---
_id: '11700'
abstract:
- lang: eng
  text: This paper contains two contributions in the study of optimal transport on
    metric graphs. Firstly, we prove a Benamou–Brenier formula for the Wasserstein
    distance, which establishes the equivalence of static and dynamical optimal transport.
    Secondly, in the spirit of Jordan–Kinderlehrer–Otto, we show that McKean–Vlasov
    equations can be formulated as gradient flow of the free energy in the Wasserstein
    space of probability measures. The proofs of these results are based on careful
    regularisation arguments to circumvent some of the difficulties arising in metric
    graphs, namely, branching of geodesics and the failure of semi-convexity of entropy
    functionals in the Wasserstein space.
acknowledgement: "ME acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG),
  Grant SFB 1283/2 2021 – 317210226. DF and JM were supported by the European Research
  Council (ERC) under the European Union’s Horizon 2020 research and innovation programme
  (grant agreement No 716117). JM also acknowledges support by the Austrian Science
  Fund (FWF), Project SFB F65. The work of DM was partially supported by the Deutsche
  Forschungsgemeinschaft\r\n(DFG), Grant 397230547. This article is based upon work
  from COST Action\r\n18232 MAT-DYN-NET, supported by COST (European Cooperation in
  Science\r\nand Technology), www.cost.eu. We wish to thank Martin Burger and Jan-Frederik\r\nPietschmann
  for useful discussions. We are grateful to the anonymous referees for\r\ntheir careful
  reading and useful suggestions."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Matthias
  full_name: Erbar, Matthias
  last_name: Erbar
- 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: Delio
  full_name: Mugnolo, Delio
  last_name: Mugnolo
citation:
  ama: Erbar M, Forkert DL, Maas J, Mugnolo D. Gradient flow formulation of diffusion
    equations in the Wasserstein space over a metric graph. <i>Networks and Heterogeneous
    Media</i>. 2022;17(5):687-717. doi:<a href="https://doi.org/10.3934/nhm.2022023">10.3934/nhm.2022023</a>
  apa: Erbar, M., Forkert, D. L., Maas, J., &#38; Mugnolo, D. (2022). Gradient flow
    formulation of diffusion equations in the Wasserstein space over a metric graph.
    <i>Networks and Heterogeneous Media</i>. American Institute of Mathematical Sciences.
    <a href="https://doi.org/10.3934/nhm.2022023">https://doi.org/10.3934/nhm.2022023</a>
  chicago: Erbar, Matthias, Dominik L Forkert, Jan Maas, and Delio Mugnolo. “Gradient
    Flow Formulation of Diffusion Equations in the Wasserstein Space over a Metric
    Graph.” <i>Networks and Heterogeneous Media</i>. American Institute of Mathematical
    Sciences, 2022. <a href="https://doi.org/10.3934/nhm.2022023">https://doi.org/10.3934/nhm.2022023</a>.
  ieee: M. Erbar, D. L. Forkert, J. Maas, and D. Mugnolo, “Gradient flow formulation
    of diffusion equations in the Wasserstein space over a metric graph,” <i>Networks
    and Heterogeneous Media</i>, vol. 17, no. 5. American Institute of Mathematical
    Sciences, pp. 687–717, 2022.
  ista: Erbar M, Forkert DL, Maas J, Mugnolo D. 2022. Gradient flow formulation of
    diffusion equations in the Wasserstein space over a metric graph. Networks and
    Heterogeneous Media. 17(5), 687–717.
  mla: Erbar, Matthias, et al. “Gradient Flow Formulation of Diffusion Equations in
    the Wasserstein Space over a Metric Graph.” <i>Networks and Heterogeneous Media</i>,
    vol. 17, no. 5, American Institute of Mathematical Sciences, 2022, pp. 687–717,
    doi:<a href="https://doi.org/10.3934/nhm.2022023">10.3934/nhm.2022023</a>.
  short: M. Erbar, D.L. Forkert, J. Maas, D. Mugnolo, Networks and Heterogeneous Media
    17 (2022) 687–717.
date_created: 2022-07-31T22:01:46Z
date_published: 2022-10-01T00:00:00Z
date_updated: 2023-08-03T12:25:49Z
day: '01'
department:
- _id: JaMa
doi: 10.3934/nhm.2022023
ec_funded: 1
external_id:
  arxiv:
  - '2105.05677'
  isi:
  - '000812422100001'
intvolume: '        17'
isi: 1
issue: '5'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2105.05677
month: '10'
oa: 1
oa_version: Preprint
page: 687-717
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
publication: Networks and Heterogeneous Media
publication_identifier:
  eissn:
  - 1556-181X
  issn:
  - 1556-1801
publication_status: published
publisher: American Institute of Mathematical Sciences
quality_controlled: '1'
scopus_import: '1'
status: public
title: Gradient flow formulation of diffusion equations in the Wasserstein space over
  a metric graph
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 17
year: '2022'
...