Abstract
We study a new class of distances between Radon measures similar to those studied in Dolbeault et al. (Calc Var Partial Differ Equ 34:193–231, 2009). These distances (more correctly pseudo-distances because can assume the value +∞) 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 Dolbeault et al. (Calc Var Partial Differ Equ 34:193–231, 2009)) 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 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.
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Lisini, S., Marigonda, A. On a class of modified Wasserstein distances induced by concave mobility functions defined on bounded intervals. manuscripta math. 133, 197–224 (2010). https://doi.org/10.1007/s00229-010-0371-3
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DOI: https://doi.org/10.1007/s00229-010-0371-3