{"type":"preprint","title":"Optimal transport of stationary point processes: Metric structure,\n  gradient flow and convexity of the specific entropy","year":"2023","external_id":{"arxiv":["2304.11145"]},"_id":"59189","publication":"arXiv:2304.11145","date_updated":"2025-03-31T07:19:53Z","abstract":[{"text":"We develop a theory of optimal transport for stationary random measures with\na focus on stationary point processes and construct a family of distances on\nthe set of stationary random measures. These induce a natural notion of\ninterpolation between two stationary random measures along a shortest curve\nconnecting them. In the setting of stationary point processes we leverage this\ntransport distance to give a geometric interpretation for the evolution of\ninfinite particle systems with stationary distribution. Namely, we characterise\nthe evolution of infinitely many Brownian motions as the gradient flow of the\nspecific relative entropy w.r.t.~the Poisson point process. Further, we\nestablish displacement convexity of the specific relative entropy along optimal\ninterpolations of point processes and establish an stationary analogue of the\nHWI inequality, relating specific entropy, transport distance, and a specific\nrelative Fisher information.","lang":"eng"}],"user_id":"113768","author":[{"full_name":"Erbar, Matthias","last_name":"Erbar","first_name":"Matthias"},{"full_name":"Huesmann, Martin","first_name":"Martin","last_name":"Huesmann"},{"first_name":"Jonas","last_name":"Jalowy","full_name":"Jalowy, Jonas"},{"full_name":"Müller, Bastian","first_name":"Bastian","last_name":"Müller"}],"date_created":"2025-03-31T07:15:22Z","citation":{"bibtex":"@article{Erbar_Huesmann_Jalowy_Müller_2023, title={Optimal transport of stationary point processes: Metric structure,   gradient flow and convexity of the specific entropy}, journal={arXiv:2304.11145}, author={Erbar, Matthias and Huesmann, Martin and Jalowy, Jonas and Müller, Bastian}, year={2023} }","short":"M. Erbar, M. Huesmann, J. Jalowy, B. Müller, ArXiv:2304.11145 (2023).","apa":"Erbar, M., Huesmann, M., Jalowy, J., &#38; Müller, B. (2023). Optimal transport of stationary point processes: Metric structure,   gradient flow and convexity of the specific entropy. In <i>arXiv:2304.11145</i>.","chicago":"Erbar, Matthias, Martin Huesmann, Jonas Jalowy, and Bastian Müller. “Optimal Transport of Stationary Point Processes: Metric Structure,   Gradient Flow and Convexity of the Specific Entropy.” <i>ArXiv:2304.11145</i>, 2023.","mla":"Erbar, Matthias, et al. “Optimal Transport of Stationary Point Processes: Metric Structure,   Gradient Flow and Convexity of the Specific Entropy.” <i>ArXiv:2304.11145</i>, 2023.","ieee":"M. Erbar, M. Huesmann, J. Jalowy, and B. Müller, “Optimal transport of stationary point processes: Metric structure,   gradient flow and convexity of the specific entropy,” <i>arXiv:2304.11145</i>. 2023.","ama":"Erbar M, Huesmann M, Jalowy J, Müller B. Optimal transport of stationary point processes: Metric structure,   gradient flow and convexity of the specific entropy. <i>arXiv:230411145</i>. Published online 2023."},"status":"public"}