Higher order averaging of linear Fokker--Planck equations with periodic forcing

We investigate linear Fokker-Planck equations or stochastic Hamiltonian systems with periodic forcing where the impact of deterministic forcing is not captured by classical stochastic averaging. To overcome this problem, a formal energy projection method is introduced, which splits the corresponding Fokker-Planck equation and allows the computation of higher order stochastic averages. Generally, the resulting averages have to be computed with a numerical scheme. We illustrate the results of our method with two examples: the linear oscillator with periodic forcing and a nonlinear oscillator