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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2504.12047 |
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| _version_ | 1866912331315806208 |
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| author | Huesmann, Martin Stange, Hanna |
| author_facet | Huesmann, Martin Stange, Hanna |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12047 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-local Wasserstein Geometry, Gradient Flows, and Functional Inequalities for Stationary Point Processes Huesmann, Martin Stange, Hanna Probability 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. |
| title | Non-local Wasserstein Geometry, Gradient Flows, and Functional Inequalities for Stationary Point Processes |
| topic | Probability |
| url | https://arxiv.org/abs/2504.12047 |