Maps on probability measures preserving certain distances --- a survey and some new results

Abstract: Borel probability measures living on metric spaces are fundamental mathematical objects. There are several meaningful distance functions that make the collection of the probability measures living on a certain space a metric space. We are interested in the description of the structure of the isometries of such metric spaces. We overview some of the recent results of the topic and we also provide some new ones concerning the Wasserstein distance. More specifically, we consider the space of all Borel probability measures on the unit sphere of a Euclidean space endowed with the Wasserstein metric $W_p$ for arbitrary $p \geq 1,$ and we show that the action of a Wasserstein isometry on the set of the Dirac measures is induced by an isometry of the underlying unit sphere.