The inviscid limit for long time statistics of the one-dimensional stochastic Ginzburg-Landau equation

We consider the long time statistics of a one-dimensional stochastic Ginzburg-Landau equation with cubic nonlinearity while being subjected to random perturbations via an additive Gaussian noise. Under the assumption that sufficiently many directions of the phase space are stochastically forced, we find that the dynamics is attractive toward the unique invariant probability measure with a polynomial rate that is independent of the vanishing viscosity. This relies on a coupling technique exploiting a Foias-Prodi argument specifically tailored to the system. Then, in the inviscid regime, we show that the sequence of invariant measures converges toward the invariant measure of the stochastic Schr\"odinger equation in a suitable Wasserstein distance. Together with the uniform polynomial mixing, we obtain the validity of the inviscid limit for the solutions on the infinite time horizon with a log log rate.