We consider a Schrödinger equation with defocusing polynomial nonlinearity and multiplicative noise.
In a bounded 2D domain, by considering Dirichlet boundary conditions we prove the existence of an invariant measure; uniqueness holds for a particular choice of the noise. Similar results hold for Neumann b.c. as well or in a bounded manifold without boundary. The technique is based on the Krylov-Bogoliubov technique in the setting of weakly topologies. This is based on a joint work with Z. Brzezniak and M. Zanella.