[{"quality_controlled":"1","type":"journal_article","external_id":{"arxiv":["2504.08658"]},"_id":"21018","language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"In this paper, we review recent results on stability and instability in logarithmic Sobolev inequalities, with a particular emphasis on strong norms. We consider several versions of these inequalities on the Euclidean space, for the Lebesgue and the Gaussian measures, and discuss their differences in terms of moments and stability. We give new and direct proofs, as well as examples and discuss the stability of a logarithmic uncertainty principle. Although we do not cover all aspects of the topic, we hope to contribute to establishing the state of the art."}],"day":"08","citation":{"ista":"Brigati G, Dolbeault J, Simonov N. 2026. Logarithmic Sobolev Inequalities: A review on stability and instability results. La Matematica. 5, 5.","ama":"Brigati G, Dolbeault J, Simonov N. Logarithmic Sobolev Inequalities: A review on stability and instability results. <i>La Matematica</i>. 2026;5. doi:<a href=\"https://doi.org/10.1007/s44007-025-00180-y\">10.1007/s44007-025-00180-y</a>","chicago":"Brigati, Giovanni, Jean Dolbeault, and Nikita Simonov. “Logarithmic Sobolev Inequalities: A Review on Stability and Instability Results.” <i>La Matematica</i>. Springer Nature, 2026. <a href=\"https://doi.org/10.1007/s44007-025-00180-y\">https://doi.org/10.1007/s44007-025-00180-y</a>.","short":"G. Brigati, J. Dolbeault, N. Simonov, La Matematica 5 (2026).","ieee":"G. Brigati, J. Dolbeault, and N. Simonov, “Logarithmic Sobolev Inequalities: A review on stability and instability results,” <i>La Matematica</i>, vol. 5. Springer Nature, 2026.","mla":"Brigati, Giovanni, et al. “Logarithmic Sobolev Inequalities: A Review on Stability and Instability Results.” <i>La Matematica</i>, vol. 5, 5, Springer Nature, 2026, doi:<a href=\"https://doi.org/10.1007/s44007-025-00180-y\">10.1007/s44007-025-00180-y</a>.","apa":"Brigati, G., Dolbeault, J., &#38; Simonov, N. (2026). Logarithmic Sobolev Inequalities: A review on stability and instability results. <i>La Matematica</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s44007-025-00180-y\">https://doi.org/10.1007/s44007-025-00180-y</a>"},"has_accepted_license":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","ddc":["510"],"year":"2026","intvolume":"         5","article_number":"5","OA_place":"publisher","date_published":"2026-01-08T00:00:00Z","file_date_updated":"2026-01-21T07:45:03Z","publication":"La Matematica","date_updated":"2026-01-21T07:48:28Z","status":"public","PlanS_conform":"1","scopus_import":"1","ec_funded":1,"publication_status":"published","file":[{"content_type":"application/pdf","success":1,"file_size":4992025,"date_created":"2026-01-21T07:45:03Z","access_level":"open_access","date_updated":"2026-01-21T07:45:03Z","file_id":"21025","relation":"main_file","creator":"dernst","checksum":"0702d8397f216555b1d5286e5d77f09c","file_name":"2026_LaMatematica_Brigati.pdf"}],"arxiv":1,"publication_identifier":{"issn":["2730-9657"]},"OA_type":"hybrid","oa_version":"Published Version","date_created":"2026-01-20T10:14:55Z","title":"Logarithmic Sobolev Inequalities: A review on stability and instability results","doi":"10.1007/s44007-025-00180-y","oa":1,"acknowledgement":"This work has been supported by the Project Conviviality (ANR-23-CE40–0003) of the French National Research Agency. G.B. has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 101034413. The authors thank a referee for a careful reading and suggestions which result in a significant improvement of the manuscript. Open access funding provided by Institute of Science and Technology (IST Austria). The work of GB has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 101034413. This work has been supported by the Project Conviviality (ANR-23-CE40–0003) of the French National Research Agency.","month":"01","department":[{"_id":"JaMa"}],"project":[{"name":"IST-BRIDGE: International postdoctoral program","call_identifier":"H2020","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","grant_number":"101034413"}],"article_type":"original","author":[{"first_name":"Giovanni","id":"63ff57e8-1fbb-11ee-88f2-f558ffc59cf1","last_name":"Brigati","full_name":"Brigati, Giovanni"},{"first_name":"Jean","last_name":"Dolbeault","full_name":"Dolbeault, Jean"},{"last_name":"Simonov","full_name":"Simonov, Nikita","first_name":"Nikita"}],"volume":5,"publisher":"Springer Nature","article_processing_charge":"Yes (via OA deal)","corr_author":"1","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)"}},{"publication_status":"epub_ahead","arxiv":1,"publication_identifier":{"eissn":["1937-5077"],"issn":["1937-5093"]},"oa_version":"Preprint","OA_type":"green","date_created":"2026-02-01T23:01:43Z","title":"Hypocoercivity meets lifts","acknowledgement":"We would like to thank Andreas Eberle and Gabriel Stoltz for many helpful discussions. GB\r\nhas received funding from the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 101034413. FL wurde gefördert durch die Deutsche Forschungsgemeinschaft (DFG) im Rahmen der Exzellenzstrategie des Bundes und der Länder – GZ2047/1, Projekt-ID 390685813. LW is supported by the National Science Foundation via grant DMS-2407166. He is also indebted to the Mathematical Sciences department at Carnegie Mellon University for partly supporting his visit to Europe in July 2024. Part of this work was completed when GB and LW were visiting the Institute for Applied Mathematics in Bonn. GB and LW would like to thank IAM for their hospitality.","oa":1,"doi":"10.3934/krm.2025020","article_type":"original","project":[{"name":"IST-BRIDGE: International postdoctoral program","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","call_identifier":"H2020","grant_number":"101034413"}],"department":[{"_id":"JaMa"}],"month":"02","author":[{"id":"63ff57e8-1fbb-11ee-88f2-f558ffc59cf1","first_name":"Giovanni","full_name":"Brigati, Giovanni","last_name":"Brigati"},{"first_name":"Francis","full_name":"Lörler, Francis","last_name":"Lörler"},{"first_name":"Lihan","last_name":"Wang","full_name":"Wang, Lihan"}],"volume":20,"publisher":"American Institute of Mathematical Sciences","article_processing_charge":"No","type":"journal_article","quality_controlled":"1","_id":"21132","language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"We unify the variational hypocoercivity framework established by D. Albritton, S. Armstrong, J.-C. Mourrat, and M. Novack [2], with the notion of second-order lifts of reversible diffusion processes, recently introduced by A. Eberle and the second author [30]. We give an abstract, yet fully constructive, presentation of the theory, so that it can be applied to a large class of linear kinetic equations. As this hypocoercivity technique does not twist the reference norm, we can recover accurate and sharp convergence rates in various models. Among those, adaptive Langevin dynamics (ALD) is discussed in full detail and we show that for near-quadratic potentials, with suitable choices of parameters, it is a near-optimal second-order lift of the overdamped Langevin dynamics. As a further consequence, we observe that the Generalised Langevin Equation (GLE) is also a second-order lift, as the standard (kinetic) Langevin dynamics are, of the overdamped Langevin dynamics. Then, convergence of (GLE) cannot exceed ballistic speed, i.e. the square root of the rate of the overdamped regime. We illustrate this phenomenon with explicit computations in a benchmark Gaussian case."}],"external_id":{"arxiv":["2412.10890"]},"citation":{"mla":"Brigati, Giovanni, et al. “Hypocoercivity Meets Lifts.” <i>Kinetic and Related Models</i>, vol. 20, American Institute of Mathematical Sciences, 2026, pp. 34–55, doi:<a href=\"https://doi.org/10.3934/krm.2025020\">10.3934/krm.2025020</a>.","short":"G. Brigati, F. Lörler, L. Wang, Kinetic and Related Models 20 (2026) 34–55.","ieee":"G. Brigati, F. Lörler, and L. Wang, “Hypocoercivity meets lifts,” <i>Kinetic and Related Models</i>, vol. 20. American Institute of Mathematical Sciences, pp. 34–55, 2026.","apa":"Brigati, G., Lörler, F., &#38; Wang, L. (2026). Hypocoercivity meets lifts. <i>Kinetic and Related Models</i>. American Institute of Mathematical Sciences. <a href=\"https://doi.org/10.3934/krm.2025020\">https://doi.org/10.3934/krm.2025020</a>","ama":"Brigati G, Lörler F, Wang L. Hypocoercivity meets lifts. <i>Kinetic and Related Models</i>. 2026;20:34-55. doi:<a href=\"https://doi.org/10.3934/krm.2025020\">10.3934/krm.2025020</a>","ista":"Brigati G, Lörler F, Wang L. 2026. Hypocoercivity meets lifts. Kinetic and Related Models. 20, 34–55.","chicago":"Brigati, Giovanni, Francis Lörler, and Lihan Wang. “Hypocoercivity Meets Lifts.” <i>Kinetic and Related Models</i>. American Institute of Mathematical Sciences, 2026. <a href=\"https://doi.org/10.3934/krm.2025020\">https://doi.org/10.3934/krm.2025020</a>."