Creation of chaos for interacting Brownian particles

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(1/N) up to an error due solely to initial pair correlations, which is damped exponentially over time. The novelty is that the initial information appears in our estimates only through pair correlations, which currently seems inaccessible to other methods. This is complemented by corresponding results on higher-order creation of chaos in the form of higher-order correlation estimates.

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