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A piecewise-deterministic Markov process, specified by random jumps andswitching semi-flows, as well as the associated Markov chain given by itspost-jump locations, are investigated in this paper. The existence of anexponentially attracting invariant measure and the strong law of large numbersare proven for the chain. Further, a one-to-one correspondence betweeninvariant measures for the chain and invariant measures for the continuous-timeprocess is established. This result, together with the aforementioned ergodicproperties of the discrete-time model, is used to derive the strong law oflarge numbers for the process. The studied random dynamical systems areinspired by certain biological models of gene expression, which are alsodiscussed within this paper.

Ergodic properties of some piecewise-deterministic Markov process with application to gene expression modelling