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

Matthias Erbar, Martin Huesmann, Jonas Jalowy, Bastian Müller

Published 2023-04-21Version 1

We develop a theory of optimal transport for stationary random measures with a particular focus on stationary point processes. This provides us with a notion of geodesic distance between distributions of stationary random measures and induces a natural displacement interpolation between 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 process. Further, we establish displacement convexity of the specific relative entropy along optimal interpolations of point processes.