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
[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. |