Wasserstein geometry and Ricci curvature bounds for Poisson spaces

Lorenzo Dello Schiavo; Ronan Herry; Kohei Suzuki

doi:10.5802/jep.270

Wasserstein geometry and Ricci curvature bounds for Poisson spaces
[Géométrie de Wasserstein et minoration de la courbure de Ricci pour les espaces de Poisson]

Lorenzo Dello Schiavo ¹ ; Ronan Herry ² ; Kohei Suzuki ³

¹ Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria
² IRMAR, Université de Rennes, 263 avenue du Général Leclerc, 35042 Rennes Cedex, France
³ Department of Mathematical Science, Durham University, Science Laboratories, South Road, DH1 3LE, United Kingdom

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 957-1010.

Résumé
Abstract

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 $𝒲$ 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.

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 $𝒲$ . 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.

Reçu le : 2023-04-07
Accepté le : 2024-08-06
Publié le : 2024-08-30

DOI : 10.5802/jep.270

Classification : 60G55, 49Q22, 30L99
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

Affiliations des auteurs :

Lorenzo Dello Schiavo ¹ ; Ronan Herry ² ; Kohei Suzuki ³

¹ Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria
² IRMAR, Université de Rennes, 263 avenue du Général Leclerc, 35042 Rennes Cedex, France
³ Department of Mathematical Science, Durham University, Science Laboratories, South Road, DH1 3LE, United Kingdom

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Wasserstein geometry and {Ricci~curvature~bounds} for {Poisson} spaces},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Lorenzo Dello Schiavo; Ronan Herry; Kohei Suzuki. Wasserstein geometry and Ricci curvature bounds for Poisson spaces. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 957-1010. doi : 10.5802/jep.270. https://jep.centre-mersenne.org/articles/10.5802/jep.270/

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