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We consider the free linear Schroedinger equation on a torus $\mathbb T^d$,perturbed by a Hamiltonian nonlinearity, driven by a random force and damped bya linear damping: $$u_t -i\Delta u +i\nu \rho |u|^{2q_*}u = - \nu f(-\Delta) u+ \sqrt\nu\,\frac{d}{d t}\sum_{k\in \mathbb Z^d} b_k\beta^k(t)e^{ik\cdot x} \ .$$ Here $u=u(t,x),\ x\in\mathbb T^d$, $0<\nu\ll1$, $q_*\in\mathbb N\cup\{0\}$,$f$ is a positive continuous function, $\rho$ is a positive parameter and$\beta^k(t)$ are standard independent complex Wiener processes. We areinterested in limiting, as $\nu\to0$, behaviour of solutions for this equationand of its stationary measure. Writing the equation in the slow time $\tau=\nut$, we prove that the limiting behaviour of the both is described by theeffective equation $$ u_\tau+ f(-\Delta) u = -iF(u)+\frac{d}{d\tau}\sumb_k\beta^k(\tau)e^{ik\cdot x} \, $$ where the nonlinearity $F(u)$ is made outof the resonant terms of the monomial $ |u|^{2q_*}u$. We explain the relevanceof this result for the problem of weak turbulence.

Resonant averaging for weakly nonlinear stochastic Schr\"odinger equations