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

TL;DR

This paper investigates the ergodic properties of a class of piecewise-deterministic Markov processes, establishing invariant measures and laws of large numbers, with applications to gene expression models.

Contribution

It introduces new ergodic results for these processes and links invariant measures between discrete and continuous models, applied to biological gene expression.

Findings

Existence of exponentially attracting invariant measure

Strong law of large numbers for the process

Correspondence between invariant measures of chain and process

Abstract

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.