Skip to main content
SpringerLink
Log in

Asymptotic Properties of Stochastic Semigroups with Applications to Piecewise Deterministic Markov Processes

  • Conference paper
  • First Online:
Semigroups of Operators – Theory and Applications (SOTA 2018)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 325))

Included in the following conference series:

  • 563 Accesses

Abstract

The paper is devoted to stochastic semigroups, i.e. semigroups of linear operators on integrable functions preserving the set of densities. We present some results concerning their asymptotic stability and asymptotic decomposition. Finally we give applications to stochastic semigroups generated by piecewise deterministic Markov processes: pure jump-type processes, stochastic billiards and to biological models of gene expressions, electrical activity of a neuron, and two-phase cell cycle.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Banasiak, J., Pichór, K., Rudnicki, R.: Asynchronous exponential growth of a general structured population model. Acta Appl. Math. 119, 149–166 (2012)

    Article  MathSciNet  Google Scholar 

  2. Bobrowski, A., Lipniacki, T., Pichór, K., Rudnicki, R.: Asymptotic behavior of distributions of mRNA and protein levels in a model of stochastic gene expression. J. Math. Anal. Appl. 333, 753–769 (2007)

    Article  MathSciNet  Google Scholar 

  3. Diekmann, O., Heijmans, H.J.A.M., Thieme, H.R.: On the stability of the cell size distribution. J. Math. Biol. 19, 227–248 (1984)

    Article  MathSciNet  Google Scholar 

  4. Davis, M.H.A.: Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models. J. Roy. Statist. Soc. Ser. B 46, 353–388 (1984)

    MathSciNet  MATH  Google Scholar 

  5. Evans, S.N.: Stochastic billiards on general tables. Ann. Appl. Probab. 11, 419–437 (2001)

    Article  MathSciNet  Google Scholar 

  6. Foguel, S.R.: The Ergodic Theory of Markov Processes. Van Nostrand Reinhold Comp., New York (1969)

    MATH  Google Scholar 

  7. Gacki, H., Lasota, A.: Markov operators defined by Volterra type integrals with advanced argument. Ann. Polon. Math. 51, 155–166 (1990)

    Article  MathSciNet  Google Scholar 

  8. Gyllenberg, M., Heijmans, H.J.A.M.: An abstract delay-differential equation modelling size dependent cell growth and division. SIAM J. Math. Anal. 18, 74–88 (1987)

    Article  MathSciNet  Google Scholar 

  9. Lasota, A., Mackey, M.C.: Chaos, Fractals and Noise. Stochastic Aspects of Dynamics. Springer Applied Mathematical Sciences, vol. 97. New York (1994)

    Google Scholar 

  10. Lebowitz, J.L., Rubinow, S.L.: A theory for the age and generation time distribution of microbial population. J. Math. Biol. 1, 17–36 (1974)

    Article  MathSciNet  Google Scholar 

  11. Lipniacki, T., Paszek, P., Marciniak-Czochra, A., Brasier, A.R., Kimmel, M.: Transcriptional stochasticity in gene expression. J. Theor. Biol. 238, 348–367 (2006)

    Article  MathSciNet  Google Scholar 

  12. Lods, B., Mokhtar-Kharroubi, M., Rudnicki, R.: Invariant density and time asymptotics for collisionless kinetic equations with partly diffuse boundary operators. Ann. I. H. Poincaré – AN (2020). https://doi.org/10.1016/j.anihpc.2020.02.004

  13. Łoskot, K., Rudnicki, R.: Sweeping of some integral operators. Bull. Pol. Ac.: Math. 37, 229–235 (1989)

    Google Scholar 

  14. Mackey, M.C., Rudnicki, R.: Global stability in a delayed partial differential equation describing cellular replication. J. Math. Biol. 33, 89–109 (1994)

    Article  MathSciNet  Google Scholar 

  15. Mokhtar-Kharroubi, M., Rudnicki, R.: On asymptotic stability and sweeping of collisionless kinetic equations. Acta Appl. Math. 147, 19–38 (2017)

