Attractors of non-autonomous stochastic lattice systems in weighted spaces

•We study a class of non-autonomous stochastic lattice systems.•We prove the existence and uniqueness of tempered random attractors.•We establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.•When deterministic external forcing terms are periodic in time, we show the random attractors are pathwise periodic.

We study the asymptotic behavior of solutions to a class of non-autonomous stochastic lattice systems driven by multiplicative white noise. We prove the existence and uniqueness of tempered random attractors in a weighted space containing all bounded sequences, and establish the upper semicontinuity of random attractors as the intensity of noise approaches zero. We also construct maximal and minimal tempered random complete solutions which bound the attractors from above and below, respectively. When deterministic external forcing terms are periodic in time, we show the random attractors are pathwise periodic. In addition, we exhibit a non-autonomous stochastic lattice system which possesses an infinite-dimensional tempered random attractor.