Abstract
Discretizations of differential equations and optimization problems to point clouds exchange information at a distance. Such discretizations are often close to nonlocal equations which serve as intermediary between the discrete and differential problems in the continuum.
Considering dynamical problems on graphs gave rise to the graph analogue of the Wasserstein distance. We will discuss the nonlocal Wasserstein distance that serves as the intermediary between the graph and the standard Wasserstein distance. In particular we will highlight how the properties of this metric are highly dependent on the kernel chosen and obtain quantitative comparison to the standard Wasserstein metric in appropriate regimes.
Finally we will introduce the nonlocal equations (among them fractional diffusion) that arise as gradient flows with respect to the nonlocal Wasserstein distance.
The talk is based on joint works with Warren and with Esposito, Patacchini, Schlichting.