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A new class of transport distances between measures

Savaré, Giuseppe; Nazaret, Bruno; Dolbeault, Jean (2009), A new class of transport distances between measures, Calculus of Variations and Partial Differential Equations, 34, 2, p. 193-231. http://dx.doi.org/10.1007/s00526-008-0182-5

Type
Article accepté pour publication ou publié
External document link
http://hal.archives-ouvertes.fr/hal-00262455/en/
Date
2009
Journal name
Calculus of Variations and Partial Differential Equations
Volume
34
Number
2
Publisher
Springer
Pages
193-231
Publication identifier
http://dx.doi.org/10.1007/s00526-008-0182-5
Metadata
Show full item record
Author(s)
Savaré, Giuseppe
Nazaret, Bruno
Dolbeault, Jean cc
Abstract (EN)
We introduce a new class of distances between nonnegative Radon measures on the euclidean space. They are modeled on the dynamical characterization of the Kantorovich-Rubinstein-Wasserstein distances proposed by Benamou and Brenier and provide a wide family interpolating between Wasserstein and homogeneous 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. 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.
Subjects / Keywords
Gradient flows; Continuity equation; Kantorovich-Rubinstein-Wasserstein distance; Optimal transport

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