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

TytułErgodic properties of some piecewise-deterministic Markov process with application to gene expression modelling
Publication TypeJournal Article
Rok publikacji2020
AutorzyCzapla D, Horbacz K, Wojewódka-Ściążko H
Journal Stochastic Processes and their Applications
Volume130
Issue5
Start Page2851-2885
Date Published05/2020
Słowa kluczoweasymptotic stability, exponential ergodicity, gene expression, invariant measure, Markov process, the strong law of large numbers
DOI10.1016/j.spa.2019.08.006
Full Text

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.