A piecewise-deterministic Markov process, specified by random jumps and switching semi-flows, as well as the associated Markov chain given by its post-jump locations, are investigated in this paper. The existence of an exponentially attracting invariant measure and the strong law of large numbers are proven for the chain. Further, a one-to-one correspondence between invariant measures for the chain and invariant measures for the continuous-time process is established. This result, together with the aforementioned ergodic properties of the discrete-time model, is used to derive the strong law of large numbers for the process. The studied random dynamical systems are inspired by certain biological models of gene expression, which are also discussed within this paper.

中文翻译：

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应用于基因表达建模的某些分段确定性马尔可夫过程的遍历特性

本文研究了由随机跳跃和切换半流指定的分段确定性马尔可夫过程，以及由其跳跃后位置给出的相关马尔可夫链。该链证明了指数吸引不变测度的存在和强大的大数定律。此外，建立了链的不变测度和连续时间过程的不变测度之间的一一对应关系。该结果与离散时间模型的上述遍历特性一起用于推导出该过程的强大数定律。所研究的随机动力系统受到某些基因表达生物模型的启发，本文也讨论了这些模型。