The isometry group of Wasserstein spaces: the Hilbertian case.

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<∞$0 < p < \infty$ 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$0<p<1$. 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$p>1$, 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<∞$0<p<\infty$. [ABSTRACT FROM AUTHOR]