Nonlocal Wasserstein Distance: Metric and Asymptotic Properties

TL;DR

This paper introduces a nonlocal version of the Wasserstein distance using jump processes, characterizes its properties, and compares it quantitatively to the classical local Wasserstein distance.

Contribution

It extends the classical Wasserstein framework to nonlocal jump processes, providing new insights into its properties and topological implications.

Findings

Characterized basic properties of nonlocal Wasserstein distances

Established conditions on kernels for weak or strong topology

Quantitative comparison between nonlocal and local Wasserstein distances

Abstract

The seminal result of Benamou and Brenier provides a characterization of the Wasserstein distance as the path of the minimal action in the space of probability measures, where paths are solutions of the continuity equation and the action is the kinetic energy. Here we consider a fundamental modification of the framework where the paths are solutions of nonlocal (jump) continuity equations and the action is a nonlocal kinetic energy. The resulting nonlocal Wasserstein distances are relevant to fractional diffusions and Wasserstein distances on graphs. We characterize the basic properties of the distance and obtain sharp conditions on the (jump) kernel specifying the nonlocal transport that determine whether the topology metrized is the weak or the strong topology. A key result of the paper are the quantitative comparisons between the nonlocal and local Wasserstein distance.