},"day":"01","year":"2026","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","intvolume":"        20","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2412.10890"}],"date_published":"2026-02-01T00:00:00Z","OA_place":"repository","date_updated":"2026-02-16T10:02:47Z","publication":"Kinetic and Related Models","page":"34-55","status":"public","scopus_import":"1","ec_funded":1},{"OA_place":"repository","date_published":"2026-03-14T00:00:00Z","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2507.11387","open_access":"1"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2026","citation":{"chicago":"Auricchio, Gennaro, Giovanni Brigati, Paolo Giudici, and Giuseppe Toscani. “From Kinetic Theory to AI: A Rediscovery of High-Dimensional Divergences and Their Properties.” <i>Mathematical Models and Methods in Applied Sciences</i>. World Scientific Publishing, 2026. <a href=\"https://doi.org/10.1142/S0218202526410010\">https://doi.org/10.1142/S0218202526410010</a>.","ista":"Auricchio G, Brigati G, Giudici P, Toscani G. 2026. From kinetic theory to AI: A rediscovery of high-dimensional divergences and their properties. Mathematical Models and Methods in Applied Sciences.","ama":"Auricchio G, Brigati G, Giudici P, Toscani G. From kinetic theory to AI: A rediscovery of high-dimensional divergences and their properties. <i>Mathematical Models and Methods in Applied Sciences</i>. 2026. doi:<a href=\"https://doi.org/10.1142/S0218202526410010\">10.1142/S0218202526410010</a>","apa":"Auricchio, G., Brigati, G., Giudici, P., &#38; Toscani, G. (2026). From kinetic theory to AI: A rediscovery of high-dimensional divergences and their properties. <i>Mathematical Models and Methods in Applied Sciences</i>. World Scientific Publishing. <a href=\"https://doi.org/10.1142/S0218202526410010\">https://doi.org/10.1142/S0218202526410010</a>","ieee":"G. Auricchio, G. Brigati, P. Giudici, and G. Toscani, “From kinetic theory to AI: A rediscovery of high-dimensional divergences and their properties,” <i>Mathematical Models and Methods in Applied Sciences</i>. World Scientific Publishing, 2026.","short":"G. Auricchio, G. Brigati, P. Giudici, G. Toscani, Mathematical Models and Methods in Applied Sciences (2026).","mla":"Auricchio, Gennaro, et al. “From Kinetic Theory to AI: A Rediscovery of High-Dimensional Divergences and Their Properties.” <i>Mathematical Models and Methods in Applied Sciences</i>, World Scientific Publishing, 2026, doi:<a href=\"https://doi.org/10.1142/S0218202526410010\">10.1142/S0218202526410010</a>."},"day":"14","external_id":{"arxiv":["2507.11387"]},"language":[{"iso":"eng"}],"_id":"21504","abstract":[{"lang":"eng","text":"Selecting an appropriate divergence measure is a critical aspect of machine learning, as it directly impacts model performance. Among the most widely used, we find the Kullback–Leibler (KL) divergence, originally introduced in kinetic theory as a measure of relative entropy between probability distributions. Just as in machine learning, the ability to quantify the proximity of probability distributions plays a central role in kinetic theory. In this paper, we present a comparative review of divergence measures rooted in kinetic theory, highlighting their theoretical foundations and exploring their potential applications in machine learning and artificial intelligence."}],"quality_controlled":"1","type":"journal_article","ec_funded":1,"scopus_import":"1","status":"public","date_updated":"2026-03-30T06:56:35Z","publication":"Mathematical Models and Methods in Applied Sciences","title":"From kinetic theory to AI: A rediscovery of high-dimensional divergences and their properties","date_created":"2026-03-29T22:07:08Z","OA_type":"green","oa_version":"Preprint","publication_identifier":{"eissn":["1793-6314"],"issn":["0218-2025"]},"arxiv":1,"publication_status":"epub_ahead","article_processing_charge":"No","publisher":"World Scientific Publishing","author":[{"full_name":"Auricchio, Gennaro","last_name":"Auricchio","first_name":"Gennaro"},{"first_name":"Giovanni","id":"63ff57e8-1fbb-11ee-88f2-f558ffc59cf1","full_name":"Brigati, Giovanni","last_name":"Brigati"},{"first_name":"Paolo","last_name":"Giudici","full_name":"Giudici, Paolo"},{"last_name":"Toscani","full_name":"Toscani, Giuseppe","first_name":"Giuseppe"}],"month":"03","department":[{"_id":"JaMa"}],"article_type":"original","project":[{"name":"IST-BRIDGE: International postdoctoral program","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","call_identifier":"H2020","grant_number":"101034413"}],"oa":1,"doi":"10.1142/S0218202526410010","acknowledgement":"This work has been written within the activities of GNCS and GNFM groups of INdAM (Italian\r\nNational Institute of High Mathematics). G.B. has been funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 101034413. P.G. has been funded by the European Union - NextGenerationEU, in the framework of the GRINSGrowing Resilient, INclusive and Sustainable (GRINS PE00000018)."