Abstract/Summary

Motivated by Kloeckner’s result on the isometry group of the quadratic Wasserstein space W2(ℝn), we describe the isometry group Isom(Wp(E))for all parameters 0<p<∞ and for all separable real Hilbert spaces E. In particular, we show that Wp(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 W1(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. As a byproduct of our methods, we also obtain the isometric rigidity of Wp(X)for all complete and separable ultrametric spaces X and parameters 0<p<∞.