Enlarged evolution system of measures of stochastic p-Laplace lattice systems with superlinear noise

Abstract This paper deals with the stochastic p -Laplace lattice systems driven by superlinear noise on Banach space ℓ p (an enlarged optimized space compared to Hilbert space ℓ 2 ) with p > 2. We use the dissipativeness of the nonlinear drift term growing at a polynomial growth rate p to control the superlinear noise, whose coefficient grows at a polynomial growth order q ∈ [ 2 , p ) . Thereby we show that the existence and uniqueness of enlarged solutions in C ( [ τ , ∞ ) , L p ( Ω , ℓ p ) ) ∩ L 2 p − 2 ( Ω , L loc 2 p − 2 ( τ , ∞ ; ℓ 2 p − 2 ) ) , and thus the existence of enlarged mean weak attractors in a higher-order Bochner space L p ( Ω , ℓ p ) . And then we show the existence of enlarged evolution system of measures along with the results developed in Da Prato and Röckner (2009 Seminar on Stochastic Analysis, Random Fields and Applications V ( Progress in Probability vol 59) pp 115–22). At last, we prove the upper semicontinuity of the set of all enlarged evolution system of measures, by establishing the convergence of enlarged solutions in probability as well as the asymptotic tightness of distribution laws for sufficiently large t , in which the proof of asymptotic tightness is based on the results of asymptotic bounded absorption and asymptotic tail property for enlarged solutions.