},{"publication_status":"published","publication_identifier":{"eissn":["1095-7154"],"issn":["0036-1410"]},"arxiv":1,"title":"How to construct explicit decay rates for kinetic Fokker–Planck equations?","date_created":"2025-08-10T22:01:29Z","oa_version":"Preprint","OA_type":"green","acknowledgement":"The first author was funded by the European Union's Horizon 2020 research andinnovation program under the Marie Sklodowska-Curie grant agreements 754362 and 101034413,and partially by Project EFI (ANR-17-CE40-0030) of the French National Research Agency (ANR).The work of the second author was partially funded by the European Research Council (ERC) underthe European Union's Horizon 2020 research and innovation programme (grant agreement 810367),and by the Agence Nationale de la Recherche under grants ANR-19-CE40-0010 (QuAMProcs) andANR-21-CE40-0006 (SINEQ).","oa":1,"doi":"10.1137/24M1700351","volume":57,"author":[{"id":"63ff57e8-1fbb-11ee-88f2-f558ffc59cf1","first_name":"Giovanni","last_name":"Brigati","full_name":"Brigati, Giovanni"},{"last_name":"Stoltz","full_name":"Stoltz, Gabriel","first_name":"Gabriel"}],"project":[{"name":"IST-BRIDGE: International postdoctoral program","call_identifier":"H2020","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","grant_number":"101034413"}],"article_type":"original","department":[{"_id":"JaMa"}],"month":"08","corr_author":"1","article_processing_charge":"No","publisher":"Society for Industrial and Applied Mathematics","_id":"20155","language":[{"iso":"eng"}],"abstract":[{"text":"We study time averages for the norm of solutions to kinetic Fokker–Planck equations associated with general Hamiltonians. We provide fully explicit and constructive decay estimates for systems subject to a confining potential, allowing fat-tail, subexponential and (super-)exponential local equilibria, which also include the classic Maxwellian case. The key step in our estimates is a modified Poincaré inequality, obtained via a Lions–Poincaré inequality and an averaging lemma.","lang":"eng"}],"external_id":{"arxiv":["2302.14506"],"isi":["001550830900006"]},"type":"journal_article","quality_controlled":"1","citation":{"chicago":"Brigati, Giovanni, and Gabriel Stoltz. “How to Construct Explicit Decay Rates for Kinetic Fokker–Planck Equations?” <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial and Applied Mathematics, 2025. <a href=\"https://doi.org/10.1137/24M1700351\">https://doi.org/10.1137/24M1700351</a>.","ama":"Brigati G, Stoltz G. How to construct explicit decay rates for kinetic Fokker–Planck equations? <i>SIAM Journal on Mathematical Analysis</i>. 2025;57(4):3587-3622. doi:<a href=\"https://doi.org/10.1137/24M1700351\">10.1137/24M1700351</a>","ista":"Brigati G, Stoltz G. 2025. How to construct explicit decay rates for kinetic Fokker–Planck equations? SIAM Journal on Mathematical Analysis. 57(4), 3587–3622.","apa":"Brigati, G., &#38; Stoltz, G. (2025). How to construct explicit decay rates for kinetic Fokker–Planck equations? <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial and Applied Mathematics. <a href=\"https://doi.org/10.1137/24M1700351\">https://doi.org/10.1137/24M1700351</a>","mla":"Brigati, Giovanni, and Gabriel Stoltz. “How to Construct Explicit Decay Rates for Kinetic Fokker–Planck Equations?” <i>SIAM Journal on Mathematical Analysis</i>, vol. 57, no. 4, Society for Industrial and Applied Mathematics, 2025, pp. 3587–622, doi:<a href=\"https://doi.org/10.1137/24M1700351\">10.1137/24M1700351</a>.","short":"G. Brigati, G. Stoltz, SIAM Journal on Mathematical Analysis 57 (2025) 3587–3622.","ieee":"G. Brigati and G. Stoltz, “How to construct explicit decay rates for kinetic Fokker–Planck equations?,” <i>SIAM Journal on Mathematical Analysis</i>, vol. 57, no. 4. Society for Industrial and Applied Mathematics, pp. 3587–3622, 2025."