Moderate deviations for fully coupled multiscale weakly interacting particle systems

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 a variational form in terms of an effective mean field control problem. We rigorously obtain equivalent representation for the moderate deviations rate function in an appropriate “negative Sobolev” form, proving their equivalence, which is reminiscent of the large deviations rate function form 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 consider the regularity of Poisson type of equations associated to McKean–Vlasov problems, both of which are topics of independent interest. A novel “doubled corrector problem” is introduced in order to control derivatives in the measure arguments of the solutions to the related Poisson equations used to control behavior of fluctuation terms.