---
OA_place: repository
OA_type: green
_id: '20571'
abstract:
- lang: eng
  text: "We prove the convergence of a modified Jordan--Kinderlehrer--Otto scheme
    to a solution to the Fokker--Planck equation in $\\Omega \\Subset \\mathbb{R}^d$
    with general, positive and temporally constant, Dirichlet boundary conditions.
    We work under mild assumptions on the domain, the drift, and the initial datum.
    \  In the special case where $\\Omega$ is an interval in $\\mathbb{R}^1$, we prove
    that such a solution is a gradient flow -- curve of maximal slope -- within a
    suitable space of measures, endowed with a modified Wasserstein distance.\r\nOur
    discrete scheme and modified distance draw inspiration from contributions by A.
    Figalli and N. Gigli [J. Math. Pures Appl. 94, (2010), pp. 107--130], and J. Morales
    [J. Math. Pures Appl. 112, (2018), pp. 41--88] on an optimal-transport approach
    to evolution equations with Dirichlet boundary conditions. Similarly to these
    works, we allow the mass to flow from/to the boundary $\\partial \\Omega$ throughout
    the evolution. However, our leading idea is to also keep track of the mass at
    the boundary by working with measures defined on the whole closure $\\overline
    \\Omega$. The driving functional is a modification of the classical relative entropy
    that also makes use of the information at the boundary. As an intermediate result,
    when $\\Omega$ is an interval in $\\mathbb{R}^1$, we find a formula for the descending
    slope of this geodesically nonconvex functional. "
acknowledgement: "The author would like to thank Jan Maas for suggesting this project
  and for many helpful\r\ncomments, Antonio Agresti, Lorenzo Dello Schiavo and Julian
  Fischer for several fruitful discussions, and Oliver Tse for pointing out the reference
  [15]. He also gratefully acknowledges support from the Austrian Science Fund (FWF)
  project 10.55776/F65.\r\n"
article_number: '2403.07803'
article_processing_charge: No
arxiv: 1
author:
- first_name: Filippo
  full_name: Quattrocchi, Filippo
  id: 3ebd6ba8-edfb-11eb-afb5-91a9745ba308
  last_name: Quattrocchi
  orcid: 0009-0000-9773-1931
citation:
  ama: Quattrocchi F. Variational structures for the Fokker-Planck equation with general
    Dirichlet boundary conditions. <i>arXiv</i>. doi:<a href="https://doi.org/10.48550/arXiv.2403.07803">10.48550/arXiv.2403.07803</a>
  apa: Quattrocchi, F. (n.d.). Variational structures for the Fokker-Planck equation
    with general Dirichlet boundary conditions. <i>arXiv</i>. <a href="https://doi.org/10.48550/arXiv.2403.07803">https://doi.org/10.48550/arXiv.2403.07803</a>
  chicago: Quattrocchi, Filippo. “Variational Structures for the Fokker-Planck Equation
    with General Dirichlet Boundary Conditions.” <i>ArXiv</i>, n.d. <a href="https://doi.org/10.48550/arXiv.2403.07803">https://doi.org/10.48550/arXiv.2403.07803</a>.
  ieee: F. Quattrocchi, “Variational structures for the Fokker-Planck equation with
    general Dirichlet boundary conditions,” <i>arXiv</i>. .
  ista: Quattrocchi F. Variational structures for the Fokker-Planck equation with
    general Dirichlet boundary conditions. arXiv, 2403.07803.
  mla: Quattrocchi, Filippo. “Variational Structures for the Fokker-Planck Equation
    with General Dirichlet Boundary Conditions.” <i>ArXiv</i>, 2403.07803, doi:<a
    href="https://doi.org/10.48550/arXiv.2403.07803">10.48550/arXiv.2403.07803</a>.
  short: F. Quattrocchi, ArXiv (n.d.).
corr_author: '1'
date_created: 2025-10-28T13:12:56Z
date_published: 2024-04-09T00:00:00Z
date_updated: 2026-05-17T22:30:22Z
day: '09'
department:
- _id: GradSch
- _id: JaMa
doi: 10.48550/arXiv.2403.07803
external_id:
  arxiv:
  - '2403.07803'
keyword:
- gradient flows
- Jordan–Kinderlehrer–Otto scheme
- curves of maximal slope
- optimal transport
- Dirichlet boundary conditions
- Fokker–Planck equation
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2403.07803
month: '04'
oa: 1
oa_version: Preprint
project:
- _id: 260482E2-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: F06504
  name: Taming Complexity in Partial Differential Systems
publication: arXiv
publication_status: draft
related_material:
  record:
  - id: '20865'
    relation: later_version
    status: public
  - id: '20563'
    relation: dissertation_contains
    status: public
status: public
title: Variational structures for the Fokker-Planck equation with general Dirichlet
  boundary conditions
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2024'
...
---
_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.
corr_author: '1'
date_created: 2022-08-07T22:01:59Z
date_published: 2022-07-18T00:00:00Z
date_updated: 2025-04-15T08:31:31Z
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
  grant_number: W1245
  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'
...