Calculus of Variations and Geometric Measure Theory

J. Dolbeault - B. Nazaret - G. Savaré

A new class of transport distances between measures

created by savare on 08 Mar 2008
modified on 12 Jan 2009


Published Paper

Inserted: 8 mar 2008
Last Updated: 12 jan 2009

Journal: Calc. Var. Partial Differential Equations
Volume: 34
Pages: 193-231
Year: 2009


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 \textsc{Benamou-Brenier} and provide a wide family interpolating between the Wasserstein and the homogeneous $W^{-1,p}$-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.

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.

Keywords: Optimal transport, Gradient flows, continuity equation