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Increasing propagation of chaos for mean field models. (English) Zbl 0928.60092
The authors consider mean-field measures \(\mu^{(N)}\) with interacting potential \(F\). The probability measure \(\mu^{(N)}\) is the law of an interacting system of size \(N\). The authors are interested in the asymptotic independence properties of these measures, as \(N\) tends to infinity. This asymptotic behavior is called asymptotic chaoticity. Under mild assumptions, one proves that the empirical measures of the system converge in law to a deterministic measure \(\mu^*\), and equivalently for any finite \(k\), the convergence of the law of \(k\)-subsystem under \(\mu^{(N)}\) to \((\mu^*)^{\otimes k}\).
The authors estimate the relative entropy distance between \(\mu^{(N)}\) and appropriate simpler exchangeable measures \(\nu^N\) which are related to the law \(\mu^*\). They prove that if the minimum of the function \(H(\cdot\mid\mu)- F(\cdot)\) is unique, say \(\mu^*\), then \(\nu^N=(\mu^*)^{\otimes N}\), and propagation of chaos holds for blocks of size \(o(N)\) if this minimum is nondegenerate. Certain degenerate situations are also studied. Next these results are applied to the Langevin dynamics of interacting particles leading to a McKean-Vlasov limit.
Reviewer: S.Meléard (Paris)

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C22 Interacting particle systems in time-dependent statistical mechanics
60G99 Stochastic processes
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
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