Momentum estimates and ergodicity for the 3D stochastic cubic Ginzburg–Landau equation with degenerate noise

In this paper, we consider the ergodicity of invariant measures for the stochastic Ginzburg–Landau equation with degenerate random forcing. First, we show the existence and pathwise uniqueness of strong solutions with H1-initial data, and then the existence of an invariant measure for the Feller semigroup by the Krylov–Bogoliubov method. Then in the case of degenerate additive noise, using the notion of asymptotically strong Feller property, we prove the uniqueness of invariant measures for the transition semigroup.