},"day":"01","intvolume":"        57","year":"2025","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2025-08-01T00:00:00Z","OA_place":"repository","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2302.14506","open_access":"1"}],"publication":"SIAM Journal on Mathematical Analysis","date_updated":"2025-11-05T13:51:40Z","isi":1,"page":"3587-3622","status":"public","ec_funded":1,"issue":"4","scopus_import":"1"},{"isi":1,"publication":"Electronic Communications in Probability","date_updated":"2025-12-01T15:08:54Z","file_date_updated":"2025-11-04T07:34:05Z","scopus_import":"1","DOAJ_listed":"1","ec_funded":1,"status":"public","PlanS_conform":"1","has_accepted_license":"1","day":"25","citation":{"chicago":"Brigati, Giovanni, and Francesco Pedrotti. “Heat Flow, Log-Concavity, and Lipschitz Transport Maps.” <i>Electronic Communications in Probability</i>. Institute of Mathematical Statistics, 2025. <a href=\"https://doi.org/10.1214/25-ECP717\">https://doi.org/10.1214/25-ECP717</a>.","ama":"Brigati G, Pedrotti F. Heat flow, log-concavity, and Lipschitz transport maps. <i>Electronic Communications in Probability</i>. 2025;30. doi:<a href=\"https://doi.org/10.1214/25-ECP717\">10.1214/25-ECP717</a>","ista":"Brigati G, Pedrotti F. 2025. Heat flow, log-concavity, and Lipschitz transport maps. Electronic Communications in Probability. 30, 71.","apa":"Brigati, G., &#38; Pedrotti, F. (2025). Heat flow, log-concavity, and Lipschitz transport maps. <i>Electronic Communications in Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/25-ECP717\">https://doi.org/10.1214/25-ECP717</a>","mla":"Brigati, Giovanni, and Francesco Pedrotti. “Heat Flow, Log-Concavity, and Lipschitz Transport Maps.” <i>Electronic Communications in Probability</i>, vol. 30, 71, Institute of Mathematical Statistics, 2025, doi:<a href=\"https://doi.org/10.1214/25-ECP717\">10.1214/25-ECP717</a>.","ieee":"G. Brigati and F. Pedrotti, “Heat flow, log-concavity, and Lipschitz transport maps,” <i>Electronic Communications in Probability</i>, vol. 30. Institute of Mathematical Statistics, 2025.","short":"G. Brigati, F. Pedrotti, Electronic Communications in Probability 30 (2025)."},"type":"journal_article","quality_controlled":"1","language":[{"iso":"eng"}],"_id":"20591","abstract":[{"lang":"eng","text":"In this paper we derive estimates for the Hessian of the logarithm (log-Hessian) for solutions to the heat equation. For initial data in the form of log-Lipschitz perturbation of strongly log-concave measures, the log-Hessian admits an explicit, uniform (in space) lower bound. This yields a new estimate for the Lipschitz constant of a transport map pushing forward the standard Gaussian to a measure in this class. On the other hand, we show that assuming only fast decay of the tails of the initial datum does not suffice to guarantee uniform log-Hessian upper bounds."}],"external_id":{"arxiv":["2404.15205"],"isi":["001611557000018"]},"date_published":"2025-09-25T00:00:00Z","OA_place":"publisher","year":"2025","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","ddc":["500"],"intvolume":"        30","article_number":"71","article_type":"original","project":[{"grant_number":"F6504","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems"},{"_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","call_identifier":"H2020","grant_number":"101034413","name":"IST-BRIDGE: International postdoctoral program"}],"month":"09","department":[{"_id":"JaMa"}],"volume":30,"author":[{"full_name":"Brigati, Giovanni","last_name":"Brigati","first_name":"Giovanni","id":"63ff57e8-1fbb-11ee-88f2-f558ffc59cf1"},{"first_name":"Francesco","id":"d3ac8ac6-dc8d-11ea-abe3-e2a9628c4c3c","full_name":"Pedrotti, Francesco","last_name":"Pedrotti"}],"acknowledgement":"This research was funded in part by the Austrian Science Fund (FWF) project 10.55776/F65 and by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 101034413. The authors thank Professors Jean Dolbeault, Jan Maas, and Nikita Simonov for many useful comments, and Professors Kazuhiro Ishige, Asuka Takatsu, and Yair Shenfeld for inspiring interactions.","oa":1,"doi":"10.1214/25-ECP717","related_material":{"record":[{"relation":"earlier_version","status":"public","id":"17353"}]},"publisher":"Institute of Mathematical Statistics","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)"},"corr_author":"1","article_processing_charge":"Yes","file":[{"relation":"main_file","creator":"dernst","date_updated":"2025-11-04T07:34:05Z","file_id":"20596","checksum":"67858edbd74658fe38955fa1216f2f18","file_name":"2025_ElectronJourProbab_Brigati.pdf","success":1,"content_type":"application/pdf","date_created":"2025-11-04T07:34:05Z","file_size":278078,"access_level":"open_access"}],"publication_status":"published","oa_version":"Published