Calculus of Variations and Geometric Measure Theory

S. Lisini - A. Marigonda

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

created by lisini on 14 Sep 2009
modified on 06 Aug 2010


Published Paper

Inserted: 14 sep 2009
Last Updated: 6 aug 2010

Journal: Manuscripta Math.
Volume: 133
Pages: 197-224
Year: 2010


We study a new class of distances between Radon measures similar to those studied in a recent paper of Dolbeault-Nazaret-Savaré 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.

Keywords: generalized Wasserstein distance, mobility function