We study the stochastic dynamics of a system of interacting species in a stochastic environment by means of a continuous-time Markov chain with transition rates depending on the state of the environment. Models of gene regulation in systems biology take this form. We characterise the finite-time distribution of the Markov chain, provide conditions for ergodicity, and characterise the stationary distribution (when it exists) as a mixture of Poisson distributions. The mixture measure is uniquely identified as the law of a fixed point of a stochastic recurrence equation. This recursion is crucial for statistical computation of moments and other distributional features.

我们通过连续时间马尔可夫链研究随机环境中相互作用物种系统的随机动力学，其中马尔可夫链的过渡速率取决于环境的状态。系统生物学中的基因调控模型采用这种形式。我们表征了马尔可夫链的有限时间分布，为遍历性提供了条件，并将稳态分布（如果存在）表征为泊松分布的混合。混合度量唯一地标识为随机递归方程的不动点定律。此递归对于矩和其他分布特征的统计计算至关重要。