The isometry group of Wasserstein spaces: the Hilbertian case

Motivated by Kloeckner's result on the isometry group of the quadratic Wasserstein space (Formula presented.), we describe the isometry group (Formula presented.) for all parameters (Formula presented.) and for all separable real Hilbert spaces (Formula presented.). In particular, we show that (Formula presented.) is isometrically rigid for all Polish space (Formula presented.) whenever (Formula presented.). This is a consequence of our more general result: we prove that (Formula presented.) is isometrically rigid if (Formula presented.) 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 (Formula presented.), 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 (Formula presented.) for all complete and separable ultrametric spaces (Formula presented.) and parameters (Formula presented.). © 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence