Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative

The asymptotic behavior of a stochastic lattice system with a Caputo fractional time derivative is investigated. In particular, the existence of a global forward attracting set in the weak mean-square topology is established. A general theorem on the existence of solutions for a fractional SDE in a Hilbert space under the assumption that the nonlinear term is weakly continuous in a given sense is established and applied to the lattice system. The existence and uniqueness of solutions for a more general fractional SDEs is also obtained under a Lipschitz condition.