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Non-local Wasserstein Geometry, Gradient Flows, and Functional Inequalities for Stationary Point Processes

Martin Huesmann, Hanna Stange

TL;DR

The paper develops a non-local transport distance per volume for stationary point processes and proves it forms an extended geodesic distance, with the Ornstein–Uhlenbeck semigroup as its gradient flow for the specific relative entropy. It establishes existence of stationary continuity-equation solutions, a large-volume relation to the classical $\mathcal{W}_0$ distance, and 1-geodesic convexity of the entropy, leading to Talagrand and HWI-type inequalities in the stationary, infinite-volume regime. The framework leverages a discrete difference operator and a logarithmic-mean-based Lagrangian to define a Benamou–Brenier-type action, ensuring convexity and lower semicontinuity to guarantee minimizers. The results connect finite-volume transport geometry to an intrinsic stationary geometry, enabling contractivity, gradient-flow structure, and functional inequalities for stationary point processes with potential applications in random geometric systems and spin-like infinite-particle dynamics.

Abstract

We construct a non-local Benamou-Brenier-type transport distance on the space of stationary point processes and analyse the induced geometry. We show that our metric is a specific variant of the transport distance recently constructed in [DSHS24]. As a consequence, we show that the Ornstein-Uhlenbeck semigroup is the gradient flow of the specific relative entropy w.r.t. the newly constructed distance. Furthermore, we show the existence of stationary geodesics, establish $1$-geodesic convexity of the specific relative entropy, and derive stationary analogues of functional inequalities such as a specific HWI inequality and a specific Talagrand inequality. One of the key technical contributions is the existence of solutions to the non-local continuity equation between arbitrary point processes.

Non-local Wasserstein Geometry, Gradient Flows, and Functional Inequalities for Stationary Point Processes

TL;DR

The paper develops a non-local transport distance per volume for stationary point processes and proves it forms an extended geodesic distance, with the Ornstein–Uhlenbeck semigroup as its gradient flow for the specific relative entropy. It establishes existence of stationary continuity-equation solutions, a large-volume relation to the classical distance, and 1-geodesic convexity of the entropy, leading to Talagrand and HWI-type inequalities in the stationary, infinite-volume regime. The framework leverages a discrete difference operator and a logarithmic-mean-based Lagrangian to define a Benamou–Brenier-type action, ensuring convexity and lower semicontinuity to guarantee minimizers. The results connect finite-volume transport geometry to an intrinsic stationary geometry, enabling contractivity, gradient-flow structure, and functional inequalities for stationary point processes with potential applications in random geometric systems and spin-like infinite-particle dynamics.

Abstract

We construct a non-local Benamou-Brenier-type transport distance on the space of stationary point processes and analyse the induced geometry. We show that our metric is a specific variant of the transport distance recently constructed in [DSHS24]. As a consequence, we show that the Ornstein-Uhlenbeck semigroup is the gradient flow of the specific relative entropy w.r.t. the newly constructed distance. Furthermore, we show the existence of stationary geodesics, establish -geodesic convexity of the specific relative entropy, and derive stationary analogues of functional inequalities such as a specific HWI inequality and a specific Talagrand inequality. One of the key technical contributions is the existence of solutions to the non-local continuity equation between arbitrary point processes.

Paper Structure

This paper contains 21 sections, 48 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Setting and Main Results
  3. Connection to the Literature
  4. Structure of the Paper
  5. Preliminaries
  6. Point Processes
  7. Transport Distance on Bounded Domains
  8. The Continuity Equation on $\mathbb {R}^d$
  9. Properties of the Continuity Equation
  10. Stationary Solutions to the Continuity Equation
  11. The Action Functional
  12. Properties of the Lagrange functional
  13. Properties of the Action Functional
  14. Transport Distance for Stationary Measures
  15. Connection between Transport Distances
  16. ...and 6 more sections

Key Result

Theorem 1.1

For any Polish set $B\in\mathcal{B}(\mathbb {R}^d)$ and any measures $\mathbf{P}_0,\mathbf{P}_1\in\mathcal{P}_1(\Gamma_B)$ there exists a solution $(\Bar{\mathbf{P}},\Bar{\nuV})\in\mathcal{CE}^B(\mathbf{P}_0,\mathbf{P}_1)$. Further, if $\mathbf{P}_0,\mathbf{P}_1\in\mathcal{P}_s(\Gamma_{\mathbb {R}^d

Theorems & Definitions (93)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Specific Talagrand inequality
  • Theorem 1.4
  • Theorem 1.5: EVI
  • Theorem 1.6: 1-contractivity of the Ornstein--Uhlenbeck semigroup
  • Theorem 1.7: $1$-geodesic convexity of the specific relative entropy
  • Theorem 1.8: $\mathrm{HWI}$ inequality
  • Proposition 2.1: schiavo2023wasserstein
  • Definition 2.2
  • ...and 83 more