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(14:00-15:00 UTC) Speaker: Matthias Erbar (Bielefeld University)

Title: Optimal Transport and Gradient Flows in Discrete or Non-Local Settings

Abstract: Optimal transport has become a powerful tool in the analysis of particle systems and the associated evolution equations. In particular the description as steepest descent or gradient flow in the Wasserstein distance gives insights on their long-time or limiting behavior. Yet many situations of interest fall outside the scope of the classical optimal transport theory.

In the talk I will try to give an overview of some recent developments concerning the construction of different geometries on the space of probability measures tailored to capture the specific features of a given system. I will focus in particular on particles systems on discrete spaces and on the collisional dynamics in the Boltzmann equation.

(15:00-16:00 UTC) Speaker: Jan Maas (IST Austria)
Title: Optimal Transport: from Discrete to Continuous
Abstract: Many stochastic systems can be viewed as gradient flow ('steepest descent') in the space of probability measures, where the driving functional is a relative entropy and the dissipation mechanism is described by a dynamical optimal transport problem. In this talk we focus on these optimal transport problems and describe recent work on the limit passage from discrete to continuous.

Surprisingly, it turns out that discrete transport metrics may fail to converge to the expected limit, even when the associated gradient flows converge. We will illustrate this phenomenon in examples, discuss a geometric criterion for convergence, and present a recent homogenisation result.

This talk is based on joint works with Matthias Erbar, Nicola Gigli, Peter Gladbach, Eva Kopfer, and Lorenzo Portinale.