Long-time asymptotics of stochastic reaction systems

Abstract

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.

References in corpus (6)

• Epidemic processes in complex networks
• A scalable computational framework for establishing long-term behavior of stochastic reaction networks
• Product-form Poisson-like distributions and complex balanced reaction systems
• M/M/$\infty$ queues in semi-Markovian random environment
• Influence of gene copy number on self-regulated gene expression
• Conditions for convergence of random coefficient AR(1) processes and perpetuities in higher dimensions