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We consider a collection of fully coupled weakly interacting diffusionprocesses moving in a two-scale environment. We study the moderate deviationsprinciple of the empirical distribution of the particles' positions in thecombined limit as the number of particles grow to infinity and the time-scaleseparation parameter goes to zero simultaneously. We make use of weakconvergence methods, which provide a convenient representation for the moderatedeviations rate function in terms of an effective mean field control problem.We rigorously obtain equivalent representations for the moderate deviationsrate function in an appropriate "negative Sobolev" form, which is reminiscentof the large deviations rate function for the empirical measure of weaklyinteracting diffusions obtained in the 1987 seminal paper byDawson-G\"{a}rtner. In the course of the proof we obtain related ergodictheorems and we rigorously study the regularity of Poisson type of equationsassociated to McKean-Vlasov problems, both of which are topics of independentinterest.

Moderate deviations for fully coupled multiscale weakly interacting particle systems