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A new class of transport distances between measures

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Date
2009
Link to item file
http://hal.archives-ouvertes.fr/hal-00262455/en/
Dewey
Probabilités et mathématiques appliquées
Sujet
Gradient flows; Continuity equation; Kantorovich-Rubinstein-Wasserstein distance; Optimal transport
Journal issue
Calculus of Variations and Partial Differential Equations
Volume
34
Number
2
Publication date
2009
Article pages
193-231
Publisher
Springer
DOI
http://dx.doi.org/10.1007/s00526-008-0182-5
URI
https://basepub.dauphine.fr/handle/123456789/837
Collections
  • CEREMADE : Publications
Metadata
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Author
Savaré, Giuseppe
Nazaret, Bruno
Dolbeault, Jean
Type
Article accepté pour publication ou publié
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.

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