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Increasing propagation of chaos for mean field models. (English) Zbl 0928.60092

The authors consider mean-field measures ${\mu }^{\left(N\right)}$ with interacting potential $F$. The probability measure ${\mu }^{\left(N\right)}$ 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 ${\mu }^{\left(N\right)}$ to ${\left({\mu }^{*}\right)}^{\otimes k}$.

The authors estimate the relative entropy distance between ${\mu }^{\left(N\right)}$ and appropriate simpler exchangeable measures ${\nu }^{N}$ which are related to the law ${\mu }^{*}$. They prove that if the minimum of the function $H\left(·\mid \mu \right)-F\left(·\right)$ is unique, say ${\mu }^{*}$, then ${\nu }^{N}={\left({\mu }^{*}\right)}^{\otimes N}$, and propagation of chaos holds for blocks of size $o\left(N\right)$ 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.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82C22 Interacting particle systems 60G99 Stochastic processes 82C31 Stochastic methods in time-dependent statistical mechanics
##### Keywords:
propagation of chaos; entropy; mean field models