Ergodicity of the stochastic real Ginzburg–Landau equation driven by [formula omitted]-stable noises

Abstract

We study the ergodicity of the stochastic real Ginzburg–Landau equation driven by additive α-stable noises, showing that as α∈(3/2,2), this stochastic system admits a unique invariant measure. After establishing the existence of invariant measures by the same method as in Dong et al. (2011) [12], we prove that the system is strong Feller and accessible to zero. These two properties imply the ergodicity by a simple but useful criterion by Hairer (2008) [14]. To establish the strong Feller property, we need to truncate the nonlinearity and apply a gradient estimate established by Priola and Zabczyk (2011) [22] (or see Priola et al. (2012) [20] for a general version for the finite dimension systems). Because the solution has discontinuous trajectories and the nonlinearity is not Lipschitz, we cannot 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, the 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.