Matthias Erbar
Title:
Optimal transport and gradient flows for point processes
Abstract
I will describe a theory of optimal transport for stationary random measures with a focus on stationary point processes. We consider two types of distances on stationary probability measures on the configuration space geared towards different dynamics of point configurations. The first one is based on continuous displacement of points and leads to a two layer optimal transport problem with a probabilistic constraint. It gives rise to a natural notion of displacement interpolation between point processes and provides e.g. a characterisation of the evolution of infinitely many (interacting) Brownian motions as the gradient flow of the specific free energy. The second notion is based on a birth-death dynamic of points and is modelled on the Benamou-Brenier formula. In a perturbative regime around the Poisson process we obtain an EVI gradient flow characterisation of the associated Glauber dynamics. This is based on joint work with Martin Huesmann, Jonas Jalowy, Bastian Müller, and Hanna Stange.