Variational structure of Fokker-Planck equations with variable mobility

TL;DR

This paper introduces a weighted Wasserstein metric for Fokker-Planck equations with variable mobility, framing them as gradient flows and establishing existence, optimal transport maps, and convergence of a variational scheme.

Contribution

It develops a novel weighted Wasserstein metric for variable mobility Fokker-Planck equations and proves existence of optimal transport maps and convergence of a variational scheme.

Findings

Weighted Wasserstein metric for variable mobility Fokker-Planck equations

Existence of optimal transport maps via Nash-Kuiper theorem

Convergence of time-discrete variational scheme to weak solutions

Abstract

We study Fokker--Planck equations with symmetric, positive definite mobility matrices capturing diffusion in heterogeneous environments. A weighted Wasserstein metric is introduced for which these equations are gradient flows. This metric is shown to emerge from an optimal control problem in the space of probability densities for a class of variable mobility matrices, with the cost function capturing the work dissipated via friction. Using the Nash-Kuiper isometric embedding theorem for Riemannian manifolds, we demonstrate the existence of optimal transport maps. Additionally, we construct a time-discrete variational scheme, establish key properties for the associated minimizing problem, and prove convergence to weak solutions of the associated Fokker-Planck equation.