Version","OA_type":"gold","title":"Heat flow, log-concavity, and Lipschitz transport maps","date_created":"2025-11-02T23:01:35Z","publication_identifier":{"eissn":["1083-589X"]},"arxiv":1},{"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2411.01052","open_access":"1"}],"OA_place":"repository","date_published":"2025-05-01T00:00:00Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","year":"2025","intvolume":"        35","citation":{"ieee":"G. Auricchio, G. Brigati, P. Giudici, and G. Toscani, “Multivariate Gini-type discrepancies,” <i>Mathematical Models and Methods in Applied Sciences</i>, vol. 35, no. 5. World Scientific Publishing, pp. 1267–1296, 2025.","short":"G. Auricchio, G. Brigati, P. Giudici, G. Toscani, Mathematical Models and Methods in Applied Sciences 35 (2025) 1267–1296.","mla":"Auricchio, Gennaro, et al. “Multivariate Gini-Type Discrepancies.” <i>Mathematical Models and Methods in Applied Sciences</i>, vol. 35, no. 5, World Scientific Publishing, 2025, pp. 1267–96, doi:<a href=\"https://doi.org/10.1142/s0218202525500174\">10.1142/s0218202525500174</a>.","apa":"Auricchio, G., Brigati, G., Giudici, P., &#38; Toscani, G. (2025). Multivariate Gini-type discrepancies. <i>Mathematical Models and Methods in Applied Sciences</i>. World Scientific Publishing. <a href=\"https://doi.org/10.1142/s0218202525500174\">https://doi.org/10.1142/s0218202525500174</a>","ista":"Auricchio G, Brigati G, Giudici P, Toscani G. 2025. Multivariate Gini-type discrepancies. Mathematical Models and Methods in Applied Sciences. 35(5), 1267–1296.","ama":"Auricchio G, Brigati G, Giudici P, Toscani G. Multivariate Gini-type discrepancies. <i>Mathematical Models and Methods in Applied Sciences</i>. 2025;35(5):1267-1296. doi:<a href=\"https://doi.org/10.1142/s0218202525500174\">10.1142/s0218202525500174</a>","chicago":"Auricchio, Gennaro, Giovanni Brigati, Paolo Giudici, and Giuseppe Toscani. “Multivariate Gini-Type Discrepancies.” <i>Mathematical Models and Methods in Applied Sciences</i>. World Scientific Publishing, 2025. <a href=\"https://doi.org/10.1142/s0218202525500174\">https://doi.org/10.1142/s0218202525500174</a>."},"day":"01","quality_controlled":"1","type":"journal_article","external_id":{"isi":["001456337300001"],"arxiv":["2411.01052"]},"language":[{"iso":"eng"}],"_id":"19565","abstract":[{"lang":"eng","text":"Measuring distances in a multidimensional setting is a challenging problem, which appears in many fields of science and engineering. In this paper, to measure the distance between two multivariate distributions, we introduce a new measure of discrepancy which is scale invariant and which, in the case of two independent copies of the same distribution, and after normalization, coincides with the scaling invariant multidimensional version of the Gini index recently proposed in [P. Giudici, E. Raffinetti and G. Toscani, Measuring multidimensional inequality: A new proposal based on the Fourier transform, preprint (2024), arXiv:2401.14012 ]. A byproduct of the analysis is an easy-to-handle discrepancy metric, obtained by application of the theory to a pair of Gaussian multidimensional densities. The obtained metric does improve the standard metrics, based on the mean squared error, as it is scale invariant. The importance of this theoretical finding is illustrated by means of a real problem that concerns measuring the importance of Environmental, Social and Governance factors for the growth of small and medium enterprises. "}],"scopus_import":"1","issue":"5","page":"1267-1296","status":"public","isi":1,"publication":"Mathematical Models and Methods in Applied Sciences","date_updated":"2025-09-30T11:36:56Z","OA_type":"green","oa_version":"Preprint","title":"Multivariate Gini-type discrepancies","date_created":"2025-04-15T13:34:00Z","publication_identifier":{"eissn":["1793-6314"],"issn":["0218-2025"]},"arxiv":1,"publication_status":"published","publisher":"World Scientific