We study Ornstein-Uhlenbeck processes whose parameters are modulated by an external two-state Markov process. The conditional means of such a process for a given modulation follow an analogue of the Langevin equation, which is controlled by a pair of telegraph processes. Stationary distributions of such processes are described. The relationship between stationary distributions and the distribution of the corresponding exponential functional is also discussed. Of interest is the limiting behaviour of these processes under conditions similar to the Kac scaling. Limiting processes turn out to be different classes of ordinary Ornstein-Uhlenbeck processes.

我们研究了 Ornstein-Uhlenbeck 过程，其参数由外部两态马尔可夫过程调制。对于给定调制，这种过程的条件方法遵循朗之万方程的类似物，该方程由一对电报过程控制。描述了此类过程的平稳分布。还讨论了平稳分布与相应指数泛函分布之间的关系。有趣的是这些过程在类似于 Kac 缩放的条件下的限制行为。限制过程原来是不同类别的普通 Ornstein-Uhlenbeck 过程。