Stability of optimal transport on metric measure spaces

TL;DR

This paper establishes a quantitative stability result for Kantorovich potentials and optimal transport maps on metric measure spaces with lower Ricci or curvature bounds, confirming a recent conjecture without relying on linear or sectional curvature assumptions.

Contribution

It proves a new stability theorem for Kantorovich potentials on metric measure spaces with curvature bounds, using heat kernel regularization, and extends results to Alexandrov spaces.

Findings

Quantitative stability of Kantorovich potentials proven

Stability of optimal transport maps on Alexandrov spaces established

Proof does not depend on linear structure or sectional curvature bounds

Abstract

We prove a quantitative stability of Kantorovich potentials on metric measure spaces with lower Ricci curvature bound, thereby confirming a recent conjecture of Kitagawa, Letrouit and M\'erigot. Our proof, which employs the heat kernel-regularized $c$-transform, does not rely on linear structure or sectional curvature bounds. As a corollary, we get a quantitative stability of optimal transport maps on Alexandrov spaces with lower curvature bound.