We establish the irreducibility of stochastic real Ginzburg-Landau e-quation with--stable noises by a maximal inequality and solving a control problem. As applications, we prove that the system converges to its equilibrium measure with exponential rate under a topology stronger than total variation and obeys the moder-ate deviation principle by constructing some Lyapunov test functions; we also establish large deviation principle by Wu's hyper-exponential recurrence criterion. It is based on joint works with Jie Xiong and Lihu Xu.
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