Asymptotic behavior of stochastic discrete complex Ginzburg–Landau equations

The complex Ginzburg–Landau equation is one of the most-studied equations in applied mathematics. We consider the discretization of complex Ginzburg–Landau equations on one dimensional lattice driven by a general Gaussian random field including the translation invariant one. The long time behavior of the sample paths and the distributions of solutions are studied respectively. Under the gauge nonlinear interaction, the dynamical behavior for the sample paths of the system is described by a global random attractor which is a random compact invariant set in a weighted Hilbert space. Furthermore the distributions of the system exponentially converge to the unique invariant measure of the system, that is the system is ergodic. The asymptotic compactness and dissipative method are important in our approach.