Long-time asymptotics of stochastic reaction systems

2019/12/01 by Cappelletti, Daniele, Majumder, Abhishek Pal, Wiuf, Carsten
#Dynamical Systems (math.DS) #FOS: Biological sciences #FOS: Mathematics #Molecular Networks (q-bio.MN)

paper · doi:10.48550/arxiv.1912.00401

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.

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