Limit measures of stochastic Schrödinger lattice systems

This paper is devoted to the existence of invariant measures and their limiting behavior of the stochastic Schrödinger lattice systems with respect to noise intensity. We prove the set of all invariant measures of the stochastic systems is weakly compact when the noise intensity varies in a bounded interval. We further show any limit of a sequence of invariant measures of the perturbed systems must be an invariant measure of the limiting system.