Creation of chaos for interacting Brownian particles

TL;DR

The paper analyzes a mean-field regime of Langevin-type Brownian particles with smooth long-range interactions and a confining potential. It introduces a two-tier strategy that first derives suboptimal mean-field and correlation bounds from Itô calculus on the empirical measure and then leverages the BBGKY hierarchy together with ergodic estimates for the linearized mean-field dynamics to obtain optimal $O(N^{-1})$ propagation errors for marginals and $O(N^{1-m})$ decay for $m$-particle correlations, up to exponentially damped initial correlations. The framework is further extended to overdamped dynamics, periodic geometries, non-quadratic confinement, and more general interaction functionals, demonstrating robustness and adaptability. The result is a precise, quantitative description of creation and propagation of chaos in stochastic mean-field systems, enabling fine-grained control of correlation structures in kinetic-type models. This advance has potential implications for stochastic kinetic theory and related mean-field models, including extensions to Kac-type or energy-cumulant analyses.

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 BBKY hierarchy.

Theorems & Definitions (16)

• Theorem 1.1: Creation of chaos
• Theorem 1.2: Creation of chaos in correlation estimates
• Remark 1.3: Smallness assumption
• Theorem 1.4: Creation of chaos --- overdamped setting
• Lemma 2.1: Dual ergodic estimates, BD_2024
• Lemma 2.2: Moment estimates
• proof
• Proposition 2.3
• proof
• proof : Proof of Theorem \ref{['th:creation']}