We investigate the long-time behavior of the Ornstein–Uhlenbeck process driven by Lévy noise with regime switching. We provide explicit criteria on the transience and recurrence of this process. Contrasted with the Ornstein–Uhlenbeck process driven simply by Brownian motion, whose stationary distribution must be light-tailed, both the jumps caused by the Lévy noise and the regime switching described by a Markov chain can derive the heavy-tailed property of the stationary distribution. The different role played by the Lévy measure and the regime-switching process is clearly characterized.

中文翻译：

---

具有状态转换的 Lévy 驱动的 Ornstein-Uhlenbeck 过程的长期行为

我们研究了由 Lévy 噪声驱动的 Ornstein-Uhlenbeck 过程的长期行为以及状态切换。我们为这一过程的短暂性和重复性提供了明确的标准。与仅由布朗运动驱动的 Ornstein-Uhlenbeck 过程（其平稳分布必须是轻尾的）相比，由 Lévy 噪声引起的跳跃和马尔可夫链描述的状态转换都可以推导出平稳分布的重尾特性. Lévy 措施和制度转换过程所起的不同作用得到了明确的描述。