Isometric rigidity of Wasserstein spaces: the graph metric case

Abstract

The aim of this paper is to prove that the $p$-Wasserstein space $\mathcal{W}_p(X)$ is isometrically rigid for all $p\geq 1$ whenever $X$ is a countable graph metric space. As a consequence, we obtain that for every countable group $H$ and any $p\geq 1$ there exists a $p$-Wasserstein space whose isometry group is isomorphic to $H$.