---
_id: '11739'
abstract:
- lang: eng
text: 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, generalizing 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.
acknowledgement: This work was supported by the European Research Council (ERC) under
the European Union's Horizon 2020 Research and Innovation Programme grant 716117
and by the AustrianScience Fund (FWF) through grants F65 and W1245.
article_processing_charge: No
article_type: original
author:
- first_name: Dominik L
full_name: Forkert, Dominik L
id: 35C79D68-F248-11E8-B48F-1D18A9856A87
last_name: Forkert
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
- first_name: Lorenzo
full_name: Portinale, Lorenzo
id: 30AD2CBC-F248-11E8-B48F-1D18A9856A87
last_name: Portinale
citation:
ama: Forkert DL, Maas J, Portinale L. Evolutionary $\Gamma$-convergence of entropic
gradient flow structures for Fokker-Planck equations in multiple dimensions. SIAM
Journal on Mathematical Analysis. 2022;54(4):4297-4333. doi:10.1137/21M1410968
apa: Forkert, D. L., Maas, J., & Portinale, L. (2022). Evolutionary $\Gamma$-convergence
of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions.
SIAM Journal on Mathematical Analysis. Society for Industrial and Applied
Mathematics. https://doi.org/10.1137/21M1410968
chicago: Forkert, Dominik L, Jan Maas, and Lorenzo Portinale. “Evolutionary $\Gamma$-Convergence
of Entropic Gradient Flow Structures for Fokker-Planck Equations in Multiple Dimensions.”
SIAM Journal on Mathematical Analysis. Society for Industrial and Applied
Mathematics, 2022. https://doi.org/10.1137/21M1410968.
ieee: D. L. Forkert, J. Maas, and L. Portinale, “Evolutionary $\Gamma$-convergence
of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions,”
SIAM Journal on Mathematical Analysis, vol. 54, no. 4. Society for Industrial
and Applied Mathematics, pp. 4297–4333, 2022.
ista: Forkert DL, Maas J, Portinale L. 2022. Evolutionary $\Gamma$-convergence of
entropic gradient flow structures for Fokker-Planck equations in multiple dimensions.
SIAM Journal on Mathematical Analysis. 54(4), 4297–4333.
mla: Forkert, Dominik L., et al. “Evolutionary $\Gamma$-Convergence of Entropic
Gradient Flow Structures for Fokker-Planck Equations in Multiple Dimensions.”
SIAM Journal on Mathematical Analysis, vol. 54, no. 4, Society for Industrial
and Applied Mathematics, 2022, pp. 4297–333, doi:10.1137/21M1410968.
short: D.L. Forkert, J. Maas, L. Portinale, SIAM Journal on Mathematical Analysis
54 (2022) 4297–4333.
date_created: 2022-08-07T22:01:59Z
date_published: 2022-07-18T00:00:00Z
date_updated: 2023-08-03T12:37:21Z
day: '18'
department:
- _id: JaMa
doi: 10.1137/21M1410968
ec_funded: 1
external_id:
arxiv:
- '2008.10962'
isi:
- '000889274600001'
intvolume: ' 54'
isi: 1
issue: '4'
keyword:
- Fokker--Planck equation
- gradient flow
- evolutionary $\Gamma$-convergence
language:
- iso: eng
main_file_link:
- open_access: '1'
url: ' https://doi.org/10.48550/arXiv.2008.10962'
month: '07'
oa: 1
oa_version: Preprint
page: 4297-4333
project:
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '716117'
name: Optimal Transport and Stochastic Dynamics
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
grant_number: F6504
name: Taming Complexity in Partial Differential Systems
- _id: 260788DE-B435-11E9-9278-68D0E5697425
call_identifier: FWF
name: Dissipation and Dispersion in Nonlinear Partial Differential Equations
publication: SIAM Journal on Mathematical Analysis
publication_identifier:
eissn:
- 1095-7154
issn:
- 0036-1410
publication_status: published
publisher: Society for Industrial and Applied Mathematics
quality_controlled: '1'
related_material:
record:
- id: '10022'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Evolutionary $\Gamma$-convergence of entropic gradient flow structures for
Fokker-Planck equations in multiple dimensions
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 54
year: '2022'
...