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A new class of transport distances between measures

By Jean Dolbeault, Bruno Nazaret, R. Mccann, W. Gangbo, L. C. Evans, F. Otto and C. Villani


Abstract We introduce a new class of distances between nonnegative Radon measures in Rd. They are modeled on the dynamical characterization of the Kantorovich-Rubinstein-Wasserstein distances proposed by BENAMOU-BRENIER [7] and provide a wide family in--Sobolev distances. From the point of view of optimal transport theory, these distances minimize a dynamical cost to move a given initial distribution of mass to a final configuration. An important difference with the classical setting in mass transport theory is that the cost not only depends on the velocity of the moving particles but also on the densities of the intermediate configurations with respect to a given reference measure γ. We study the topological and geometric properties of these new distances, comparing them with the notion of weak convergence of measures and the well established Kantorovich-Rubinstein-Wasserstein theory. An example of possible applications to the geometric theory of gradient flows is also given. terpolating between the Wasserstein and the homogeneous W −1,p

Year: 2013
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