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We study the ergodicity of stochastic real Ginzburg-Landau equation driven byadditive $\alpha$-stable noises, showing that as $\alpha \in (3/2,2)$, thisstochastic system admits a unique invariant measure. After establishing theexistence of invariant measures by the same method as in [9], we prove that thesystem is strong Feller and accessible to zero. These two properties imply theergodicity by a simple but useful criterion in [16]. To establish the strongFeller property, we need to truncate the nonlinearity and apply a gradientestimate established in [26] (or see [24]} for a general version for the finitedimension systems). Because the solution has discontinuous trajectories and thenonlinearity is not Lipschitz, we can not solve a control problem to getirreducibility. Alternatively, we use a replacement, i.e., the fact that thesystem is accessible to zero. In section 3, we establish a maximal inequalityfor stochastic $\alpha$-stable convolution, which is crucial for studying thewell-posedness, strong Feller property and the accessibility of the mildsolution. We hope this inequality will also be useful for studying other SPDEsforced by $\alpha$-stable noises.

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