Publishing","article_processing_charge":"No","department":[{"_id":"JaMa"}],"month":"05","article_type":"original","volume":35,"author":[{"first_name":"Gennaro","last_name":"Auricchio","full_name":"Auricchio, Gennaro"},{"full_name":"Brigati, Giovanni","last_name":"Brigati","id":"63ff57e8-1fbb-11ee-88f2-f558ffc59cf1","first_name":"Giovanni"},{"last_name":"Giudici","full_name":"Giudici, Paolo","first_name":"Paolo"},{"full_name":"Toscani, Giuseppe","last_name":"Toscani","first_name":"Giuseppe"}],"oa":1,"doi":"10.1142/s0218202525500174"},{"corr_author":"1","article_processing_charge":"No","related_material":{"record":[{"relation":"dissertation_contains","status":"public","id":"20563"}]},"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.","doi":"10.48550/arXiv.2502.15665","oa":1,"author":[{"id":"63ff57e8-1fbb-11ee-88f2-f558ffc59cf1","first_name":"Giovanni","last_name":"Brigati","full_name":"Brigati, Giovanni"},{"id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","first_name":"Jan","orcid":"0000-0002-0845-1338","full_name":"Maas, Jan","last_name":"Maas"},{"full_name":"Quattrocchi, Filippo","last_name":"Quattrocchi","id":"3ebd6ba8-edfb-11eb-afb5-91a9745ba308","first_name":"Filippo","orcid":"0009-0000-9773-1931"}],"project":[{"_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","grant_number":"F6504","name":"Taming Complexity in Partial Differential Systems"},{"name":"IST-BRIDGE: International postdoctoral program","grant_number":"101034413","call_identifier":"H2020","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c"}],"month":"08","department":[{"_id":"GradSch"},{"_id":"JaMa"}],"arxiv":1,"keyword":["optimal transport","kinetic theory","second-order discrepancy","Vlasov equation","Wasserstein splines."],"title":"Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between probability measures","date_created":"2025-10-28T13:12:08Z","oa_version":"Preprint","OA_type":"green","publication_status":"draft","status":"public","ec_funded":1,"publication":"arXiv","date_updated":"2026-04-27T22:30:15Z","article_number":"2502.15665","year":"2025","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2025-08-10T00:00:00Z","OA_place":"repository","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2502.15665"}],"language":[{"iso":"eng"}],"_id":"20569","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."}],"external_id":{"arxiv":["2502.15665"]},"type":"preprint","citation":{"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>","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.).","ieee":"G. Brigati, J. Maas, and F. Quattrocchi, “Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between probability measures,” <i>arXiv</i>. .","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>.","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>","ista":"Brigati G, Maas J, Quattrocchi F. Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between probability measures. arXiv, 2502.15665."},"day":"10"},{"OA_type":"green","oa_version":"Preprint","title":"Stability for the logarithmic Sobolev inequality","date_created":"2024-07-21T22:01:00Z","arxiv":1,"publication_identifier":{"eissn":["1096-0783"],"issn":["0022-1236"]},"publication_status":"published","publisher":"Elsevier","article_processing_charge":"No","corr_author":"1","month":"10","department":[{"_id":"JaMa"}],"article_type":"original","author":[{"id":"63ff57e8-1fbb-11ee-88f2-f558ffc59cf1","first_name":"Giovanni","full_name":"Brigati, Giovanni","last_name":"Brigati"},{"last_name":"Dolbeault","full_name":"Dolbeault, Jean","first_name":"Jean"},{"full_name":"Simonov, Nikita","last_name":"Simonov","first_name":"Nikita"}],"volume":287,"oa":1,"doi":"10.1016/j.jfa.2024.110562","acknowledgement":"The authors thank Max Fathi and Pierre Cardaliaguet for fruitful discussions and Emanuel Indrei for stimulating interactions. They also thank an anonymous referee for useful comments and suggestions which have led to an improvement of the manuscript. They also want to express their gratitude to the managing editor, L. Gross, for his encouragements and questions. G.B. has been funded by the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 754362. This work has been (partially) supported by the Project Conviviality ANR-23-CE40-0003 of the French National Research Agency.","main_file_link":[{"url":"10.48550/arXiv.2303.12926","open_access":"1"}],"OA_place":"repository","date_published":"2024-10-15T00:00:00Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","year":"2024","article_number":"110562","intvolume":"       287","citation":{"apa":"Brigati, G., Dolbeault, J., &#38; Simonov, N. (2024). Stability for the logarithmic Sobolev inequality. <i>Journal of Functional Analysis</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jfa.2024.110562\">https://doi.org/10.1016/j.jfa.2024.110562</a>","ieee":"G. Brigati, J. Dolbeault, and N. Simonov, “Stability for the logarithmic Sobolev inequality,” <i>Journal of Functional Analysis</i>, vol. 287, no. 8. Elsevier, 