Coarse Ricci curvature on the space of probability measures

TL;DR

This paper explores the properties of coarse Ricci curvature on probability measure spaces, linking it to Wasserstein spaces, random walks, and measure concentration phenomena, with implications for geometric analysis.

Contribution

It establishes the equivalence of infimum coarse Ricci curvature across Wasserstein spaces and original random walks, and investigates convergence and measure concentration.

Findings

Infimum coarse Ricci curvature matches across Wasserstein and original spaces

Relation between Gromov-Hausdorff convergence and coarse Ricci curvature

Connection between coarse Ricci curvature and measure concentration phenomena

Abstract

In this paper we study the coarse Ricci curvature on the space of probability measures on a metric space. The infimum of the $p$-coarse Ricci curvature on the $L^p$-Wasserstein space coincides with that with respect to the original random walk. Considering a random walk as a map, we investigate the relation between Gromov-Hausdorff convergence and the $p$-coarse Ricci curvature. We also study the concentration of measure phenomenon related to the coarse Ricci curvature.