AI 中文总结
研究由有界混合噪声驱动的动力系统，通过将连续时间系统简化为离散时间随机动力系统，引入一类满足特定假设的混合随机力，在一定假设下证明有限维和无限维相空间的指数混合，还给出了对常微分方程和大气动力学偏微分方程的应用。
AI 中文摘要
我们研究由有界混合随机力驱动的离散时间和连续时间耗散动力系统的混合性质。连续时间系统被简化为由时间-1映射生成的离散时间随机动力系统，以便在离散设置中进行主要分析。我们引入了一类混合随机力，其关于过去的条件分布满足自然的正则性、递归性和非退化性假设，扩展了Kuksin-Shirikyan在GAFA（2025）中为更具限制性的过程类先前开发的框架。在系统的线性化可控性假设下，我们证明了有限维相空间在全变差度量下的指数混合。然后，在对系统和随机力的限制进行适当修正的情况下，我们建立了无限维对应物，在对偶-利普希茨度量下产生指数混合。我们的方法基于将动力学提升到无限维历史空间上的适当马尔可夫过程，并通过康托罗维奇泛函方法应用多布林耦合论证。作为应用，我们推导了由有界混合随机过程驱动的一类广泛的常微分方程的指数混合。作为我们结果对偏微分方程的应用，我们讨论了大气动力学的随机扰动原始方程。
英文摘要
We study the mixing properties of discrete-time and continuous-time dissipative dynamical systems driven by bounded mixing random forces. The continuous-time systems are reduced to discrete-time random dynamical systems generated by time-one maps, so that the main analysis is carried out in the discrete setting. We introduce a class of mixing random forcings whose regular conditional distributions with respect to the past satisfy natural regularity, recurrence, and non-degeneracy assumptions, extending the framework previously developed for more restrictive classes of processes in a paper by Kuksin-Shirikyan in GAFA (2025). Under a linearised controllability assumptions on the system, we prove exponential mixing in the total variation metric for finite-dimensional phase spaces. We then establish an infinite-dimensional counterpart yielding exponential mixing in the dual-Lipschitz metric under suitable amendments of restrictions on the system and the random forcing. Our approach is based on lifting the dynamics to an appropriate Markov process on an infinite-dimensional history space and applying a Doeblin coupling argument through the method of Kantorovich functional. As applications, we derive exponential mixing for a broad class of ordinary differential equations driven by mixing random processes with bounded continuous trajectories. As an application of our result to PDEs we discuss the randomly perturbed primitive equations of atmospheric dynamics.
