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We consider a stochastic nonlinear defocusing Schr\"{o}dinger equation withzero-order linear damping, where the stochastic forcing term is given by acombination of a linear multiplicative noise in the Stratonovich form and anonlinear noise in the It\^o form. We work at the same time on compactRiemannian manifolds without boundary and on relatively compact smooth domainswith either the Dirichlet or the Neumann boundary conditions, always indimension 2. We construct a martingale solution using a modifiedFaedo-Galerkin's method, following arXiv:1707.05610. Then by means of theStrichartz estimates deduced from arXiv:math/0609455 but modified for ourstochastic setting we show the pathwise uniqueness of solutions. Finally, weprove the existence of an invariant measure by means of a version of theKrylov-Bogoliubov method, which involves the weak topology, as proposed byMaslowski and Seidler. This is the first result of this type for stochastic NLSon compact Riemannian manifolds without boundary and on relatively compactsmooth domains even for an additive noise. Some remarks on the uniqueness in aparticular case are provided as well.

Invariant measures for a stochastic nonlinear and damped 2D Schr\"odinger equation