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Higher-order propagation of chaos in $L^2$ for interacting diffusions

Elias Hess-Childs and Keefer Rowan

Vol. 6 (2025), No. 2, 581–646
DOI: 10.2140/pmp.2025.6.581
Abstract

We study diffusions with bounded pairwise interaction. We show for the first time propagation of chaos on arbitrary time horizons in a stronger L2-based distance, as opposed to the usual Wasserstein or relative entropy distances. The estimate is based on iterating inequalities derived from the BBGKY hierarchy and does not follow directly from bounds on the full N-particle density. This argument gives the optimal rate in N, showing the distance between the j-particle marginal density and the tensor product of the mean-field limit is O(N1). We use cluster expansions to give perturbative higher-order corrections to the mean-field limit. For an arbitrary order i, these provide “low-dimensional” approximations to the j-particle marginal density with error O(N(i+1)).

Keywords
propagation of chaos, mean-field interacting diffusions, BBGKY hierarchy
Mathematical Subject Classification
Primary: 35Q70, 82C31
Milestones
Received: 22 May 2024
Revised: 17 October 2024
Accepted: 2 November 2024
Published: 12 March 2025
Authors
Elias Hess-Childs
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA
United States
Keefer Rowan
Courant Institute of Mathematical Sciences
New York University
New York, NY
United States