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We investigate a piecewise-deterministic Markov process, evolving on a Polishmetric space, whose deterministic behaviour between random jumps is governed bysome semi-flow, and any state right after the jump is attained by a randomlyselected continuous transformation. It is assumed that the jumps appear atrandom moments, which coincide with the jump times of a Poisson process withintensity $\lambda$. The model of this type, although in a more generalversion, was examined in our previous papers, where we have shown, amongothers, that the Markov process under consideration possesses a uniqueinvariant probability measure, say $\nu_{\lambda}^*$. The aim of this paper isto prove that the map $\lambda\mapsto\nu_{\lambda}^*$ is continuous (in thetopology of weak convergence of probability measures). The studied dynamicalsystem is inspired by certain stochastic models for cell division and geneexpression.

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