Optimal transport of stationary point processes: Metric structure, gradient flow and convexity of the specific entropy
We develop a theory of optimal transport for stationary random measures with a focus on stationary point processes and construct a family of distances on the set of stationary random measures. These induce a natural notion of interpolation between two stationary random measures along a shortest curv...
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| Published in | Journal of functional analysis Vol. 289; no. 4; p. 110974 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Inc
15.08.2025
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0022-1236 |
| DOI | 10.1016/j.jfa.2025.110974 |
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| Abstract | We develop a theory of optimal transport for stationary random measures with a focus on stationary point processes and construct a family of distances on the set of stationary random measures. These induce a natural notion of interpolation between two stationary random measures along a shortest curve connecting them. In the setting of stationary point processes we leverage this transport distance to give a geometric interpretation for the evolution of infinite particle systems with stationary distribution. Namely, we characterise the evolution of infinitely many Brownian motions as the gradient flow of the specific relative entropy w.r.t. the Poisson point process. Further, we establish displacement convexity of the specific relative entropy along optimal interpolations of point processes and establish a stationary analogue of the HWI inequality, relating specific entropy, transport distance, and a specific relative Fisher information. |
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| AbstractList | We develop a theory of optimal transport for stationary random measures with a focus on stationary point processes and construct a family of distances on the set of stationary random measures. These induce a natural notion of interpolation between two stationary random measures along a shortest curve connecting them. In the setting of stationary point processes we leverage this transport distance to give a geometric interpretation for the evolution of infinite particle systems with stationary distribution. Namely, we characterise the evolution of infinitely many Brownian motions as the gradient flow of the specific relative entropy w.r.t. the Poisson point process. Further, we establish displacement convexity of the specific relative entropy along optimal interpolations of point processes and establish a stationary analogue of the HWI inequality, relating specific entropy, transport distance, and a specific relative Fisher information. |
| ArticleNumber | 110974 |
| Author | Erbar, Matthias Müller, Bastian Huesmann, Martin Jalowy, Jonas |
| Author_xml | – sequence: 1 givenname: Matthias surname: Erbar fullname: Erbar, Matthias email: erbar@math.uni-bielefeld.de organization: Faculty of Mathematics, University of Bielefeld, Universitätsstrasse 25, 33615 Bielefeld, Germany – sequence: 2 givenname: Martin surname: Huesmann fullname: Huesmann, Martin email: martin.huesmann@uni-muenster.de organization: Institute for Mathematical Stochastics, University of Münster, Orléans-Ring 10, 48149 Münster, Germany – sequence: 3 givenname: Jonas orcidid: 0000-0001-9624-2685 surname: Jalowy fullname: Jalowy, Jonas email: jjalowy@math.upb.de organization: Institute of Mathematics, Paderborn University, Warburger Str. 100, 33098 Paderborn, Germany – sequence: 4 givenname: Bastian surname: Müller fullname: Müller, Bastian email: bastian.mueller@uni-muenster.de organization: Institute for Mathematical Stochastics, University of Münster, Orléans-Ring 10, 48149 Münster, Germany |
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| Cites_doi | 10.1214/08-AOP420 10.1007/s00039-007-0613-z 10.1016/j.spa.2022.09.008 10.4007/annals.2010.172.617 10.1137/08071346X 10.1214/22-AIHP1317 10.1214/14-AOP966 10.1137/16M1080197 10.1006/jfan.1997.3215 10.1006/jfan.1999.3557 10.1088/0951-7715/24/4/016 10.1007/s00440-022-01151-y 10.3934/dcds.2019304 10.1214/15-AOP1001 10.1007/s00440-015-0643-7 10.1017/S1474748015000201 10.1007/s00526-014-0790-1 10.1007/s00440-018-0837-x 10.1016/j.spa.2012.11.002 10.5488/CMP.11.4.701 10.1103/PhysRevE.90.012118 10.1214/11-AOP736 10.1137/S0036141096303359 10.1007/s10955-015-1425-4 10.1214/009117904000000603 10.1006/aima.1997.1634 10.1007/s00440-011-0352-9 10.1214/21-AOP1562 10.1214/009117906000000098 10.1016/j.anihpb.2004.05.002 10.1081/PDE-100002243 10.1007/s00780-013-0205-8 10.1007/s00222-017-0738-0 10.1214/12-AOP814 10.1007/s00440-020-00993-8 10.1214/aoms/1177698120 10.1007/s00222-016-0692-2 10.1002/cpa.21977 10.1016/j.jfa.2021.108998 10.1063/5.0086835 10.1016/j.jfa.2011.06.009 10.1214/08-AIHP170 10.1007/PL00008749 10.1006/jfan.1997.3183 10.1214/12-AIHP537 |
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| Keywords | Specific relative entropy Wasserstein distance Gradient flow Stationary point processes |
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