A new class of transport distances between measures

被引:135
作者
Dolbeault, Jean [2 ]
Nazaret, Bruno [2 ]
Savare, Giuseppe [1 ]
机构
[1] Univ Pavia, Dept Math, I-27100 Pavia, Italy
[2] Univ Paris 09, CNRS, CEREMADE, UMR 7534, F-75775 Paris 16, France
关键词
GENERATING CONVEX-FUNCTIONS; EULERIAN CALCULUS; FUNCTIONALS; EVOLUTION; GEOMETRY;
D O I
10.1007/s00526-008-0182-5
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We introduce a new class of distances between nonnegative Radon measures in R-d. They are modeled on the dynamical characterization of the Kantorovich-Rubinstein-Wasserstein distances proposed by Benamou and Brenier (Numer Math 84:375-393, 2000) and provide a wide family interpolating between the Wasserstein and the homogeneous W-gamma(-1,p)-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 gamma. 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.
引用
收藏
页码:193 / 231
页数:39
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