Global well-posedness and long-term behavior of discrete reaction-diffusion equations driven by superlinear noise

The global well-posedness as well as long-term behavior in terms of mean random attractors and invariant measures are investigated for a class of stochastic discrete reaction-diffusion equations defined on Zk with a family of superlinear noise. The existence and uniqueness of weak pullback mean random attractors for the mean random dynamical system associated with the non-autonomous equations are established in L2(Ω,ℓ2). The existence of invariant measures for the autonomous equations is established in ℓ2 by Krylov-Bogolyubov's method. The idea of uniform estimates on the tails of solutions is employed to establish the tightness of a family of distribution laws of the solutions. It seems that this is the first time to study the random attractors and invariant measures of stochastic equations with superlinear noise.