# zbMATH — the first resource for mathematics

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
##### Keywords:
propagation of chaos; entropy; mean field models
Full Text: