Randomly forced CGL equation: Stationary measures and the inviscid limit

We study a complex Ginzburg-Landau (CGL) equation perturbed by a random force which is white in time and smooth in the space variable x. Assuming that dim = 4, we prove that this equation has a unique solution and discuss its asymptotic in time properties. Next we consider the case when the random force is proportional to the square root of the viscosity and study the behaviour of stationary solutions as the viscosity goes to zero. We show that, under this limit, a subsequence of solutions in question converges to a nontrivial stationary process formed by global strong solutions of the nonlinear Schrödinger equation.