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 $R^d$. We first prove the global-in-time well-posedness of the equation in $L^{2\vartheta }(\Omega ,H)$ with $H≔L^2\left(R^d\right)$ for any $\vartheta ⩾1$, and then demonstrate the existence and uniqueness of PMAs in $(P_{2\vartheta }\left(H\right),d_{P\left(H\right)})$ 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 $(P_{2\vartheta }\left(H\right),d_{P\left(H\right)})$. The upper semicontinuity of the PMAs is established as the noise intensity $ϵ$ converges to ${ϵ}_0$ 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 $\sqrt{2\beta +1}⩾\left|\eta \beta \right|$ 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.