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Ergodicity of stochastic real Ginzburg-Landau equation driven by $α$-stable noises

We study the ergodicity of stochastic real Ginzburg-Landau equation driven by additive $α$-stable noises, showing that as $α\in (3/2,2)$, this stochastic system admits a unique invariant measure. After establishing the existence of invariant measures by the same method as in [9], we prove that the system is strong Feller and accessible to zero. These two properties imply the ergodicity by a simple but useful criterion in [16]. To establish the strong Feller property, we need to truncate the nonlinearity and apply a gradient estimate established in [26] (or see [24]} for a general version for the finite dimension systems). Because the solution has discontinuous trajectories and the nonlinearity is not Lipschitz, we can not solve a control problem to get irreducibility. Alternatively, we use a replacement, i.e., the fact that the system is accessible to zero. In section 3, we establish a maximal inequality for stochastic $α$-stable convolution, which is crucial for studying the well-posedness, strong Feller property and the accessibility of the mild solution. We hope this inequality will also be useful for studying other SPDEs forced by $α$-stable noises.