Article ID Journal Published Year Pages File Type
1156718 Stochastic Processes and their Applications 2013 27 Pages PDF
Abstract

We study the ergodicity of the stochastic real Ginzburg–Landau equation driven by additive αα-stable noises, showing that as α∈(3/2,2)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 αα-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 αα-stable noises.

Related Topics
Physical Sciences and Engineering Mathematics Mathematics (General)
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