Curvature and transport inequalities for Markov chains in discrete spaces

TL;DR

This paper investigates how different notions of Ricci curvature in discrete spaces influence transport-information inequalities, establishing new links and applications such as a discrete Bonnet-Meyer's theorem.

Contribution

It demonstrates that various curvature conditions imply specific transport-information inequalities and applies these results to prove a discrete version of Bonnet-Meyer's theorem.

Findings

Under curvature-dimension or coarse Ricci curvature, an L1 transport-information inequality holds.

Exponential curvature-dimension condition leads to weak-transport information inequalities.

A discrete Bonnet-Meyer's theorem is established under CD(κ,∞) condition.

Abstract

We study various transport-information inequalities under three different notions of Ricci curvature in the discrete setting: the curvature-dimension condition of Bakry and \'Emery, the exponential curvature-dimension condition of Bauer \textit{et al.} and the coarse Ricci curvature of Ollivier. We prove that under a curvature-dimension condition or coarse Ricci curvature condition, an $L_1$ transport-information inequality holds; while under an exponential curvature-dimension condition, some weak-transport information inequalities hold. As an application, we establish a Bonnet-Meyer's theorem under the curvature-dimension condition CD$(\kappa,\infty)$ of Bakry and \'Emery.