Quantitative rigidity of the Wasserstein contraction under convolution

TL;DR

This paper explores how convolution affects the contraction properties of p-Wasserstein distances in Euclidean spaces, linking it to the uniform convexity of the Kantorovich functional and extending results to p=1.

Contribution

It extends the understanding of Wasserstein contraction properties under convolution, especially by establishing uniform convexity results for p=1.

Findings

Contraction properties of p-Wasserstein distances under convolution are characterized.

Uniform convexity of the Kantorovich functional is established for p=1.

Connections between Wasserstein contraction and functional convexity are clarified.

Abstract

The aim of this paper is to investigate the contraction properties of $p$-Wasserstein distances with respect to convolution in Euclidean spaces both qualitatively and quantitatively. We connect this question to the question of uniform convexity of the Kantorovich functional on which there was substantial recent progress (mostly for $p=2$ and partially for $p>1$). Motivated by this connection we extend these uniform convexity results to the case $p=1$, which is of independent interest.