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We study the limiting behavior of the two-dimensional singular stochasticstochastic cubic nonlinear complex Ginzburg-Landau with Gibbs measure initialdata. We show that in the appropriate small viscosity and small noise regimes,the limiting dynamics is given by the deterministic cubic nonlinearSchr\"odinger equation at Gibbs equilibrium. In order to obtain thisconvergence, our approach combines both heat and Schr\"odinger analysis, withinthe framework of the Fourier restriction norm method of Bourgain (1993).

On the inviscid limit of the singular stochastic complex Ginzburg-Landau equation at statistical equilibrium