Search Advanced

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 2: 1059-1073. doi: 10.3934/mbe.2020056
Previous Article Next Article
Research article Special Issues

Continuous dependence of an invariant measure on the jump rate of a piecewise-deterministic Markov process

  • 1.
    Institute of Mathematics, University of Silesia in Katowice, Bankowa 14, 40-007 Katowice, Poland
  • 2.
    Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands
  • Received: 31 July 2019 Accepted: 29 October 2019 Published: 12 November 2019

  • We investigate a piecewise-deterministic Markov process, evolving on a Polish metric space, whose deterministic behaviour between random jumps is governed by some semi-flow, and any state right after the jump is attained by a randomly selected continuous transformation. It is assumed that the jumps appear at random moments, which coincide with the jump times of a Poisson process with intensity λ. The model of this type, although in a more general version, was examined in our previous papers, where we have shown, among others, that the Markov process under consideration possesses a unique invariant probability measure, say $\nu_{\lambda}^*$. The aim of this paper is to prove that the map $\lambda\mapsto\nu_{\lambda}^*$ is continuous (in the topology of weak convergence of probability measures). The studied dynamical system is inspired by certain stochastic models for cell division and gene expression.

    Citation: Dawid Czapla, Sander C. Hille, Katarzyna Horbacz, Hanna Wojewódka-Ściążko. Continuous dependence of an invariant measure on the jump rate of a piecewise-deterministic Markov process[J]. Mathematical Biosciences and Engineering, 2020, 17(2): 1059-1073. doi: 10.3934/mbe.2020056

    Related Papers:

  • We investigate a piecewise-deterministic Markov process, evolving on a Polish metric space, whose deterministic behaviour between random jumps is governed by some semi-flow, and any state right after the jump is attained by a randomly selected continuous transformation. It is assumed that the jumps appear at random moments, which coincide with the jump times of a Poisson process with intensity λ. The model of this type, although in a more general version, was examined in our previous papers, where we have shown, among others, that the Markov process under consideration possesses a unique invariant probability measure, say $\nu_{\lambda}^*$. The aim of this paper is to prove that the map $\lambda\mapsto\nu_{\lambda}^*$ is continuous (in the topology of weak convergence of probability measures). The studied dynamical system is inspired by certain stochastic models for cell division and gene expression.


