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Hierarchies, entropy, and quantitative propagation of chaos for mean field diffusions

Daniel Lacker

Vol. 4 (2023), No. 2, 377–432
DOI: 10.2140/pmp.2023.4.377
Abstract

This paper develops a nonasymptotic, local approach to quantitative propagation of chaos for a wide class of mean field diffusive dynamics. For a system of n interacting particles, the relative entropy between the marginal law of k particles and its limiting product measure is shown to be O((kn)2) at each time, as long as the same is true at time zero. A simple Gaussian example shows that this rate is optimal. The main assumption is that the limiting measure obeys a certain functional inequality, which is shown to encompass many potentially irregular but not too singular finite-range interactions, as well as some infinite-range interactions. This unifies the previously disparate cases of Lipschitz versus bounded measurable interactions, improving the best prior bounds of O(kn) which were deduced from global estimates involving all n particles. We also cover a class of models for which qualitative propagation of chaos and even well-posedness of the McKean–Vlasov equation were previously unknown. At the center of our new approach is a differential inequality, derived from a form of the BBGKY hierarchy, which bounds the k-particle entropy in terms of the (k + 1)-particle entropy.

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Keywords
mean field limits, propagation of chaos
Mathematical Subject Classification
Primary: 82C22
Secondary: 60F17, 60H10
Milestones
Received: 22 March 2022
Revised: 28 October 2022
Accepted: 28 November 2022
Published: 31 May 2023
Authors
Daniel Lacker
Industrial Engineering & Operations Research
Columbia University
New York, NY
United States