|Ergodic properties of some piecewise-deterministic Markov process with application to gene expression modelling
|Year of Publication
|Czapla D, Horbacz K, Wojewódka-Ściążko H
|Stochastic Processes and their Applications
|asymptotic stability, exponential ergodicity, gene expression, invariant measure, Markov process, the strong law of large numbers
We investigate a piecewise-deterministic Markov process with a Polish state space, whose deterministic behaviour between random jumps is governed by a finite number of semiflows. We provide tractable conditions ensuring a form of exponential ergodicity and the strong law of large numbers for the chain given by the post-jump locations. Further, we establish a one-to-one correspondence between invariant measures of the chain and those of the continuous-time process. These results enable us to derive the strong law of large numbers for the latter. The studied dynamical system is inspired by certain models of gene expression, which are also discussed here.