Multi-scale McKean–Vlasov SDEs: Moderate deviation principle in different regimes

The main aim of this paper is to study the moderate deviation principle for McKean-Vlasov stochastic differential equations with multiple scales. Specifically, we are interested in the asymptotic estimates of the deviation processes $\frac{X^δ-\bar{X}}{λ(δ)}$ as $δ\to 0$ in different regimes (i.e. $\varepsilon=o(δ)$ and $\varepsilon=O(δ)$), where $δ$ stands for the intensity of the noise and $\varepsilon:=\varepsilon(δ)$ stands for the time scale separation. The rate functions in two regimes are different, in particular, we show that it is strongly affected by the noise of the fast component in latter regime, which is essentially different from the former one and the case of large deviations (cf. \cite{HLLS}). As a by-product, the explicit representation formulas of the rate functions in all of regimes are also given. The main techniques are based on the weak convergence approach and the functional occupation measure approach.