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Ergodic properties of some piecewise-deterministic Markov process with application to gene expression modelling

Author

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  • Czapla, Dawid
  • Horbacz, Katarzyna
  • Wojewódka-Ściążko, Hanna

Abstract

We investigate a piecewise-deterministic Markov process with a Polish state space, whose deterministic behaviour between random jumps is governed by a finite number of semiflows. We provide tractable conditions ensuring a form of exponential ergodicity and the strong law of large numbers for the chain given by the post-jump locations. Further, we establish a one-to-one correspondence between invariant measures of the chain and those of the continuous-time process. These results enable us to derive the strong law of large numbers for the latter. The studied dynamical system is inspired by certain models of gene expression, which are also discussed here.

Suggested Citation

  • Czapla, Dawid & Horbacz, Katarzyna & Wojewódka-Ściążko, Hanna, 2020. "Ergodic properties of some piecewise-deterministic Markov process with application to gene expression modelling," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 2851-2885.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:5:p:2851-2885
    DOI: 10.1016/j.spa.2019.08.006
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    References listed on IDEAS

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    1. Lasota, Andrzej & Traple, Janusz, 2003. "Invariant measures related with Poisson driven stochastic differential equation," Stochastic Processes and their Applications, Elsevier, vol. 106(1), pages 81-93, July.
    2. Stenflo, Örjan, 2001. "A note on a theorem of Karlin," Statistics & Probability Letters, Elsevier, vol. 54(2), pages 183-187, September.
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