---
_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
arxiv: 1
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. <i>SIAM
    Journal on Mathematical Analysis</i>. 2022;54(4):4297-4333. doi:<a href="https://doi.org/10.1137/21M1410968">10.1137/21M1410968</a>
  apa: Forkert, D. L., Maas, J., &#38; Portinale, L. (2022). Evolutionary $\Gamma$-convergence
    of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions.
    <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial and Applied
    Mathematics. <a href="https://doi.org/10.1137/21M1410968">https://doi.org/10.1137/21M1410968</a>
  chicago: Forkert, Dominik L, Jan Maas, and Lorenzo Portinale. “Evolutionary $\Gamma$-Convergence
    of Entropic Gradient Flow Structures for Fokker-Planck Equations in Multiple Dimensions.”
    <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial and Applied
    Mathematics, 2022. <a href="https://doi.org/10.1137/21M1410968">https://doi.org/10.1137/21M1410968</a>.
  ieee: D. L. Forkert, J. Maas, and L. Portinale, “Evolutionary $\Gamma$-convergence
    of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions,”
    <i>SIAM Journal on Mathematical Analysis</i>, 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.”
    <i>SIAM Journal on Mathematical Analysis</i>, vol. 54, no. 4, Society for Industrial
    and Applied Mathematics, 2022, pp. 4297–333, doi:<a href="https://doi.org/10.1137/21M1410968">10.1137/21M1410968</a>.
  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'
...