Asymptotic behavior of a class of multiple time scales stochastic kinetic equations

We consider a class of stochastic kinetic equations, depending on two time scale separation parameters $\epsilon$ and $\delta$: the evolution equation contains singular terms with respect to $\epsilon$, and is driven by a fast ergodic process which evolves at the time scale $t/\delta^2$. We prove that when $(\epsilon,\delta)\to (0,0)$ the density converges to the solution of a linear diffusion PDE. This is a mixture of diffusion approximation in the PDE sense (with respect to the parameter $\epsilon$) and of averaging in the probabilistic sense (with respect to the parameter $\delta$). The proof employs stopping times arguments and a suitable perturbed test functions approach which is adapted to consider the general regime $\epsilon\neq \delta$.