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

We consider the free linear Schroedinger equation on a torus $\T^d$, perturbed by a hamiltonian nonlinearity, driven by a random force and damped by a linear damping: $$ u_t -i\Delta u +i\nu \rho |u|^{2q_*}u = - \nu f(-\Delta) u + \sqrt\nu\,\frac{d}{d t}\sum_{\bk\in \Z^d} b_\bk\bb^\bk(t)e^{i\bk\cdot x} \ . $$ Here $u=u(t,x),\ x\in\T^d$, $0<\nu\ll1$, $q_*\in\N$, $f$ is a positive continuous function, $\rho$ is a positive parameter and $\bb^\bk(t)$ are standard independent complex Wiener processes. We are interested in limiting, as $\nu\to0$, behaviour of distributions of solutions for this equation and of its stationary measure. Writing the equation in the slow time $\tau=\nu t$, we prove that the limiting behaviour of the both % of solutions and of the stationary measure is described by the effective equation $$ u_\tau+ f(-\Delta) u = -iF(u)+\frac{d}{d\tau}\sum b_\bk\bb^\bk(\tau)e^{i\bk\cdot x} \ , $$ where the nonlinearity $F(u)$ is made out of the resonant terms of the monomial $ |u|^{2q_*}u$.