropies. This convexity condition corresponds to Lott and Villani's version of the curvature-dimension condition. As applications, we obtain appropriate variants of the Talagrand, HWI and logarithmic Sobolev inequalities, as well as the concentration of measures. We also investigate the gradient flow of our generalized relative entropy.