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 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 random force which affects only a few Fourier modes and belongs to a certain class of random processes. 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 coefficients for high Haar modes decay sufficiently fast. 查看全文>>