I will discuss stochastic \epsilon-perturbations of deterministic integrable Hamiltonian systems in R^{2n}, linear and non-linear. I will show that, firstly, on time intervals of order 1/\epsilon the actions of solutions for perturbed equations are close to those of solutions for specially constructed effective stochastic equations. Secondly, if the effective equation is mixing, then the approximation of the actions of solutions for perturbed equations, provided by it, is uniform in time. The required mixing assumption admits an easy sufficient condition, as well as a sufficient condition, close to necessary, given in terms of control theory.