2024.","short":"G. Brigati, J. Dolbeault, N. Simonov, Journal of Functional Analysis 287 (2024).","mla":"Brigati, Giovanni, et al. “Stability for the Logarithmic Sobolev Inequality.” <i>Journal of Functional Analysis</i>, vol. 287, no. 8, 110562, Elsevier, 2024, doi:<a href=\"https://doi.org/10.1016/j.jfa.2024.110562\">10.1016/j.jfa.2024.110562</a>.","chicago":"Brigati, Giovanni, Jean Dolbeault, and Nikita Simonov. “Stability for the Logarithmic Sobolev Inequality.” <i>Journal of Functional Analysis</i>. Elsevier, 2024. <a href=\"https://doi.org/10.1016/j.jfa.2024.110562\">https://doi.org/10.1016/j.jfa.2024.110562</a>.","ista":"Brigati G, Dolbeault J, Simonov N. 2024. Stability for the logarithmic Sobolev inequality. Journal of Functional Analysis. 287(8), 110562.","ama":"Brigati G, Dolbeault J, Simonov N. Stability for the logarithmic Sobolev inequality. <i>Journal of Functional Analysis</i>. 2024;287(8). doi:<a href=\"https://doi.org/10.1016/j.jfa.2024.110562\">10.1016/j.jfa.2024.110562</a>"},"day":"15","quality_controlled":"1","type":"journal_article","external_id":{"isi":["001271814000001"],"arxiv":["2303.12926"]},"abstract":[{"lang":"eng","text":"This paper is devoted to stability results for the Gaussian logarithmic Sobolev inequality, with explicit stability constants.\r\n\r\n"}],"_id":"17277","language":[{"iso":"eng"}],"scopus_import":"1","issue":"8","status":"public","isi":1,"publication":"Journal of Functional Analysis","date_updated":"2025-09-08T08:25:34Z"},{"year":"2024","arxiv":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_number":"2404.15205","oa_version":"Preprint","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2404.15205","open_access":"1"}],"date_published":"2024-05-08T00:00:00Z","OA_place":"repository","title":"Heat flow, log-concavity, and Lipschitz transport maps","date_created":"2024-07-31T08:17:14Z","type":"preprint","publication_status":"draft","abstract":[{"lang":"eng","text":"In this paper we derive estimates for the Hessian of the logarithm\r\n(log-Hessian) for solutions to the heat equation. For initial data in the form\r\nof log-Lipschitz perturbation of strongly log-concave measures, the log-Hessian\r\nadmits an explicit, uniform (in space) lower bound. This yields a new estimate\r\nfor the Lipschitz constant of a transport map pushing forward the standard\r\nGaussian to a measure in this class. Further connections are discussed with\r\nscore-based diffusion models and improved Gaussian logarithmic Sobolev\r\ninequalities. Finally, we show that assuming only fast decay of the tails of\r\nthe initial datum does not suffice to guarantee uniform log-Hessian upper\r\nbounds."}],"_id":"17353","language":[{"iso":"eng"}],"external_id":{"arxiv":["2404.15205"]},"citation":{"ieee":"G. Brigati and F. Pedrotti, “Heat flow, log-concavity, and Lipschitz transport maps,” <i>arXiv</i>. .","short":"G. Brigati, F. Pedrotti, ArXiv (n.d.).","mla":"Brigati, Giovanni, and Francesco Pedrotti. “Heat Flow, Log-Concavity, and Lipschitz Transport Maps.” <i>ArXiv</i>, 2404.15205, doi:<a href=\"https://doi.org/10.48550/arXiv.2404.15205\">10.48550/arXiv.2404.15205</a>.","apa":"Brigati, G., &#38; Pedrotti, F. (n.d.). Heat flow, log-concavity, and Lipschitz transport maps. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2404.15205\">https://doi.org/10.48550/arXiv.2404.15205</a>","ista":"Brigati G, Pedrotti F. Heat flow, log-concavity, and Lipschitz transport maps. arXiv, 2404.15205.","ama":"Brigati G, Pedrotti F. Heat flow, log-concavity, and Lipschitz transport maps. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2404.15205\">10.48550/arXiv.2404.15205</a>","chicago":"Brigati, Giovanni, and Francesco Pedrotti. “Heat Flow, Log-Concavity, and Lipschitz Transport Maps.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2404.15205\">https://doi.org/10.48550/arXiv.2404.15205</a>."},"day":"08","status":"public","corr_author":"1","article_processing_charge":"No","related_material":{"record":[{"relation":"later_version","id":"20591","status":"public"},{"status":"public","id":"17336","relation":"dissertation_contains"}]},"doi":"10.48550/arXiv.2404.15205","oa":1,"month":"05","department":[{"_id":"JaMa"}],"publication":"arXiv","date_updated":"2026-04-07T13:00:02Z","author":[{"full_name":"Brigati, Giovanni","last_name":"Brigati","id":"63ff57e8-1fbb-11ee-88f2-f558ffc59cf1","first_name":"Giovanni"},{"last_name":"Pedrotti","full_name":"Pedrotti, Francesco","id":"d3ac8ac6-dc8d-11ea-abe3-e2a9628c4c3c","first_name":"Francesco"}]}]