Wasserstein geometry and Ricci curvature bounds for Poisson spaces

Dello Schiavo, Lorenzo; Herry, Ronan; Suzuki, Kohei

Abstract

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.

Citation

Dello Schiavo, L., Herry, R., & Suzuki, K. (2024). Wasserstein geometry and Ricci curvature bounds for Poisson spaces. Journal de l’École polytechnique — Mathématiques, 11, 957-1010. https://doi.org/10.5802/jep.270

• Journal Article Type: Article
• Acceptance Date: Jun 8, 2024
• Online Publication Date: Aug 30, 2024
• Publication Date: Aug 30, 2024
• Deposit Date: Oct 20, 2024
• Publicly Available Date: Oct 21, 2024
• Journal: Journal de l’École polytechnique — Mathématiques
• Print ISSN: 2429-7100
• Electronic ISSN: 2270-518X
• Publisher: École polytechnique
• Peer Reviewed: Peer Reviewed
• Volume: 11
• Pages: 957-1010
• DOI: https://doi.org/10.5802/jep.270
• Public URL: https://durham-repository.worktribe.com/output/2977746