Creation of chaos for interacting Brownian particles

Armand Bernou, Mitia Duerinckx and Matthieu Ménard

Stochastic Processes and their Applications, 2026, vol. 193, issue C

Abstract: 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 BBGKY hierarchy.

Date: 2026
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DOI: 10.1016/j.spa.2025.104849

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