Evolutionary $\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions

TL;DR

This paper demonstrates the convergence of finite-volume approximations of Fokker-Planck equations in multiple dimensions using Evolutionary $ ext{Gamma}$-convergence, extending previous one-dimensional results and employing variational methods.

Contribution

It generalizes the convergence proof of discrete to continuous Fokker-Planck equations to multiple dimensions via Evolutionary $ ext{Gamma}$-convergence, including arbitrary regular meshes.

Findings

Proves convergence of discrete Fokker-Planck to continuous equations in multiple dimensions.

Introduces a Mosco convergence result for functionals in the discrete-to-continuum limit.

Applicable to arbitrary regular meshes, even when discrete transport distances do not converge to Wasserstein distance.

Abstract

We consider finite-volume approximations of Fokker-Planck equations on bounded convex domains in $\mathbb{R}^d$ and study the corresponding gradient flow structures. We reprove the convergence of the discrete to continuous Fokker-Planck equation via the method of Evolutionary $\Gamma$-convergence, i.e., we pass to the limit at the level of the gradient flow structures, generalising the one-dimensional result obtained by Disser and Liero. The proof is of variational nature and relies on a Mosco convergence result for functionals in the discrete-to-continuum limit that is of independent interest. Our results apply to arbitrary regular meshes, even though the associated discrete transport distances may fail to converge to the Wasserstein distance in this generality.