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A new class of transport distances between measures

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Abstract

We introduce a new class of distances between nonnegative Radon measures in \({\mathbb{R}^d}\) . 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 \({W^{-1,p}_\gamma}\) -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.

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Authors and Affiliations

  1. Ceremade (UMR CNRS no. 7534), Université Paris-Dauphine, place de Lattre de Tassigny, 75775, Paris Cédex 16, France

    Jean Dolbeault & Bruno Nazaret

  2. Department of Mathematics, Università degli studi di Pavia, Via Ferrata 1, 27100, Pavia, Italy

    Giuseppe Savaré

Authors
  1. Jean Dolbeault
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  2. Bruno Nazaret
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  3. Giuseppe Savaré
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Corresponding author

Correspondence to Giuseppe Savaré.

Additional information

J. Dolbeault and B. Nazaret have been partially supported by the ANR project IFO. The second author has also been partially supported by the ANR project OTARIE. G. Savaré has been partially supported by grants of M.I.U.R., PRIN ’06. Part of this research was carried out while the third author was visiting professor at Ceremade, Université Paris-Dauphine, whose hospitality and support are also gratefully acknowledged.

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Dolbeault, J., Nazaret, B. & Savaré, G. A new class of transport distances between measures. Calc. Var. 34, 193–231 (2009). https://doi.org/10.1007/s00526-008-0182-5

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