In the paper we consider some piecewise deterministic Markov process whose continuous component evolves according to semiflows, which are switched at the jump times of a Poisson process. The associated Markov chain describes the states of this process directly after the jumps. Certain ergodic properties of these two objects have already been investigated in our recent papers. We now aim to establish the law of the iterated logarithm for the aforementioned continuous-time process. Moreover, we intend to do this using the already proven properties of the discrete-time system. The abstract model under consideration has interesting interpretations in real-life sciences, such as biology. Among others, it can be used to describe the stochastic dynamics of gene expression.

中文翻译：

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分段确定性马尔可夫过程的迭代对数定律由其跳跃后位置给出的马尔可夫链的属性保证

在论文中，我们考虑了一些分段确定性马尔可夫过程，其连续分量根据半流演化，半流在泊松过程的跳跃时间切换。关联的马尔可夫链直接描述了跳转之后这个过程的状态。我们最近的论文已经研究了这两个物体的某些遍历特性。我们现在的目标是建立上述连续时间过程的迭代对数定律。此外，我们打算使用离散时间系统已经证明的特性来做到这一点。正在考虑的抽象模型在现实生命科学（例如生物学）中有有趣的解释。其中，它可用于描述基因表达的随机动态。