Czapla Dawid, Hille Sander C, Horbacz Katarzyna, Wojewódka-Ściążko Hanna
Institute of Mathematics, University of Silesia in Katowice, Bankowa 14, 40-007 Katowice, Poland.
Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands.
Math Biosci Eng. 2019 Nov 12;17(2):1059-1073. doi: 10.3934/mbe.2020056.
We investigate a piecewise-deterministic Markov process, evolving on a Polish metric space, whose deterministic behaviour between random jumps is governed by some semi-flow, and any state right after the jump is attained by a randomly selected continuous transformation. It is assumed that the jumps appear at random moments, which coincide with the jump times of a Poisson process with intensity . The model of this type, although in a more general version, was examined in our previous papers, where we have shown, among others, that the Markov process under consideration possesses a unique invariant probability measure, say $\nu_{\lambda}^$. The aim of this paper is to prove that the map $\lambda\mapsto\nu_{\lambda}^$ is continuous (in the topology of weak convergence of probability measures). The studied dynamical system is inspired by certain stochastic models for cell division and gene expression.
我们研究一个在波兰度量空间上演变的分段确定性马尔可夫过程，其随机跳跃之间的确定性行为由某种半流控制，并且跳跃后的任何状态通过随机选择的连续变换获得。假设跳跃出现在随机时刻，这些时刻与强度为 的泊松过程的跳跃时间一致。这种类型的模型，尽管是更一般的版本，在我们之前的论文中已经研究过，在那里我们除了其他结果之外还表明，所考虑的马尔可夫过程具有唯一的不变概率测度，记为 $\nu_{\lambda}^$。本文的目的是证明映射 $\lambda\mapsto\nu_{\lambda}^$ 是连续的（在概率测度弱收敛的拓扑中）。所研究的动力系统受到细胞分裂和基因表达的某些随机模型的启发。
