Continuous dependence of an invariant measure on the jump rate of a piecewise-deterministic Markov process
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...
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AIMS Press
2020-01-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2020056?viewType=HTML |
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| author | Dawid Czapla Sander C. Hille Katarzyna Horbacz Hanna Wojewódka-Ściążko |
| author_facet | Dawid Czapla Sander C. Hille Katarzyna Horbacz Hanna Wojewódka-Ściążko |
| author_sort | Dawid Czapla |
| collection | DOAJ |
| description | 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 <i>λ</i>. 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. |
| first_indexed | 2024-12-24T00:38:32Z |
| format | Article |
| id | doaj.art-7a5cf196e4a94821b7003fe9783007e3 |
| institution | Directory Open Access Journal |
| issn | 1551-0018 |
| language | English |
| last_indexed | 2024-12-24T00:38:32Z |
| publishDate | 2020-01-01 |
| publisher | AIMS Press |
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| series | Mathematical Biosciences and Engineering |
| spelling | doaj.art-7a5cf196e4a94821b7003fe9783007e32022-12-21T17:24:02ZengAIMS PressMathematical Biosciences and Engineering1551-00182020-01-011721059107310.3934/mbe.2020056Continuous dependence of an invariant measure on the jump rate of a piecewise-deterministic Markov processDawid Czapla0Sander C. Hille1Katarzyna Horbacz2Hanna Wojewódka-Ściążko31. Institute of Mathematics, University of Silesia in Katowice, Bankowa 14, 40-007 Katowice, Poland2. Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands1. Institute of Mathematics, University of Silesia in Katowice, Bankowa 14, 40-007 Katowice, Poland1. Institute of Mathematics, University of Silesia in Katowice, Bankowa 14, 40-007 Katowice, PolandWe 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 <i>λ</i>. 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.https://www.aimspress.com/article/doi/10.3934/mbe.2020056?viewType=HTMLinvariant measurepiecewise-deterministic markov processrandom dynamical systemjump ratecontinuous dependence |
| spellingShingle | 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 Mathematical Biosciences and Engineering invariant measure piecewise-deterministic markov process random dynamical system jump rate continuous dependence |
| title | Continuous dependence of an invariant measure on the jump rate of a piecewise-deterministic Markov process |
| title_full | Continuous dependence of an invariant measure on the jump rate of a piecewise-deterministic Markov process |
| title_fullStr | Continuous dependence of an invariant measure on the jump rate of a piecewise-deterministic Markov process |
| title_full_unstemmed | Continuous dependence of an invariant measure on the jump rate of a piecewise-deterministic Markov process |
| title_short | Continuous dependence of an invariant measure on the jump rate of a piecewise-deterministic Markov process |
| title_sort | continuous dependence of an invariant measure on the jump rate of a piecewise deterministic markov process |
| topic | invariant measure piecewise-deterministic markov process random dynamical system jump rate continuous dependence |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2020056?viewType=HTML |
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