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We consider the free linear Schr\"odinger equation on a torus $\mathbb T^d$, 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_{k\in \mathbb Z^d} b_l\beta^k(t)e^{ik\cdot x} \ .
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