The isometry group of Wasserstein spaces: the Hilbertian case

Abstract

Motivated by Kloeckner's result on the isometry group of the quadratic Wasserstein space ${\mathcal{W}}_2\left({\mathbb{R}}^n\right)$, we describe the isometry group $\mathrm{Isom}\left({\mathcal{W}}_p(E)\right)$ for all parameters $0<p<\infty$ and for all separable real Hilbert spaces $E.$ In particular, we show that ${\mathcal{W}}_p(X)$ is isometrically rigid for all Polish space $X$ whenever $0<p<1$. This is a consequence of our more general result: we prove that ${\mathcal{W}}_1(X)$ is isometrically rigid if $X$ is a complete separable metric space that satisfies the strict triangle inequality. Furthermore, we show that this latter rigidity result does not generalise to parameters $p>1$, by solving Kloeckner's problem affirmatively on the existence of mass-splitting isometries.

arXiv comment30 pages, 2 figures. v2: minor changes

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• Isometric rigidity of the Wasserstein space ${\mathcal{W}}_1(\mathbf{G})$ over Carnot groups
• Isometric Rigidity of compact Wasserstein spaces