Many systems in applied sciences involve multiple interacting scales, and a natural goal is to derive effective descriptions capturing the influence of fast variables on slower ones.

We consider stochastic systems with slow and fast components. It is well known that the evolution of their law is described by a Fokker-Planck equation, yielding an equivalent PDE formulation. In the classical case where the fast process admits a unique invariant measure, time-scale separation allows one to replace it in the effective dynamics by the equilibrium behavior. The averaged dynamics can be derived using PDE techniques, in particular multiscale expansions and the associated Poisson equation.

We finally discuss the case where the fast dynamics admits multiple invariant measures. In this regime, standard averaging techniques break down. This issue arises, for instance, in McKean-Vlasov type equations.