Asymptotic Behavior of Stochastic Complex Lattice Systems Driven by Superlinear Noise

This paper deals with the asymptotic behavior of complex stochastic lattice systems driven by superlinear noise. We first prove the well-posedness and the existence of weak mean random attractors of the systems. We then show the existence of periodic measures and the weak compactness of the set of all periodic measures when a parameter varies in a bounded interval. We finally examine the limiting behavior of the periodic measures and in particular prove that every limit of periodic measures of the stochastic Ginzburg-Landau lattice system must be a periodic measure of the stochastic Schrodinger lattice system.