Abstract
Consider the metric space of square integrable laws on with the topology induced by the 2-Wasserstein distance . Let and . In this work, we consider (a) being the empirical measure of N-samples from μ, and the other case in which (b) is the empirical measure of marginal laws of the particle system of a McKean–Vlasov PDE . The main result of this paper is to show that under suitable regularity conditions, we have
for some positive constants that do not depend on N, where k corresponds to the degree of smoothness. The case where the samples are i.i.d. is studied using functional derivatives on the space of measures. The case of particle systems relies on an Itô-type formula for the flow of probability measures and is intimately connected to PDEs on the space of measures, called the master equation in the literature of mean-field games. We state general regularity conditions required for each case and analyze the regularity in the case of functionals of the laws of McKean–Vlasov PDEs. Ultimately, this work reveals quantitative estimates of propagation of chaos for interacting particle systems. Furthermore, we are able to provide weak propagation of chaos estimates for ensembles of interacting particles and show that these may have some remarkable properties.
Funding Statement
Jean-François Chassagneux was partially supported by the Agence Nationale de la Recherche project ANR-16-CE40-0015-01.
Lukasz Szpruch acknowledges the support of The Alan Turing Institute under the Engineering and Physical Sciences Research Council grant EP/N510129/1.
Citation
Jean-François Chassagneux. Lukasz Szpruch. Alvin Tse. "Weak quantitative propagation of chaos via differential calculus on the space of measures." Ann. Appl. Probab. 32 (3) 1929 - 1969, June 2022. https://doi.org/10.1214/21-AAP1725
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