Random attractor of stochastic complex Ginzburg-Landau equation with multiplicative noise on unbounded domain

The existence of a compact random attractor for the stochastic complex Ginzburg-Landau equation with multiplicative noise has been investigated on unbounded domain. The solutions are considered in suitable spaces with weights. According to crucial properties of Ornstein-Uhlenbeck process, using the tail-estimates method, the key uniform a priori estimates for the tail of solutions have been obtained, which give the asymptotic compactness of random attractors. Then the existence of a compact random attractor for the corresponding dynamical system is proved in suitable spaces with weights.