[{"arxiv":1,"keyword":["gradient flows","Jordan–Kinderlehrer–Otto scheme","curves of maximal slope","optimal transport","Dirichlet boundary conditions","Fokker–Planck equation"],"oa_version":"Preprint","OA_type":"green","title":"Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions","date_created":"2025-10-28T13:12:56Z","publication_status":"draft","corr_author":"1","article_processing_charge":"No","related_material":{"record":[{"relation":"later_version","status":"public","id":"20865"},{"relation":"dissertation_contains","status":"public","id":"20563"}]},"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","oa":1,"doi":"10.48550/arXiv.2403.07803","project":[{"grant_number":"F06504","_id":"260482E2-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Taming Complexity in Partial Differential Systems"}],"department":[{"_id":"GradSch"},{"_id":"JaMa"}],"month":"04","author":[{"full_name":"Quattrocchi, Filippo","last_name":"Quattrocchi","orcid":"0009-0000-9773-1931","first_name":"Filippo","id":"3ebd6ba8-edfb-11eb-afb5-91a9745ba308"}],"year":"2024","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_number":"2403.07803","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2403.07803"}],"date_published":"2024-04-09T00:00:00Z","OA_place":"repository","type":"preprint","_id":"20571","language":[{"iso":"eng"}],"abstract":[{"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. ","lang":"eng"}],"external_id":{"arxiv":["2403.07803"]},"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>","ista":"Quattrocchi F. Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions. arXiv, 2403.07803.","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>.","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.).","ieee":"F. Quattrocchi, “Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions,” <i>arXiv</i>. .","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>"},"day":"09","status":"public","date_updated":"2026-04-27T22:30:15Z","publication":"arXiv"},{"_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."}],"language":[{"iso":"eng"}],"external_id":{"isi":["000889274600001"],"arxiv":["2008.10962"]},"type":"journal_article","quality_controlled":"1","day":"18","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>","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.","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>.","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>.","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.","short":"D.L. Forkert, J. Maas, L. Portinale, SIAM Journal on Mathematical Analysis 54 (2022) 4297–4333.","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>"},"intvolume":"        54","year":"2022","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","date_published":"2022-07-18T00:00:00Z","main_file_link":[{"url":" https://doi.org/10.48550/arXiv.2008.10962","open_access":"1"}],"date_updated":"2025-04-15T08:31:31Z","publication":"SIAM Journal on Mathematical Analysis","isi":1,"page":"4297-4333","status":"public","issue":"4","ec_funded":1,"scopus_import":"1","publication_status":"published","publication_identifier":{"issn":["0036-1410"],"eissn":["1095-7154"]},"arxiv":1,"keyword":["Fokker--Planck equation","gradient flow","evolutionary $\\Gamma$-convergence"],"title":"Evolutionary $\\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions","date_created":"2022-08-07T22:01:59Z","oa_version":"Preprint","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.","oa":1,"doi":"10.1137/21M1410968","author":[{"id":"35C79D68-F248-11E8-B48F-1D18A9856A87","first_name":"Dominik L","last_name":"Forkert","full_name":"Forkert, Dominik L"},{"first_name":"Jan","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-0845-1338","full_name":"Maas, Jan","last_name":"Maas"},{"full_name":"Portinale, Lorenzo","last_name":"Portinale","id":"30AD2CBC-F248-11E8-B48F-1D18A9856A87","first_name":"Lorenzo"}],"volume":54,"project":[{"name":"Optimal Transport and Stochastic Dynamics","grant_number":"716117","_id":"256E75B8-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"name":"Taming Complexity in Partial Differential Systems","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","grant_number":"F6504"},{"_id":"260788DE-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"W1245","name":"Dissipation and dispersion in nonlinear partial differential equations"}],"article_type":"original","month":"07","department":[{"_id":"JaMa"}],"corr_author":"1","article_processing_charge":"No","publisher":"Society for Industrial and Applied Mathematics","related_material":{"record":[{"relation":"earlier_version","id":"10022","status":"public"}]}}]