Small mass limit for stochastic N-interacting particles system in L2(Rd) in mean field limit

An $L^2\left(R^d\right)$-valued stochastic N-interacting particles system with small mass is investigated. Mean field limit and the propagation of chaos are derived. Moreover the small mass limit of the solution is also built, which can be seen as a Smoluchowski–Kramers approximation on unbounded domain. Here a key step is the asymptotic compactness of the distribution of the solution, which is derived via a splitting technique of the domain $R^d$ and some estimation of the solution for the mean field limit equation. We also show that the limits $N\to \infty$ and $ϵ\to 0$ commute.