Weak Mean Attractors and Invariant Measures of Fractional Stochastic Delay FitzHugh-Nagumo Systems on R N

Abstract

This article is devoted to the study of mean attractors and invariant measures for the fractional stochastic FitzHugh-Nagumo systems on ${\mathbb{R}}^N$ with locally Lipschitz noise. When the diffusion coefficients are locally Lipschitz, we establish the global-in-time well-posedness and the existence of Itô’s energy equalities for the stochastic systems. In addition, we establish the existence and uniqueness of weak pullback mean random attractors of the mean random dynamical systems generated by the solution operators in the product Bochner space: $$L^2(\mathrm{\Omega },\mathcal{F},L^2({\mathbb{R}}^N)\times L^2({\mathbb{R}}^N))\times L^2(\mathrm{\Omega },\mathcal{F};L^2((-\rho ,0);L^2({\mathbb{R}}^N)\times L^2({\mathbb{R}}^N)))$$ over the complete filtered probability space $(\mathrm{\Omega },\mathcal{F},\{{\mathcal{F}}_t{\}}_{t\in \mathbb{R}},P)$ , where $\rho \ge 0$ is the time-delay parameter. When the diffusion and drift terms are globally Lipschitz continuous, we address the existence, uniqueness and exponentially mixing of invariant measures for the autonomous stochastic FitzHugh-Nagumo systems on ${\mathbb{R}}^N$ under a large damping condition. This work deepens and extends the results of Wang, Guo and Wang (Science China Mathematics, 64 (11): 2395–2436, 2021).