Quantitative rigidity of the Wasserstein contraction under convolution

TL;DR

This work analyzes how the Wasserstein distance $W_p$ contracts under convolution in Euclidean space, introducing the deficit $\delta_\varepsilon(\lambda,\mu)$ and proving a rigidity dichotomy: for $p>1$ equality forces a translate relationship between the measures, while for $p=1$ equality corresponds to a fixed transport direction. It develops both qualitative (rigidity) and quantitative (stability) results, including a two-point stability framework and a stability theory for Kantorovich potentials, and provides concrete quantitative bounds in the quadratic and linear-cost cases, with special attention to Gaussian settings and non-compact scenarios. The paper blends optimal transport duality, nonlocal Sobolev perspectives (BBM-type), and gluing arguments to derive sharp estimates linking dual gaps to transport-map regularity. Overall, it advances understanding of how noise (convolution) affects Wasserstein distances and when the original transport structure remains detectable from noisy data.

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.

Figures (1)

• Figure 1: For every point $x$ in the upper shaded region and every $x'$ in the bottom shaded region $z\in I(x,x')$.

Theorems & Definitions (56)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Remark 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10