No abstract

We introduce a new class of distances between nonnegative Radon measures in R d . They are modeled on the dynamical characterization of the Kantorovich-RubinsteinWasserstein distances proposed by BENAMOU-BRENIER [7] and provide a wide family interpolating between the Wasserstein and the homogeneous WFrom 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 KantorovichRubinstein-Wasserstein theory. An example of possible applications to the geometric theory of gradient flows is also given.

In this paper, we study the characterization of geodesics for a class of distances between probability measures introduced by Dolbeault, Nazaret and Savaré. We first prove the existence of a potential function and then give necessary and sufficient optimality conditions that take the form of a coupled system of PDEs somehow similar to the MeanField-Games system of Lasry and Lions. We also consider an equivalent formulation posed in a set of probability measures over curves.