Creation of chaos for interacting Brownian particles
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908359032045568 |
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| author | Bernou, Armand Duerinckx, Mitia Ménard, Matthieu |
| author_facet | Bernou, Armand Duerinckx, Mitia Ménard, Matthieu |
| contents | We consider a system of $N$ Brownian particles, with or without inertia, interacting in the mean-field regime via a weak, smooth, long-range potential, and starting initially from an arbitrary exchangeable $N$-particle distribution. In this model framework, we establish a fine version of the so-called creation-of-chaos phenomenon: in weak norms, the mean-field approximation for a typical particle is shown to hold with an accuracy $O(N^{-1})$ up to an error due solely to initial pair correlations, which is damped exponentially over time. Corresponding higher-order results are also derived in the form of higher-order correlation estimates. The approach is new and easily adaptable: we start from suboptimal correlation estimates obtained from an elementary use of Itô's calculus on moments of the empirical measure, together with ergodic properties of the mean-field dynamics, and these bounds are then made optimal after combination with PDE estimates on the BBKY hierarchy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_09917 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Creation of chaos for interacting Brownian particles Bernou, Armand Duerinckx, Mitia Ménard, Matthieu Probability Mathematical Physics Analysis of PDEs We consider a system of $N$ Brownian particles, with or without inertia, interacting in the mean-field regime via a weak, smooth, long-range potential, and starting initially from an arbitrary exchangeable $N$-particle distribution. In this model framework, we establish a fine version of the so-called creation-of-chaos phenomenon: in weak norms, the mean-field approximation for a typical particle is shown to hold with an accuracy $O(N^{-1})$ up to an error due solely to initial pair correlations, which is damped exponentially over time. Corresponding higher-order results are also derived in the form of higher-order correlation estimates. The approach is new and easily adaptable: we start from suboptimal correlation estimates obtained from an elementary use of Itô's calculus on moments of the empirical measure, together with ergodic properties of the mean-field dynamics, and these bounds are then made optimal after combination with PDE estimates on the BBKY hierarchy. |
| title | Creation of chaos for interacting Brownian particles |
| topic | Probability Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2504.09917 |