We study a class of discrete-time random dynamical systems with compact phase space. Assuming that the deterministic counterpart of the system in question possesses a dissipation property, its linearisation is approximately controllable, and the driving noise is bounded and has a decomposable structure, we prove that the corresponding family of Markov processes has a unique stationary measure, which is exponentially mixing in the dual-Lipschitz metric. The abstract result is applicable to nonlinear dissipative PDEs perturbed by a bounded random force which affects only a few Fourier modes. We assume that the nonlinear PDE in question is well posed, its nonlinearity is non-degenerate in the sense of the control theory, and the random force is a regular and bounded function of time which satisfies some decomposability and observability hypotheses. This class of forces includes random Haar series, where the coefficients for high Haar modes decay sufficiently fast. In particular, the result applies to the 2D Navier–Stokes system and the nonlinear complex Ginzburg–Landau equations. The proof of the abstract theorem uses the coupling method, enhanced by the Newton–Kantorovich–Kolmogorov fast convergence.

中文翻译：

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一类具有有限简并噪声的耗散PDE的指数混合

我们研究了一类具有紧凑相空间的离散时间随机动力系统。假设所讨论系统的确定性对等方具有耗散性质，其线性化是可控制的，并且驱动噪声是有界的并且具有可分解的结构，我们证明相应的马尔可夫过程族具有唯一的平稳度量，即以对偶的Lipschitz度量进行指数混合。该抽象结果适用于受有限随机力干扰的非线性耗散PDE，该有限力仅影响少数傅立叶模式。我们假设所讨论的非线性PDE具有适当的位置，从控制理论的意义上讲，其非线性是非退化的，随机力是时间的有规律和有界函数，满足一些可分解性和可观察性假设。此类力包括随机Haar级数，其中高Haar模式的系数衰减足够快。尤其是，该结果适用于二维Navier–Stokes系统和非线性复杂的Ginzburg–Landau方程。抽象定理的证明使用了耦合方法，牛顿-坎托罗维奇-柯尔莫哥罗夫快速收敛得到了增强。