A new class of transport distances

TL;DR

This paper introduces a new class of transport distances that interpolate between Wasserstein and Sobolev distances, with unique cost dependencies on densities and velocities, expanding optimal transport theory.

Contribution

It proposes a novel family of distances based on a dynamical formulation, incorporating density-dependent costs, and explores their properties and applications in gradient flows.

Findings

Distances interpolate between Wasserstein and Sobolev metrics.

They depend on both velocity and intermediate densities.

Applications to gradient flow geometry are demonstrated.

Abstract

We introduce a new class of distances between nonnegative Radon measures in Euclidean spaces. They are modeled on the dynamical characterization of the Kantorovich-Rubinstein-Wasserstein distances proposed by Benamou-Brenier and provide a wide family interpolating between the Wasserstein and the homogeneous (dual) 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.