    Article  MathSciNet  Google Scholar 

  16. Othmer, H.G., Dunbar, S.R., Alt, W.: Models of dispersal in biological systems. J. Math. Biol. 26, 263–298 (1988)

    Article  MathSciNet  Google Scholar 

  17. Pichór, K., Rudnicki, R.: Continuous Markov semigroups and stability of transport equations. J. Math. Anal. Appl. 249, 668–685 (2000)

    Article  MathSciNet  Google Scholar 

  18. Pichór, K., Rudnicki, R.: Asymptotic decomposition of substochastic operators and semigroups. J. Math. Anal. Appl. 436, 305–321 (2016)

    Article  MathSciNet  Google Scholar 

  19. Pichór, K., Rudnicki, R.: Asymptotic decomposition of substochastic semigroups and applications. Stochast. Dyn. 18, 1850001 (2018)

    Article  MathSciNet  Google Scholar 

  20. Pichór, K., Rudnicki, R.: Stability of stochastic semigroups and applications to Stein’s neuronal model. Discret. Contin. Dyn. Syst. B 23, 377–385 (2018)

    MathSciNet  MATH  Google Scholar 

  21. Pichór, K., Rudnicki, R.: Applications of stochastic semigroups to cell cycle models. Discret. Contin. Dyn. Syst. B 24, 2365–2381 (2019)

    MathSciNet  MATH  Google Scholar 

  22. Rotenberg, M.: Transport theory for growing cell populations. J. Theor. Biol. 103, 181–199 (1983)

    Article  MathSciNet  Google Scholar 

  23. Rudnicki, R.: Stochastic operators and semigroups and their applications in physics and biology. In: Banasiak, J., Mokhtar-Kharroubi, M. (eds.) Evolutionary Equations with Applications in Natural Sciences, Lecture Notes in Mathematics, vol. 2126, pp. 255–318. Springer, Heidelberg (2015)

    Google Scholar 

  24. Rudnicki, R., Pichór, K.: Markov semigroups and stability of the cell maturation distribution. J. Biol. Syst. 8, 69–94 (2000)

    Article  Google Scholar 

  25. Rudnicki, R., Tomski, A.: On a stochastic gene expression with pre-mRNA, mRNA and protein contribution. J. Theor. Biol. 387, 54–67 (2015)

    Article  MathSciNet  Google Scholar 

  26. Rudnicki, R., Tyran-Kamińska, M.: Piecewise Deterministic Processes in Biological Models. SpringerBriefs in Applied Sciences and Technology, Mathematical Methods. Springer, Cham, Switzerland (2017)

    Google Scholar 

  27. Stein, R.B.: Some models of neuronal variability. Biophys. J. 7, 37–68 (1967)

    Article  Google Scholar 

  28. Tyrcha, J.: Asymptotic stability in a generalized probabilistic/deterministic model of the cell cycle. J. Math. Biol. 26, 465–475 (1988)

    Google Scholar 

Download references

Acknowledgements

This research was partially supported by the National Science Centre (Poland) Grant No. 2017/27/B/ST1/00100.

Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Silesia, Bankowa 14, 40-007, Katowice, Poland

    Katarzyna Pichór

  2. Institute of Mathematics, Polish Academy of Sciences, Bankowa 14, 40-007, Katowice, Poland

    Ryszard Rudnicki

Authors
  1. Katarzyna Pichór
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ryszard Rudnicki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ryszard Rudnicki .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield, South Africa

    Jacek Banasiak

  2. Department of Mathematics, Lublin University of Technology, Lublin, Poland

    Adam Bobrowski

  3. Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw, Poland

    Mirosław Lachowicz

  4. Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland

    Yuri Tomilov

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Pichór, K., Rudnicki, R. (2020). Asymptotic Properties of Stochastic Semigroups with Applications to Piecewise Deterministic Markov Processes. In: Banasiak, J., Bobrowski, A., Lachowicz, M., Tomilov, Y. (eds) Semigroups of Operators – Theory and Applications. SOTA 2018. Springer Proceedings in Mathematics & Statistics, vol 325. Springer, Cham. https://doi.org/10.1007/978-3-030-46079-2_19

Download citation

Publish with us

Policies and ethics

Search

Navigation