Long Term Behavior for a Class of Stochastic Delay Lattice Systems in Xρ Space

In this paper, we focus on the asymptotic behavior of solutions to stochastic delay lattice equations with additive noise and deterministic forcing. We first show the existence of a continuous random dynamical system for the equations. Then we investigate the pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractor in Xρ space. Finally, ergodicity of the systems is achieved