{"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2105.05677","open_access":"1"}],"abstract":[{"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.","lang":"eng"}],"department":[{"_id":"JaMa"}],"issue":"5","publication_status":"published","date_published":"2022-10-01T00:00:00Z","date_updated":"2023-08-03T12:25:49Z","month":"10","article_processing_charge":"No","day":"01","type":"journal_article","status":"public","quality_controlled":"1","title":"Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graph","scopus_import":"1","publisher":"American Institute of Mathematical Sciences","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","date_created":"2022-07-31T22:01:46Z","oa_version":"Preprint","oa":1,"_id":"11700","language":[{"iso":"eng"}],"citation":{"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.","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.","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>","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.","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>."},"article_type":"original","ec_funded":1,"publication_identifier":{"issn":["1556-1801"],"eissn":["1556-181X"]},"external_id":{"arxiv":["2105.05677"],"isi":["000812422100001"]},"publication":"Networks and Heterogeneous Media","author":[{"full_name":"Erbar, Matthias","first_name":"Matthias","last_name":"Erbar"},{"last_name":"Forkert","first_name":"Dominik L","id":"35C79D68-F248-11E8-B48F-1D18A9856A87","full_name":"Forkert, Dominik L"},{"orcid":"0000-0002-0845-1338","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","full_name":"Maas, Jan","last_name":"Maas","first_name":"Jan"},{"full_name":"Mugnolo, Delio","first_name":"Delio","last_name":"Mugnolo"}],"doi":"10.3934/nhm.2022023","page":"687-717","project":[{"call_identifier":"H2020","name":"Optimal Transport and Stochastic Dynamics","_id":"256E75B8-B435-11E9-9278-68D0E5697425","grant_number":"716117"},{"_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems","grant_number":"F6504"}],"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.","isi":1,"year":"2022","volume":17,"intvolume":"        17"}