Random uniform attractors for non-autonomous stochastic lattice dynamical systems in weighted spaces

In this paper, we first study the existence and uniqueness of random uniform attractors to a class of non-autonomous stochastic lattice dynamical systems driven by multiplicative noise in a weighted space. Then, we establish the upper semicontinuity of random uniform attractors of the corresponding limiting system as the intensity of noise approaches ɛ0 > 0 or as the finite-dimensional approximation approaches the infinite-dimensional case. The study of random uniform attractors in weighted spaces brings about essential difficulties. Some special estimates will overcome the difficulties. Uniform-tail estimates are needed to prove the asymptotic compactness of solution operators, which is a major step towards proving the existence of attractors. The upper semicontinuity of random uniform attractors is proved when the union of all perturbed random uniform attractors is tempered.