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 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 . 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 ; (c) satisfies an HWI inequality.
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 riemannienne sur l’espace des processus ponctuels et la distance associée 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 , 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 ; (c) satisfait une inégalité HWI.
Accepted:
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Keywords: Poisson point process, optimal transportation, Wasserstein distance, gradient flows, Ricci curvature
Mot clés : Processus ponctuel de Poisson, transport optimal, distance de Wasserstein, flots de gradient, courbure de Ricci
Lorenzo Dello Schiavo 1; Ronan Herry 2; Kohei Suzuki 3
@article{JEP_2024__11__957_0, author = {Lorenzo Dello Schiavo and Ronan Herry and Kohei Suzuki}, title = {Wasserstein geometry and {Ricci~curvature~bounds} for {Poisson} spaces}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {957--1010}, publisher = {\'Ecole polytechnique}, volume = {11}, year = {2024}, doi = {10.5802/jep.270}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.270/} }
TY - JOUR AU - Lorenzo Dello Schiavo AU - Ronan Herry AU - Kohei Suzuki TI - Wasserstein geometry and Ricci curvature bounds for Poisson spaces JO - Journal de l’École polytechnique — Mathématiques PY - 2024 SP - 957 EP - 1010 VL - 11 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.270/ DO - 10.5802/jep.270 LA - en ID - JEP_2024__11__957_0 ER -
%0 Journal Article %A Lorenzo Dello Schiavo %A Ronan Herry %A Kohei Suzuki %T Wasserstein geometry and Ricci curvature bounds for Poisson spaces %J Journal de l’École polytechnique — Mathématiques %D 2024 %P 957-1010 %V 11 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.270/ %R 10.5802/jep.270 %G en %F JEP_2024__11__957_0
Lorenzo Dello Schiavo; Ronan Herry; Kohei Suzuki. Wasserstein geometry and Ricci curvature bounds for Poisson spaces. Journal de l’École polytechnique — Mathématiques, Volume 11 (2024), pp. 957-1010. doi : 10.5802/jep.270. https://jep.centre-mersenne.org/articles/10.5802/jep.270/
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