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

2025/04/16 by Martin Huesmann, Huesmann, Martin, Hanna Stange +1 · 3 citations
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Statistical Mechanics and Entropy

paper · pdf · doi:10.48550/arxiv.2504.12047

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.

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