Diffusion Approximation for Multi-Scale McKean-Vlasov SDEs Through Different Methods

In this paper, we aim to study the diffusion approximation for multi-scale McKean-Vlasov stochastic differential equations. More precisely, we prove the weak convergence of slow process $X^\varepsilon$ in $C([0,T];\mathbb{R}^n)$ towards the limiting process $X$ that is the solution of a distribution dependent stochastic differential equation in which some new drift and diffusion terms compared to the original equation appear. The main contribution is to use two different methods to explicitly characterize the limiting equations respectively. The obtained diffusion coefficients in the limiting equations have different form through these two methods, however it will be asserted that they are essential the same by a comparison.