Weak Mean Attractors of Fractional Stochastic Lattice Systems Driven by Nonlinear Delay Noise

This paper deals with the existence and uniqueness of weak pullback mean random attractors for fractional stochastic lattice systems driven by nonlinear delay noise defined on the entire integer set \(\mathbb{Z}\). We first establish the global well-posedness to stochastic lattice system in \(C([\tau,\infty), L^2(\Omega, \ell^2))\) when the nonlinear diffusion terms and drift terms are locally Lipschitz continuous functions. Then we define a mean random dynamical system through the solution operators and prove the existence and uniqueness of weak pullback mean random attractors in \(L^2(\Omega,\mathcal{F};\ell^2)\times L^2\big(\Omega,\mathcal{F};L^2((-\rho,0), \ell^2)\big)\) under certain conditions.