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

International audienceWe consider the free linear Schrödinger equation on a torus T d , perturbed by a Hamiltonian nonlinearity, driven by a random force and subject to a linear damping: ut − i∆u + iνρ|u| 2q * u = −νf (−∆)u + √ ν d dt k∈Z d b k β k (t)e ik·x. Here u = u(t, x), x ∈ T d , 0 < ν 1, q * ∈ N, f is a positive continuous function, ρ is a positive parameter and β k (t) 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 uτ + f (−∆)u = −iF (u) + d dτ b k β k (τ)e ik·x , where the nonlinearity F (u) is made out of the resonant terms of the monomial |u| 2q * u