{"title":"Heat flow, log-concavity, and Lipschitz transport maps","PlanS_conform":"1","abstract":[{"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.","lang":"eng"}],"author":[{"id":"63ff57e8-1fbb-11ee-88f2-f558ffc59cf1","last_name":"Brigati","full_name":"Brigati, Giovanni","first_name":"Giovanni"},{"first_name":"Francesco","full_name":"Pedrotti, Francesco","last_name":"Pedrotti","id":"d3ac8ac6-dc8d-11ea-abe3-e2a9628c4c3c"}],"quality_controlled":"1","oa_version":"Published Version","oa":1,"corr_author":"1","date_updated":"2025-11-04T07:36:53Z","license":"https://creativecommons.org/licenses/by/4.0/","publication":"Electronic Communications in Probability","related_material":{"record":[{"status":"public","id":"17353","relation":"earlier_version"}]},"article_number":"71","file":[{"checksum":"67858edbd74658fe38955fa1216f2f18","file_id":"20596","success":1,"access_level":"open_access","creator":"dernst","content_type":"application/pdf","file_name":"2025_ElectronJourProbab_Brigati.pdf","date_created":"2025-11-04T07:34:05Z","relation":"main_file","file_size":278078,"date_updated":"2025-11-04T07:34:05Z"}],"DOAJ_listed":"1","scopus_import":"1","status":"public","language":[{"iso":"eng"}],"citation":{"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>","short":"G. Brigati, F. Pedrotti, Electronic Communications in Probability 30 (2025).","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.","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>","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>.","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>."},"ddc":["500"],"year":"2025","OA_type":"gold","has_accepted_license":"1","department":[{"_id":"JaMa"}],"month":"09","tmp":{"image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"publisher":"Institute of Mathematical Statistics","ec_funded":1,"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.","publication_identifier":{"eissn":["1083-589X"]},"file_date_updated":"2025-11-04T07:34:05Z","arxiv":1,"publication_status":"published","project":[{"_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","grant_number":"F6504","name":"Taming Complexity in Partial Differential Systems"},{"grant_number":"101034413","name":"IST-BRIDGE: International postdoctoral program","call_identifier":"H2020","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","volume":30,"intvolume":"        30","external_id":{"arxiv":["2404.15205"]},"_id":"20591","article_processing_charge":"Yes","doi":"10.1214/25-ECP717","OA_place":"publisher","day":"25","date_published":"2025-09-25T00:00:00Z","date_created":"2025-11-02T23:01:35Z","article_type":"original","type":"journal_article"}