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

Abstract

A piecewise-deterministic Markov process, specified by random jumps and switching semiflows, 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.

Continuous dependence of an invariant measure on the jump rate of a piecewise-deterministic Markov process

“…Proof. In view of [6,Theorem 4.1], it is sufficient to prove that the convergence is uniform with respect to λ.…”

“…In this paper, we are concerned with a special case of the PDMP described in [6,8], whose deterministic motion between jumps depends on a single continuous semi-flow, and any post-jump location is attained by a continuous transformation of the pre-jump state, randomly selected (with a place-dependent probability) among all possible ones. The jumps in this model occur at random time points according to a homogeneous Poisson process.…”

The Law of the Iterated Logarithm for Random Dynamical System with Jumps and State-Dependent Jump Intensity

“…These processes are governed by deterministic semiflows which are intermittent by jumps. PDMP's have been introduced by Davis in [5] and have found their application, among others, in modeling phenomena in biology, as stochastic model for gene expression ( [2,14,15]).…”

The Strassen invariance principle for certain non-stationary Markov–Feller chains

“…The motivation to establish such a result derives from our research on certain random dynamical systems, developed mainly in molecular biology (see eg. the models for gene expression investigated in [11,3,17] or the model for cell cycle discussed in [16,22]), to which we were not able to apply [1,Theorem 1] directly.…”

“…Theorem 3.7 is formulated in the same spirit as [13, Theorem 2.1] and [4, Theorem 2.1], which were both applied to a particular discrete-time Markov dynamical system (cf. [3]) in order to verify its exponential ergodicity (in the context of weak convergance of probability measures) and the CLT, respectively. Theorem 3.7 can be used to establish the functional LIL for such a system (cf.…”