Measure attractors and measure evolution systems in high-order moment metric space for fractional stochastic Ginzburg–Landau equations on unbounded domains

We initially study pullback measure attractors (PMAs) and measure evolution systems (MESs) of fractional stochastic complex-valued Ginzburg–Landau equations on Rd. We first prove the global-in-time well-posedness of the equation in L2ϑ(Ω,H) with H≔L2(Rd) for any ϑ⩾1, and then demonstrate the existence and uniqueness of PMAs in (P2ϑ(H),dP(H)) the high-order moment metric space of probability measures on H. By the structures of PMAs rather than the Krylov–Bogolyubov method, we prove that the time-inhomogeneous transition operator has a MES in (P2ϑ(H),dP(H)). The upper semicontinuity of the PMAs is established as the noise intensity ϵ converges to ϵ0 in [0,1]. Under a large damping condition, we prove that the PMA is a singleton set, and establish the uniqueness, exponentially mixing and stability of MESs, periodic measures, and invariant measures of the corresponding systems. The difficulties caused by the non-monotonic drift term and the non-compactness of the standard Sobolev embedding on unbounded domains are surmounted by a balance condition 2β+1⩾|ηβ| and the idea of uniform tail-ends estimates. Several numerical simulations are provided for above qualitative analysis. The methods used in this paper can be applied to other stochastic real or complex-valued PDEs.