{"scopus_import":"1","article_number":"23","article_type":"original","publication_status":"published","publisher":"Springer Nature","oa":1,"doi":"10.1007/s00526-025-03193-1","date_created":"2025-12-29T12:06:26Z","arxiv":1,"publication_identifier":{"issn":["0944-2669"],"eissn":["1432-0835"]},"quality_controlled":"1","intvolume":"        65","abstract":[{"text":"We prove the convergence of a modified Jordan–Kinderlehrer–Otto scheme to a solution\r\nto the Fokker–Planck equation in Ω e R^d with general—strictly positive and temporally\r\nconstant—Dirichlet boundary conditions. We work under mild assumptions on the domain,\r\nthe drift, and the initial datum. In the special case where Ω is an interval in R1, we prove\r\nthat such a solution is a gradient flow—curve of maximal slope—within a suitable space of\r\nmeasures, endowed with a modified Wasserstein distance. Our discrete scheme and modified\r\ndistance draw inspiration from contributions by A. Figalli and N. Gigli [J. Math. Pures\r\nAppl. 94, (2010), pp. 107–130], and J. Morales [J. Math. Pures Appl. 112, (2018), pp. 41–88]\r\non an optimal-transport approach to evolution equations with Dirichlet boundary conditions.\r\nSimilarly to these works, we allow the mass to flow from/to the boundary ∂Ω throughout\r\nthe evolution. However, our leading idea is to also keep track of the mass at the boundary\r\nby working with measures defined on the whole closure Ω . The driving functional is a\r\nmodification of the classical relative entropy that also makes use of the information at the\r\nboundary. As an intermediate result, when Ω is an interval in R1, we find a formula for the\r\ndescending slope of this geodesically nonconvex functional.","lang":"eng"}],"PlanS_conform":"1","_id":"20865","title":"Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions","language":[{"iso":"eng"}],"related_material":{"record":[{"relation":"earlier_version","id":"20571","status":"public"}]},"citation":{"ieee":"F. Quattrocchi, “Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions,” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 65, no. 1. Springer Nature, 2026.","mla":"Quattrocchi, Filippo. “Variational Structures for the Fokker-Planck Equation with General Dirichlet Boundary Conditions.” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 65, no. 1, 23, Springer Nature, 2026, doi:<a href=\"https://doi.org/10.1007/s00526-025-03193-1\">10.1007/s00526-025-03193-1</a>.","ama":"Quattrocchi F. Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions. <i>Calculus of Variations and Partial Differential Equations</i>. 2026;65(1). doi:<a href=\"https://doi.org/10.1007/s00526-025-03193-1\">10.1007/s00526-025-03193-1</a>","short":"F. Quattrocchi, Calculus of Variations and Partial Differential Equations 65 (2026).","chicago":"Quattrocchi, Filippo. “Variational Structures for the Fokker-Planck Equation with General Dirichlet Boundary Conditions.” <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature, 2026. <a href=\"https://doi.org/10.1007/s00526-025-03193-1\">https://doi.org/10.1007/s00526-025-03193-1</a>.","apa":"Quattrocchi, F. (2026). Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions. <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00526-025-03193-1\">https://doi.org/10.1007/s00526-025-03193-1</a>","ista":"Quattrocchi F. 2026. Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions. Calculus of Variations and Partial Differential Equations. 65(1), 23."},"file_date_updated":"2026-01-05T12:36:39Z","type":"journal_article","date_published":"2026-01-01T00:00:00Z","issue":"1","oa_version":"Published Version","tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"department":[{"_id":"JaMa"}],"ddc":["510"],"OA_type":"hybrid","article_processing_charge":"Yes (via OA deal)","status":"public","file":[{"success":1,"file_name":"2026_CalculusVariations_Quattrocchi.pdf","checksum":"635370d64abaf444f50f5cca60bba1be","creator":"dernst","date_updated":"2026-01-05T12:36:39Z","file_size":958382,"relation":"main_file","access_level":"open_access","date_created":"2026-01-05T12:36:39Z","content_type":"application/pdf","file_id":"20945"}],"has_accepted_license":"1","date_updated":"2026-01-05T12:37:38Z","volume":65,"day":"01","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","external_id":{"arxiv":["2403.07803"]},"publication":"Calculus of Variations and Partial Differential Equations","corr_author":"1","year":"2026","OA_place":"publisher","month":"01","author":[{"last_name":"Quattrocchi","first_name":"Filippo","orcid":"0009-0000-9773-1931","full_name":"Quattrocchi, Filippo","id":"3ebd6ba8-edfb-11eb-afb5-91a9745ba308"}],"acknowledgement":"The author would like to thank Jan Maas for suggesting this project and for many helpful comments, Antonio Agresti, Lorenzo Dello Schiavo and Julian Fischer for several fruitful discussions, Oliver Tse for pointing out the reference [10], and the anonymous reviewer for carefully reading this manuscript and providing valuable suggestions. He also gratefully acknowledges support from the Austrian Science Fund (FWF) project 10.55776/F65.Open access funding provided by Institute of Science and Technology (IST Austria).","project":[{"name":"Taming Complexity in Partial Differential Systems","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","grant_number":"F6504"}]}