    加载中


    [1] M. H. A. Davis, Piecewise-deterministic Markov processes: a general class of non-diffusion stochastic models, J. Roy. Statist. Soc. Ser. B, 46 (1984), 353-388.
    [2] M. C. Mackey, M. Tyran-Kamińska, R. Yvinec, Dynamic behaviour of stochastic gene expression models in the presence of bursting, SIAM J. Appl. Math., 73 (2013), 1830-1852.
    [3] S. C. Hille, K. Horbacz, T. Szarek, Existence of a unique invariant measure for a class of equicontinuous Markov operators with application to a stochastic model for an autoregulated gene, Ann. Math. Blaise Pascal, 23 (2016), 171-217.
    [4] A. Lasota and M. C. Mackey, Cell division and the stability of cellular populations, J. Math. Biol., 38 (1999), 241-261.
    [5] M. G. Riedler, M. Thieullen, G. Wainrib, Limit theorems for infinite-dimensional piecewise deterministic Markov processes. Applications to stochastic excitable membrane models, Electron. J. Probab., 17 (2012), 1-48.
    [6] T. Alkurdi, S. C. Hille, O. Van Gaans, Persistence of stability for equilibria of map iterations in Banach spaces under small perturbations, Potential Anal., 42 (2015), 175-201.
    [7] M. Benaïm, C. Lobry, Lotka Volterra with randomly fluctuationg environments or 'how switching between benefcial environments can make survival harder', Ann. Appl. Probab., 26 (2016), 3754-3785.
    [8] M. Benaïm, S. Le Borgne, F. Malrieu, P. A. Zitt, Qualitative properties of certain piecewise deterministic Markov processes, Ann. Inst. Henri Poincar Probab., 51 (2014), 1040-1075.
    [9] M. Benaïm, S. Le Borgne, F. Malrieu, P. A. Zitt, Quantitative ergodicity for some switched dynamical systems, Electron. Commun. Probab., 17 (2012), 1-14.
    [10] F. Dufour, O. L. V. Costa, Stability of piecewise-deterministic Markov processes, SIAM J. Control Optim., 37 (1999), 1483-1502.
    [11] O. L. V. Costa, F. Dufour, Stability and ergodicity of piecewise deterministic Markov processes, SIAM J. Control Optim., 47 (2008), 1053-1077.
    [12] B. Cloez, M. Hairer, Exponential ergodicity for Markov processes with random switching, Bernoulli, 21 (2015), 505-536.
    [13] D. Czapla, K. Horbacz, H. Wojewódka-Ściążko, Ergodic properties of some piecewise-deterministic Markov process with application to gene expression modelling, Stochastic Process. Appl., 2019, doi: 10.1016/j.spa.2019.08.006.
    [14] D. Czapla, J. Kubieniec, Exponential ergodicity of some Markov dynamical systems with application to a Poisson driven stochastic differential equation, Dyn. Syst., 34 (2019), 130-156.
    [15] H. Wojewódka, Exponential rate of convergence for some Markov operators, Statist. Probab. Lett., 83 (2013), 2337-2347.
    [16] S. C. Hille, K. Horbacz, T. Szarek, H. Wojewódka, Limit theorems for some Markov chains, J. Math. Anal. Appl., 443 (2016), 385-408.
    [17] S. C. Hille, K. Horbacz, T. Szarek, H. Wojewódka, Law of the iterated logarithm for some Markov operators, Asymptotic Anal., 97 (2016), 91-112.
    [18] D. Czapla, K. Horbacz, H. Wojewódka-Ściążko, A useful version of the central limit theorem for a general class of Markov chains, preprint, arXiv:1804.09220v2.
    [19] D. Czapla, K. Horbacz, H. Wojewódka-Ściążko, The Strassen invariance principle for certain non-stationary Markov-Feller chains, preprint, arXiv:1810.07300v2.
    [20] T. Komorowski, C. Landim, S. Olla, Fluctuations in Markov processes. Time symmetry and martingale approximation, Springer-Verlag, Heidelberg, 2012.
    [21] V. I. Bogachev, Measure Theory, vol. II, Springer-Verlag, Berlin, 2007.
    [22] P. Gwiazda, S. C. Hille, K. Łyczek, A. Świerczewska-Gwiazda, Differentiability in perturbation parameter of measure solutions to perturbed transport equation, Kinet. Relat. Mod., (2019), in press, preprint arXiv:18l06.00357.
    [23] N. Weaver, Lipschitz Algebras, World Scientific Publishing Co. Pte Ltd., Singapore, 1999.
    [24] V. I. Bogachev, Measure Theory, vol. I, Springer-Verlag, Berlin, 2007.
    [25] R. M. Dudley, Convergence of Baire measures, Stud. Math., 27 (1966), 251-268.
    [26] T. Szarek, Invariant measures for Markov operators with application to function systems, Studia Math., 154 (2003), 207-222.
    [27] D. T. H. Worm, Semigroups on spaces of measures, Ph.D thesis, Leiden University, The Netherlands, 2010. Available from: www.math.leidenuniv.nl/nl/theses/PhD/.
    [28] J. Diestel, Jr. J. J. Uhl, Vector measures, American Mathematical Society, Providence, R.I., 1977.
    [29] J. Evers, S. C. Hille, A. Muntean, Mild solutions to a measure-valued mass evolution problem with flux boundary conditions, J. Diff. Equ., 259 (2015), 1068-1097.
    [30] M. Crandall, T. Ligget, Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265-298.
    [31] R. Kapica, M. Ślęczka, Random iterations with place dependent probabilities, to appear in Probab. Math. Statist. (2019).
    [32] A. Lasota, J. A. Yorke, Lower bound technique for Markov operators and iterated function systems, Random Comput. Dyn., 2 (1994), 41-77.
    [33] O. Stenflo, A note on a theorem of Karlin, Stat. Probab. Lett., 54 (2001), 183-187.
    [34] J. J. Tyson, K. B. Hannsgen, Cell growth and division: a deterministic/probabilistic model of the cell cycle, J. Math. Biol., 23 (1986), 231-246.
    [35] W. Rudin, Principles of mathematical analysis, McGraw-Hill, Inc., New York, 1976.
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematical Biosciences and Engineering
2.6 3.9

Metrics

Article views(3124) PDF downloads(448) Cited by(1)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog