Isometric Rigidity of compact Wasserstein spaces

TL;DR

This paper investigates the isometric properties of Wasserstein spaces over compact metric measure spaces, establishing invariance of Dirac measures and identifying conditions under which the isometry groups of the base space and Wasserstein space coincide.

Contribution

It proves invariance of Dirac measures under Wasserstein isometries and characterizes when the isometry group of the base space matches that of the Wasserstein space for specific geometric conditions.

Findings

Dirac measures are invariant under Wasserstein isometries

Isometry groups of the base manifold and Wasserstein space coincide under certain curvature conditions

Results apply to compact Riemannian manifolds with positive sectional curvature or symmetric spaces

Abstract

Let $(X,d,\mathfrak{m})$ be a metric measure space. The study of the Wasserstein space $(\mathbb{P}_p(X),\mathbb{W}_p)$ associated to $X$ has proved useful in describing several geometrical properties of $X.$ In this paper we focus on the study of isometries of $\mathbb{P}_p(X)$ for $p \in (1,\infty)$ under the assumption that there is some characterization of optimal maps between measures, the so called Good transport behaviour $GTB_p$. Our first result states that the set of Dirac deltas is invariant under isometries of the Wasserstein space. Additionally we obtain that the isometry groups of the base Riemannian manifold $M$ coincides with the one of the Wasserstein space $\mathbb{P}_p(M)$ under assumptions on the manifold; namely, for $p=2$ that the sectional curvature is strictly positive and for general $p\in (1,\infty)$ that $M$ is a Compact Rank One Symmetric Space.