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

We study a new class of distances between Radon measures similar to those studied in J. Dolbeault, B. Nazaret, G. Savaré, "A new class of transport distances between measures",Calc. Var. Partial Differential Equations, 34 (2009), pp. 193--231. 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 D.-N.-S.) 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 $mathbb R^d$ with finite moments and the set of measuresdefined 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.