IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v130y2020i5p2851-2885.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414918303661
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2019.08.006?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Jaroszewska, Joanna, 2008. "How to construct asymptotically stable iterated function systems," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1570-1576, September.
    2. Dong, Zhao & Xu, Tiange & Zhang, Tusheng, 2009. "Invariant measures for stochastic evolution equations of pure jump type," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 410-427, February.
    3. Guido Cozzi & Fabio Privileggi, 2002. "Wealth Polarization and Pulverization in Fractal Societies," ICER Working Papers - Applied Mathematics Series 39-2002, ICER - International Centre for Economic Research.
    4. Jaroszewska, Joanna, 2013. "A note on iterated function systems with discontinuous probabilities," Chaos, Solitons & Fractals, Elsevier, vol. 49(C), pages 28-31.
    5. Horbacz, Katarzyna, 2016. "Strong law of large numbers for continuous random dynamical systems," Statistics & Probability Letters, Elsevier, vol. 118(C), pages 70-79.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:130:y:2020:i:5:p:2851-2885. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.