Limiting Dynamics for Stochastic FitzHugh–Nagumo Lattice Systems in Weighted Spaces

In this paper, stochastic FitzHugh–Nagumo lattice system with nonlinear noise in weighted spaces is considered. Firstly, the well-posedness of solution of such system in a weighted space L^2(Ω ,l^2_σ× l^2_σ ) is established, based on which we further prove the existence and uniqueness of weak pullback mean random attractor in the weighted space. Then the existence and uniqueness of invariant measure are proved in the weighted space l^2_σ× l^2_σ as well as exponentially mixing property in the sense of Wasserstein metric. Moreover, the limit behaviors of invariant measure in the weighted space l^2_σ× l^2_σ are also investigated with respect to noise intensity.