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Wasserstein geometry and Ricci curvature bounds for Poisson spaces

Lorenzo Dello Schiavo, Ronan Herry, Kohei Suzuki

TL;DR

This work develops a rigorous non-local Wasserstein geometry on the Poisson configuration space to study synthetic Ricci curvature bounds in infinite dimensions. By introducing a Benamou–Brenier–type transport distance $W$ defined via a CE driven by the discrete difference operator, the authors show that the entropy domain is a complete geodesic space and prove entropy–Fisher-type inequalities. The Ornstein–Uhlenbeck semigroup is shown to be a gradient flow of the relative entropy with respect to $W$, yielding an $EVI(0)$ and $EVI(1)$ formulation, geodesic convexity of entropy, a Talagrand inequality, and a Poisson-analogue of the HWI inequality. Overall, the paper extends synthetic Ricci curvature concepts to configuration spaces with non-local, infinite-dimensional structures, providing foundational tools for further geometric analysis of point processes.

Abstract

Let $\varUpsilon$ be the configuration space over a complete and separable metric base space, endowed with the Poisson measure $π$. We study the geometry of $\varUpsilon$ from the point of view of optimal transport and Ricci-lower bounds. To do so, we define a formal Riemannian structure on $\mathscr{P}_{1}(\varUpsilon)$, the space of probability measures over $\varUpsilon$ with finite first moment, and we construct an extended distance $\mathcal{W}$ on $\mathscr{P}_{1}(\varUpsilon)$. The distance $\mathcal{W}$ corresponds, in our setting, to the Benamou--Brenier variational formulation of the Wasserstein distance. Our main technical tool is a non-local continuity equation defined via the difference operator on the Poisson space. We show that the closure of the domain of the relative entropy is a complete geodesic space, when endowed with $\mathcal{W}$. We establish non-local infinite-dimensional analogues of results regarding the geometry of the Wasserstein space over a metric measure space with synthetic Ricci curvature bounded below. In particular, we obtain that: (a) the Ornstein--Uhlenbeck semi-group is the gradient flow of the relative entropy; (b) the Poisson space has a Ricci curvature, in the entropic sense, bounded below by $1$; (c) the distance $\mathcal{W}$ satisfies an HWI inequality.

Wasserstein geometry and Ricci curvature bounds for Poisson spaces

TL;DR

This work develops a rigorous non-local Wasserstein geometry on the Poisson configuration space to study synthetic Ricci curvature bounds in infinite dimensions. By introducing a Benamou–Brenier–type transport distance defined via a CE driven by the discrete difference operator, the authors show that the entropy domain is a complete geodesic space and prove entropy–Fisher-type inequalities. The Ornstein–Uhlenbeck semigroup is shown to be a gradient flow of the relative entropy with respect to , yielding an and formulation, geodesic convexity of entropy, a Talagrand inequality, and a Poisson-analogue of the HWI inequality. Overall, the paper extends synthetic Ricci curvature concepts to configuration spaces with non-local, infinite-dimensional structures, providing foundational tools for further geometric analysis of point processes.

Abstract

Let be the configuration space over a complete and separable metric base space, endowed with the Poisson measure . We study the geometry of from the point of view of optimal transport and Ricci-lower bounds. To do so, we define a formal Riemannian structure on , the space of probability measures over with finite first moment, and we construct an extended distance on . The distance corresponds, in our setting, to the Benamou--Brenier variational formulation of the Wasserstein distance. Our main technical tool is a non-local continuity equation defined via the difference operator on the Poisson space. We show that the closure of the domain of the relative entropy is a complete geodesic space, when endowed with . We establish non-local infinite-dimensional analogues of results regarding the geometry of the Wasserstein space over a metric measure space with synthetic Ricci curvature bounded below. In particular, we obtain that: (a) the Ornstein--Uhlenbeck semi-group is the gradient flow of the relative entropy; (b) the Poisson space has a Ricci curvature, in the entropic sense, bounded below by ; (c) the distance satisfies an HWI inequality.
Paper Structure (58 sections, 52 theorems, 230 equations)

This paper contains 58 sections, 52 theorems, 230 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Summary of our construction
  4. Motivation
  5. Related works
  6. Entropic Ricci curvature for Markov chains and jump processes
  7. Other transportation costs for the configuration space
  8. Other geometries on the configuration space
  9. Curvature of the Wiener space
  10. Outline of the paper
  11. Acknowledgments
  12. Topological results for point processes
  13. Topological preliminaries for spaces of functions and measures
  14. The weak topology
  15. The vague topology
  16. ...and 43 more sections

Key Result

Theorem 1

The distance $\mathcal{W}$ satisfies the following properties: Furthermore, the non-extended metric space $(\mathop{\mathrm{\mathscr{D}om}}\nolimits \mathop{\mathrm{\mathcal{H}}}\nolimits, \mathcal{W})$ captures the Ricci-curvature lower bounds of $(\varUpsilon, \pi)$ in the following sense:

Theorems & Definitions (109)

  • Theorem
  • Remark 1.1
  • Remark 2.1
  • Theorem 2.2
  • proof
  • Lemma 2.3: GHP
  • Theorem 2.4: Kallenberg
  • Lemma 2.5
  • Remark 2.6
  • proof
  • ...and 99 more