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Erschienen in:
2021 | OriginalPaper | Buchkapitel
14. Ergodicity of Stochastic Hydrodynamical-Type Evolution Equations Driven by \(\alpha \)-Stable Noise
verfasst von : Jianhua Huang, Tianlong Shen, Yuhong Li
Erschienen in: Contemporary Approaches and Methods in Fundamental Mathematics and Mechanics
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Abstract
The present paper is devoted to the ergodicity of stochastic 2D hydro-dynamical-type evolution equation driven by \(\alpha \)-stable noise with \(\alpha \in (\frac{3}{2},2)\), which covers stochastic Navier–Stokes equation, magneto-hydrodynamic equation, Boussinesq equation, magnetic Benard equation and so on. The existence and uniqueness of the invariant measure of this stochastic system are established by the strong Feller property and accessibility of the transition semigroup. The novel to overcome those difficulties caused by the trajectory discontinuity and lower regularity of the corresponding Ornstein–Uhlenbeck process for \(\alpha \)-stable noise. As applications of the abstract result, the existence and uniqueness of the invariant measure for the stochastic Boussinesq equation and stochastic 2D Magneto-hydrodynamic equation are given.