Gradient flows of the entropy for jump processes

TL;DR

This paper introduces a novel transportation distance based on Lévy jump kernels, explores its geometric properties, and establishes its connection to the gradient flow of relative entropy for jump processes.

Contribution

It defines a new non-local transportation distance using a Lévy jump kernel and proves its geometric properties and relation to entropy gradient flows.

Findings

Existence of geodesics in the new distance space.

Identification of the semigroup as the gradient flow of entropy.

Convexity of entropy along geodesics.

Abstract

We introduce a new transportation distance between probability measures that is built from a L\'evy jump kernel. It is defined via a non-local variant of the Benamou-Brenier formula. We study geometric and topological properties of this distance, in particular we prove existence of geodesics. For translation invariant jump kernels we identify the semigroup generated by the associated non-local operator as the gradient flow of the relative entropy w.r.t. the new distance and show that the entropy is convex along geodesics.