Resonant averaging for small-amplitude solutions of stochastic nonlinear Schrödinger equations

We consider the free linear Schrödinger equation on a torus d, perturbed by a Hamiltonian nonlinearity, driven by a random force and subject to a linear damping: (equation presented) Here u = u(t, x), x ϵ d, 0 <, 0 < v <, qϵ ℕ, f is a positive continuous function, ρ is a positive parameter and are standard independent complex Wiener processes. We are interested in limiting, as ν → 0, behaviour of distributions of solutions for this equation and of its stationary measure. Writing the equation in the slow time τ = νt, we prove that the limiting behaviour of them both is described by the effective equation (equation presented) where the nonlinearity F(u) is made out of the resonant terms of the monomial |u|2q-u.