Diffusion approximation for multi-scale McKean-Vlasov SDEs through different methods

In this paper, our objective is to investigate the diffusion approximation for multi-scale McKean-Vlasov stochastic differential equations. More precisely, we first establish the tightness of the law of ${\left\{X^{\epsilon }\right\}}_{0<\epsilon ⩽1}$ in $C\left(\right[0,T];R^n)$. Subsequently, we demonstrate that any accumulation point of ${\left\{X^{\epsilon }\right\}}_{0<\epsilon ⩽1}$ can be regarded as a solution to the martingale problem or a weak solution of a distribution-dependent stochastic differential equation, which incorporates new drift and diffusion terms compared to the original equation. Our main contribution lies in employing two different methods to explicitly characterize the accumulation point. The diffusion matrices obtained through these two methods have different forms, however we assert their essential equivalence through a comparison.