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

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We consider a class of stochastic kinetic equations, depending on two time scale separation parameters ɛ and δ: the evolution equation contains singular terms with respect to ɛ, and is driven by a fast ergodic process which evolves at the time scale t/δ2. We prove that when (ɛ,δ)→(0,0) the spatial density converges to the solution of a linear diffusion PDE. This result is a mixture of diffusion approximation in the PDE sense (with respect to the parameter ɛ) and of averaging in the probabilistic sense (with respect to the parameter δ). The proof employs stopping times arguments and a suitable new perturbed test functions approach which is adapted to consider the general regime ɛ≠δ.