  A piecewise-deterministic Markov process, specified by random jumps and
switching semi-flows, as well as the associated Markov chain given by its
post-jump locations, are investigated in this paper. The existence of an
exponentially attracting invariant measure and the strong law of large numbers
are proven for the chain. Further, a one-to-one correspondence between
invariant measures for the chain and invariant measures for the continuous-time
process is established. This result, together with the aforementioned ergodic
properties of the discrete-time model, is used to derive the strong law of
large numbers for the process. The studied random dynamical systems are
inspired by certain biological models of gene expression, which are also
discussed within this paper.
