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 Z k 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 L 2 (Ω , ℓ 2). The existence of invariant measures for the autonomous equations is established in ℓ 2 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. [ABSTRACT FROM AUTHOR]