On a class of modified Wasserstein distances induced by concave mobility functions defined on bounded intervals

TL;DR

This paper introduces a new class of pseudo-distances between Radon measures based on concave mobility functions on bounded intervals, extending the dynamical formulation of Wasserstein distances and analyzing their properties in specific measure spaces.

Contribution

It generalizes the dynamical formulation of Wasserstein distances using concave mobility functions on bounded intervals and studies their properties in measure spaces with finite moments and bounded convex sets.

Findings

Defined new pseudo-distances with properties similar to Wasserstein distances.

Established convergence conditions for sequences of measures under these distances.

Proved boundedness properties in specific measure spaces.

Abstract

We study a new class of distances between Radon measures similar to those studied in a recent paper of Dolbeault-Nazaret-Savar\'e [DNS]. These distances (more correctly pseudo-distances because can assume the value $+\infty$) are defined generalizing the dynamical formulation of the Wasserstein distance by means of a concave mobility function. We are mainly interested in the physical interesting case (not considered in [DNS]) of a concave mobility function defined in a bounded interval. We state the basic properties of the space of measures endowed with this pseudo-distance. Finally, we study in detail two cases: the set of measures defined in $R^d$ with finite moments and the set of measures defined in a bounded convex set. In the two cases we give sufficient conditions for the convergence of sequences with respect to the distance and we prove a property of boundedness.