{"oa":1,"external_id":{"arxiv":["2502.15665"]},"corr_author":"1","publication_status":"draft","ec_funded":1,"date_published":"2025-08-10T00:00:00Z","department":[{"_id":"GradSch"},{"_id":"JaMa"}],"OA_place":"repository","keyword":["optimal transport","kinetic theory","second-order discrepancy","Vlasov equation","Wasserstein splines."],"status":"public","type":"preprint","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_number":"2502.15665","day":"10","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2502.15665","open_access":"1"}],"oa_version":"Preprint","date_updated":"2025-11-25T15:27:46Z","acknowledgement":"This work was partially inspired by an unpublished note from 2014 by Guillaume Carlier,\r\nJean Dolbeault, and Bruno Nazaret. GB deeply thanks Jean Dolbeault for proposing\r\nthis problem to him, guiding him into the subject, and sharing the aforementioned note.\r\nWe are grateful to Karthik Elamvazhuthi for making us aware of the work [20].\r\nThe work of GB has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement\r\nNo 101034413.\r\nJM and FQ gratefully acknowledge support from the Austrian Science Fund (FWF)\r\nproject 10.55776/F65.","citation":{"ama":"Brigati G, Maas J, Quattrocchi F. Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between probability measures. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2502.15665\">10.48550/arXiv.2502.15665</a>","ieee":"G. Brigati, J. Maas, and F. Quattrocchi, “Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between probability measures,” <i>arXiv</i>. .","ista":"Brigati G, Maas J, Quattrocchi F. Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between probability measures. arXiv, 2502.15665.","apa":"Brigati, G., Maas, J., &#38; Quattrocchi, F. (n.d.). Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between probability measures. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2502.15665\">https://doi.org/10.48550/arXiv.2502.15665</a>","chicago":"Brigati, Giovanni, Jan Maas, and Filippo Quattrocchi. “Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-Order Discrepancies between Probability Measures.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2502.15665\">https://doi.org/10.48550/arXiv.2502.15665</a>.","mla":"Brigati, Giovanni, et al. “Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-Order Discrepancies between Probability Measures.” <i>ArXiv</i>, 2502.15665, doi:<a href=\"https://doi.org/10.48550/arXiv.2502.15665\">10.48550/arXiv.2502.15665</a>.","short":"G. Brigati, J. Maas, F. Quattrocchi, ArXiv (n.d.)."},"abstract":[{"lang":"eng","text":"This is the first part of a general description in terms of mass transport for time-evolving interacting particles systems, at a mesoscopic level. Beyond kinetic theory, our framework naturally applies in biology, computer vision, and engineering. The central object of our study is a new discrepancy d between two probability distributions in position and velocity states, which is reminiscent of the 2-Wasserstein distance, but of second-order nature. We construct d in two steps. First, we optimise over transport plans. The cost function is given by the minimal acceleration between two coupled states on a fixed time horizon T. Second, we further optimise over the time horizon T > 0. We prove the existence of optimal transport plans and maps, and study two time-continuous characterisations of d. One is given in terms of dynamical transport plans. The other one -- in the spirit of the Benamou--Brenier formula -- is formulated as the minimisation of an action of the acceleration field, constrained by Vlasov's equations. Equivalence of static and dynamical formulations of d holds true. While part of this result can be derived from recent, parallel developments in optimal control between measures, we give an original proof relying on two new ingredients: Galilean regularisation of Vlasov's equations and a kinetic Monge--Mather shortening principle. Finally, we establish a first-order differential calculus in the geometry induced by d, and identify solutions to Vlasov's equations with curves of measures satisfying a certain d-absolute continuity condition. One consequence is an explicit formula for the d-derivative of such curves."}],"author":[{"id":"63ff57e8-1fbb-11ee-88f2-f558ffc59cf1","full_name":"Brigati, Giovanni","last_name":"Brigati","first_name":"Giovanni"},{"full_name":"Maas, Jan","last_name":"Maas","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","first_name":"Jan","orcid":"0000-0002-0845-1338"},{"orcid":"0009-0000-9773-1931","first_name":"Filippo","id":"3ebd6ba8-edfb-11eb-afb5-91a9745ba308","last_name":"Quattrocchi","full_name":"Quattrocchi, Filippo"}],"month":"08","title":"Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between probability measures","publication":"arXiv","related_material":{"record":[{"relation":"dissertation_contains","id":"20563","status":"for_moderation"}]},"_id":"20569","doi":"10.48550/arXiv.2502.15665","date_created":"2025-10-28T13:12:08Z","project":[{"name":"Taming Complexity in Partial Differential Systems","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","grant_number":"F6504"},{"call_identifier":"H2020","name":"IST-BRIDGE: International postdoctoral program","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","grant_number":"101034413"}],"article_processing_charge":"No","OA_type":"green","year":"2025","language":[{"iso":"eng"}],"arxiv":1}