The isometry group of Wasserstein spaces: the Hilbertian case

Abstract

Motivated by Kloeckner's result on the isometry group of the quadratic Wasserstein space W2⁡(Rn)$\mathcal {W}_2(\mathbb {R}^n)$, we describe the isometry group Isom⁡(Wp⁡(E))$\mathrm{Isom}(\mathcal {W}_p(E))$ for all parameters 0<p<∞$0 &lt; p &lt; \infty$ and for all separable real Hilbert spaces E$E$. In particular, we show that Wp⁡(X)$\mathcal {W}_p(X)$ is isometrically rigid for all Polish space X$X$ whenever 0<p<1$0&lt;p&lt;1$. This is a consequence of our more general result: we prove that W1⁡(X)$\mathcal {W}_1(X)$ is isometrically rigid if X$X$ 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$p&gt;1$, by solving Kloeckner's problem affirmatively on the existence of mass-splitting isometries. As a byproduct of our methods, we also obtain the isometric rigidity of Wp⁡(X)$\mathcal {W}_p(X)$ for all complete and separable ultrametric spaces X$X$ and parameters 0<p<∞$0&lt;p&lt;\infty$.

Topics & Concepts