Random Attractor, Invariant Measures, and Ergodicity of Lattice p -Laplacian Equations Driven by Superlinear Noise

highly nonlinear lattice p -Laplacian equation driven by superlinear noise is considered. By using an appropriate stopping time technique and the dissipativeness of the nonlinear drift terms, we establish the global existence and uniqueness of the solutions in C ([τ ,∞ ), L^2(Ω , ℓ ^2) )∩ L^p (Ω , L_loc^p((τ ,∞ ), ℓ ^p ) ) for any p>2 when the coefficient of the noise has a superlinear growth order q∈ [2,p) . By the theory of mean random dynamical systems recently developed in Kloeden and Lorenz (J Differ Equ 253:1422–1438, 2012) and Wang (J Dyn Differ Equ 31:2177–2204, 2019), we prove that the nonautonomous system has a unique mean random attractor in the Bochner space L^2(Ω , ℓ ^2) . When the drift and diffusion terms satisfy certain conditions, we show that the autonomous system has a unique, ergodic, mixing, and stable invariant probability measure in ℓ ^2 . The idea of uniform tail-estimates is employed to establish the tightness of a family of distribution laws of the solutions in order to overcome the lack of compactness in infinite lattice as well as the infinite-dimensionalness of ℓ ^2 . This work deepens and extends the results in Wang and Wang (Stoch Process Appl 130:7431–7462, 2020) where the coefficient of the noise grows linearly rather than superlinearly.