Optimal transport of stationary point processes: Metric structure, gradient flow and convexity of the specific entropy

TL;DR

This paper develops a geometric and analytical framework for optimal transport of stationary point processes, linking entropy, gradient flows, and convexity to understand the evolution of infinite particle systems.

Contribution

It introduces a new transport distance for stationary random measures, characterizes the evolution of infinite particle systems as gradient flows, and proves convexity properties of the specific entropy.

Findings

Transport distance induces natural interpolation between stationary measures.

Evolution of infinite Brownian particles is a gradient flow of entropy.

Displacement convexity of entropy and a stationary HWI inequality are established.

Abstract

We develop a theory of optimal transport for stationary random measures with a focus on stationary point processes and construct a family of distances on the set of stationary random measures. These induce a natural notion of interpolation between two stationary random measures along a shortest curve connecting them. In the setting of stationary point processes we leverage this transport distance to give a geometric interpretation for the evolution of infinite particle systems with stationary distribution. Namely, we characterise the evolution of infinitely many Brownian motions as the gradient flow of the specific relative entropy w.r.t.~the Poisson point process. Further, we establish displacement convexity of the specific relative entropy along optimal interpolations of point processes and establish an stationary analogue of the HWI inequality, relating specific entropy, transport distance, and a specific relative Fisher information.