Calculus of Variations and Geometric Measure Theory

D. Forkert - J. Maas - L. Portinale

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

created by portinale on 22 Jun 2021
modified on 05 Oct 2022


Published Paper

Inserted: 22 jun 2021
Last Updated: 5 oct 2022

Journal: SIAM Journal on Mathematical Analysis (SIMA)
Year: 2022

ArXiv: 2008.10962 PDF


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.