Exponential mixing of 2D stochastic Ginzburg–Landau–Newell equations driven by degenerate noise

Abstract

The current paper is devoted to stochastic Ginzburg–Landau–Newell equation with degenerate stochastic forcing. First, we establish a type of gradient inequality, which is also essential to proving asymptotic strong Feller property. Then, we prove that the corresponding dynamical system possesses a strong type of Lyapunov structure and is of a relatively weak form of irreducibility. Finally, we prove that the corresponding Markov semigroup possesses a unique, exponentially mixing invariant measure.