We construct stationary statistical solutions of a deterministic unforced nonlinear Schrödinger equation, by perturbing it by adding a linear damping \(\gamma u\) and a stochastic force whose intensity is proportional to \(\sqrt{\gamma }\) , and then letting \(\gamma \rightarrow 0^+\) . We prove indeed that the family of stationary solutions \(\{U_\gamma \}_{\gamma >0}\) of the perturbed equation possesses an accumulation point for any vanishing sequence \(\gamma _j\rightarrow 0^+\) and this stationary limit solves the deterministic unforced nonlinear Schrödinger equation and is not a trivial process. This technique has been introduced in Kuksin and Shirikyan (J Phys A: Math Gen 37:1–18, 2004), using a different dissipation. However, considering a linear damping of zero order and weaker solutions, we can deal with larger ranges of the nonlinearity and of the spatial dimension; moreover we consider the focusing equation and the defocusing equation as well.
