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We consider a collection of fully coupled weakly interacting diffusion processes moving in a two-scale environment. We study the moderate deviations principle of the empirical distribution of the particles' positions in the combined limit as the number of particles grow to infinity and the time-scale separation parameter goes to zero simultaneously. We make use of weak convergence methods, which provide a convenient representation for the moderate deviations rate function in terms of an effective mean field control problem. We rigorously obtain equivalent representations for the moderate deviations rate function in an appropriate "negative Sobolev" form, which is reminiscent of the large deviations rate function for the empirical measure of weakly interacting diffusions obtained in the 1987 seminal paper by Dawson-Gärtner. In the course of the proof we obtain related ergodic theorems and we rigorously study the regularity of Poisson type of equations associated to McKean-Vlasov problems, both of which are topics of independent interest.

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We consider a fully-coupled slow-fast system of McKean-Vlasov SDEs with full dependence on the slow and fast component and on the law of the slow component and derive convergence rates to its homogenized limit. We do not make periodicity assumptions, but we impose conditions on the fast motion to guarantee ergodicity. In the course of the proof we obtain related ergodic theorems and we gain results on the regularity of Poisson type of equations and of the associated Cauchy-Problem on the Wasserstein space that are of independent interest.

In this paper, we consider a fully-coupled slow–fast system of McKean–Vlasov stochastic differential equations with full dependence on the slow and fast component and on the law of the slow component and derive convergence rates to its homogenized limit. We do not make periodicity assumptions, but we impose conditions on the fast motion to guarantee ergodicity. In the course of the proof we obtain related ergodic theorems and we gain results on the regularity of Poisson type of equations and of the associated Cauchy problem on the Wasserstein space that are of independent interest.