{"month":"01","article_type":"original","publication_status":"published","citation":{"ista":"Dello Schiavo L, Herry R, Suzuki K. 2024. Wasserstein geometry and Ricci curvature bounds for Poisson spaces. Journal de l’Ecole Polytechnique - Mathematiques. 11, 957–1010.","ieee":"L. Dello Schiavo, R. Herry, and K. Suzuki, “Wasserstein geometry and Ricci curvature bounds for Poisson spaces,” <i>Journal de l’Ecole Polytechnique - Mathematiques</i>, vol. 11. Ecole Polytechnique, pp. 957–1010, 2024.","apa":"Dello Schiavo, L., Herry, R., &#38; Suzuki, K. (2024). Wasserstein geometry and Ricci curvature bounds for Poisson spaces. <i>Journal de l’Ecole Polytechnique - Mathematiques</i>. Ecole Polytechnique. <a href=\"https://doi.org/10.5802/jep.270\">https://doi.org/10.5802/jep.270</a>","mla":"Dello Schiavo, Lorenzo, et al. “Wasserstein Geometry and Ricci Curvature Bounds for Poisson Spaces.” <i>Journal de l’Ecole Polytechnique - Mathematiques</i>, vol. 11, Ecole Polytechnique, 2024, pp. 957–1010, doi:<a href=\"https://doi.org/10.5802/jep.270\">10.5802/jep.270</a>.","ama":"Dello Schiavo L, Herry R, Suzuki K. Wasserstein geometry and Ricci curvature bounds for Poisson spaces. <i>Journal de l’Ecole Polytechnique - Mathematiques</i>. 2024;11:957-1010. doi:<a href=\"https://doi.org/10.5802/jep.270\">10.5802/jep.270</a>","short":"L. Dello Schiavo, R. Herry, K. Suzuki, Journal de l’Ecole Polytechnique - Mathematiques 11 (2024) 957–1010.","chicago":"Dello Schiavo, Lorenzo, Ronan Herry, and Kohei Suzuki. “Wasserstein Geometry and Ricci Curvature Bounds for Poisson Spaces.” <i>Journal de l’Ecole Polytechnique - Mathematiques</i>. Ecole Polytechnique, 2024. <a href=\"https://doi.org/10.5802/jep.270\">https://doi.org/10.5802/jep.270</a>."},"date_published":"2024-01-01T00:00:00Z","status":"public","author":[{"last_name":"Dello Schiavo","orcid":"0000-0002-9881-6870","first_name":"Lorenzo","id":"ECEBF480-9E4F-11EA-B557-B0823DDC885E","full_name":"Dello Schiavo, Lorenzo"},{"first_name":"Ronan","last_name":"Herry","full_name":"Herry, Ronan"},{"full_name":"Suzuki, Kohei","first_name":"Kohei","last_name":"Suzuki"}],"language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"We study the geometry of Poisson point processes from the point of view of optimal transport and Ricci lower bounds. We construct a Riemannian structure on the space of point processes and the associated distance W that corresponds to the Benamou–Brenier variational formula. Our main tool is a non-local continuity equation formulated with the difference operator. The closure of the domain of the relative entropy is a complete geodesic space, when endowed with \r\nW. The geometry of this non-local infinite-dimensional space is analogous to that of spaces with positive Ricci curvature. Among others: (a) the Ornstein–Uhlenbeck semi-group is the gradient flow of the relative entropy; (b) the Poisson space has an entropic Ricci curvature bounded from below by 1; (c) W satisfies an HWI inequality."},{"text":"Nous étudions la géométrie des processus ponctuels de Poisson à travers le prisme du transport optimal et de la minoration de la courbure de Ricci. Nous construisons une structure\r\nriemannienne sur l’espace des processus ponctuels et la distance associée W qui concorde avec la formulation variationnelle de Benamou–Brenier. Notre analyse repose sur une équation de continuité non locale définie à l’aide de l’opérateur de différence. La fermeture du domaine de l’entropie relative, équipé de W, est un espace géodésique complet. La géométrie de cet espace non local et de dimension infinie est analogue à celle des espaces à courbure de Ricci strictement positive. Entre autres : (a) le semi-groupe d’Ornstein–Uhlenbeck est le flot du gradient de l’entropie relative ; (b) l’espace de Poisson a une courbure de Ricci entropique minorée par 1 ; (c) W satisfait une inégalité HWI.","lang":"fre"}],"external_id":{"arxiv":["2303.00398"]},"title":"Wasserstein geometry and Ricci curvature bounds for Poisson spaces","_id":"18158","article_processing_charge":"Yes","date_updated":"2024-10-01T07:34:00Z","scopus_import":"1","license":"https://creativecommons.org/licenses/by/4.0/","tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"corr_author":"1","year":"2024","doi":"10.5802/jep.270","department":[{"_id":"JaMa"}],"oa_version":"Published Version","day":"01","publication":"Journal de l'Ecole Polytechnique - Mathematiques","publisher":"Ecole Polytechnique","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","file_date_updated":"2024-10-01T07:31:56Z","type":"journal_article","ddc":["510"],"intvolume":"        11","date_created":"2024-09-29T22:01:38Z","has_accepted_license":"1","quality_controlled":"1","publication_identifier":{"issn":["2429-7100"],"eissn":["2270-518X"]},"file":[{"relation":"main_file","success":1,"access_level":"open_access","date_updated":"2024-10-01T07:31:56Z","creator":"dernst","file_id":"18164","date_created":"2024-10-01T07:31:56Z","checksum":"5a51da5fb5f7fcaada378d43444cced8","file_name":"2024_JourEcolePolytechniqueMath_DelloSchiavo.pdf","file_size":1250553,"content_type":"application/pdf"}],"volume":11,"oa":1,"page":"957-1010"}