Compactness and attracting of random attractors for non-autonomous stochastic lattice dynamical systems in weighted space ℓρp

In this article, some sufficient conditions on the regularity of random attractors are first provided for general random dynamical systems in the weighted space ℓρp(p>2) of infinite sequences. They are then used to study the asymptotic dynamics of a class of non-autonomous stochastic lattice differential equations with spatially valued additive noises. The existences of tempered random attractors for this in both spaces ℓρ2 and ℓρp are proved respectively, which implies that the obtained ℓρ2-random attractor is compact and attracting in the topology of ℓρp space. To solve this, a common embedding space of ℓρ2 and ℓρp is constructed and some new estimates are also developed here.