Stochastic gene regulatory networks with bursting dynamics can be modeled mesocopically as a generalized density-dependent Markov chain (GDDMC) or macroscopically as a piecewise deterministic Markov process (PDMP). Here we prove a limit theorem showing that each family of GDDMCs will converge to a PDMP as the system size tends to infinity. Moreover, under a simple dissipative condition, we prove the existence and uniqueness of the stationary distribution and the exponential ergodicity for the PDMP limit via the coupling method. Further extensions and applications to single-cell stochastic gene expression kinetics and bursty stochastic gene regulatory networks are also discussed and the convergence of the stationary distribution of the GDDMC model to that of the PDMP model is also proved.

中文翻译：

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广义密度依赖马尔可夫链和突发随机基因调控网络的极限定理。

具有爆发动力学的随机基因调控网络可以介观地建模为广义的密度依赖性马尔可夫链（GDDMC），或者宏观地建模为分段确定性马尔可夫过程（PDMP）。在这里，我们证明了一个极限定理，该定理表明随着系统规模趋于无穷大，每个GDDMC系列都将收敛到PDMP。此外，在简单的耗散条件下，我们通过耦合方法证明了PDMP极限的平稳分布的存在性和唯一性以及指数遍历性。还讨论了单细胞随机基因表达动力学和突发性随机基因调控网络的进一步扩展和应用，还证明了GDDMC模型与PDMP模型的平稳分布具有收敛性。