No abstract

We consider a class of stochastic kinetic equations, depending on two time scale separation parameters ε and δ: the evolution equation contains singular terms with respect to ε, and is driven by a fast ergodic process which evolves at the time scale t{δ 2 . We prove that when pε, δq Ñ p0, 0q 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 ε) and of averaging in the probabilistic sense (with respect to the parameter δ). The proof employs stopping times arguments and a suitable perturbed test functions approach which is adapted to consider the general regime ε ‰ δ.