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

TL;DR

This survey and new results explore the structure of isometries on probability measure spaces with various distances, focusing on Wasserstein metrics and their relation to underlying space isometries.

Contribution

The paper reviews recent findings and introduces new results on isometries of probability measure spaces, especially regarding Wasserstein distances and their connection to underlying metric space isometries.

Findings

Wasserstein isometries on probability measures are induced by underlying space isometries.

The structure of isometries on measure spaces depends on the metric used.

New results relate Wasserstein isometries to isometries of the unit sphere in Euclidean space.

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.