Abstract
Since the seminal work of Jordan, Kinderlehrer and Otto on Fokker--Planck equations, it is well-known that a broad class of evolution equations can be recast in terms of gradient flows (or "steepest descent") in the space of probability measures, where the driving functional is a relative entropy and underlying geometry is described by an optimal transport problem. Not only has this given mathematicians a deeper understanding of the PDEs, but it has also provided useful approximation results, in both time and in space.
In this talk, we focus on discrete-to-continuum limits of optimal transport problems. We introduce a natural discretization of a broad class of dynamical transport problems and analyze their convergence in the framework of $\Z^d$-periodic graphs in $\R^d$. Surprisingly, it turns out that discrete transport metric mail fail to converge to the expected continuous limit, despite the convergence of the associated gradient flow.
The talk is based on joint works with Peter Gladbach, Eva Kopfer, and Jan Maas.