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

preprint Evolutionary Γ-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions draft Dominik L Forkert author 35C79D68-F248-11E8-B48F-1D18A9856A87 Jan Maas author 4C5696CE-F248-11E8-B48F-1D18A9856A870000-0002-0845-1338 Lorenzo Portinale author 30AD2CBC-F248-11E8-B48F-1D18A9856A87 JaMa department Optimal Transport and Stochastic Dynamics project Taming Complexity in Partial Differential Systems project We consider finite-volume approximations of Fokker-Planck equations on bounded convex domains in 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 Γ-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. 2020 eng arXiv 2008.1096210.48550/arXiv.2008.10962 https://research-explorer.ista.ac.at/record/11739 https://research-explorer.ista.ac.at/record/10030 D.L. Forkert, J. Maas, L. Portinale, ArXiv (n.d.). Forkert, Dominik L., et al. “Evolutionary Γ-Convergence of Entropic Gradient Flow Structures for Fokker-Planck Equations in Multiple Dimensions.” ArXiv, 2008.10962, doi:<a href="https://doi.org/10.48550/arXiv.2008.10962">10.48550/arXiv.2008.10962</a>. Forkert, Dominik L, Jan Maas, and Lorenzo Portinale. “Evolutionary Γ-Convergence of Entropic Gradient Flow Structures for Fokker-Planck Equations in Multiple Dimensions.” ArXiv, n.d. <a href="https://doi.org/10.48550/arXiv.2008.10962">https://doi.org/10.48550/arXiv.2008.10962</a>. Forkert DL, Maas J, Portinale L. Evolutionary Γ-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions. arXiv, 2008.10962. Forkert, D. L., Maas, J., & Portinale, L. (n.d.). Evolutionary Γ-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions. arXiv. <a href="https://doi.org/10.48550/arXiv.2008.10962">https://doi.org/10.48550/arXiv.2008.10962</a> D. L. Forkert, J. Maas, and L. Portinale, “Evolutionary Γ-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions,” arXiv. . Forkert DL, Maas J, Portinale L. Evolutionary Γ-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions. arXiv. doi:<a href="https://doi.org/10.48550/arXiv.2008.10962">10.48550/arXiv.2008.10962</a> 100222021-09-17T10:57:27Z2025-05-15T10:54:10Z