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The dynamics of stochastic mono-molecular reaction systems in stochastic environments

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  • Cappelletti, Daniele
  • Pal Majumder, Abhishek
  • Wiuf, Carsten

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.

Suggested Citation

  • Cappelletti, Daniele & Pal Majumder, Abhishek & Wiuf, Carsten, 2021. "The dynamics of stochastic mono-molecular reaction systems in stochastic environments," Stochastic Processes and their Applications, Elsevier, vol. 137(C), pages 106-148.
    • Handle: RePEc:eee:spapps:v:137:y:2021:i:c:p:106-148
    DOI: 10.1016/j.spa.2021.03.010
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    References listed on IDEAS

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    1. Thomas B. Kepler & Timothy C. Elston, 2001. "Stochasticity in Transcriptional Regulation: Origins, Consequences and Mathematical Representations," Working Papers 01-06-033, Santa Fe Institute.
    2. Ankit Gupta & Corentin Briat & Mustafa Khammash, 2014. "A Scalable Computational Framework for Establishing Long-Term Behavior of Stochastic Reaction Networks," PLOS Computational Biology, Public Library of Science, vol. 10(6), pages 1-16, June.
    3. Johan Paulsson, 2004. "Summing up the noise in gene networks," Nature, Nature, vol. 427(6973), pages 415-418, January.
    4. Shao, Jinghai, 2015. "Ergodicity of regime-switching diffusions in Wasserstein distances," Stochastic Processes and their Applications, Elsevier, vol. 125(2), pages 739-758.
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    Cited by:

    1. Lee, Julian, 2023. "Poisson distributions in stochastic dynamics of gene expression: What events do they count?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 630(C).

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