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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

Special Issues: Mathematical Modeling with Measures

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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.
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