Dawid Czapla

Equicontinuity and Stability Properties of Markov Chains Arising from Iterated Function Systems on Polish Spaces

Iterated function systems with place-dependent probabilities are considered in this article. Assumptions ensuring the equicontinuity of iteration orbits {P n f: n ∈ ℕ} of bounded continuous functions f, where P is the dual of transition operator corresponding to an IFS, are presented. The results are applied in the study of asymptotic stability properties of these systems.

Limit theorems for certain stochastic models for gene regulatory networks

In the paper we propose conditions ensuring the central limit theorem (CLT) and the law of the iterated logarithm (LIL) for a certain class of Markov chains. We further use this general criteria to verify the aforementioned limit theorems for a particular disrete-time Markov system. The piecewise-deterministic Markov process defined via interpolation of the explored Markov chain can be used e.g. to describe a model for gene expression. The aim for the future work is to establish the CLT and the LIL for the continuous-time process too.

Exponential ergodicity of some Markov dynamical system with application to a Poisson driven stochastic differential equation

ABSTRACT We are concerned with the asymptotics of the Markov chain given by the post-jump locations of a certain piecewise-deterministic Markov process with a state-dependent jump intensity. We provide sufficient conditions for such a model to possess a unique invariant distribution, which is exponentially attracting in the dual-bounded Lipschitz distance. Having established this, we generalize a result of J. Kazak on the jump process defined by a Poisson-driven stochastic differential equation with a solution-dependent intensity of perturbations.

The strong law of large numbers for certain piecewise-deterministic Markov processes with application to a gene expression model

The main goal of this paper is to establish the strong law of large numbers (SLLN) for a subclass of piecewise-deterministic Markov processes (PDMPs). On the way to this result, we provide sufficient conditions for the existence of an exponentially attracting invariant distribution for the Markov chain given by the post-jump locations of a PDMP. Furthermore, we obtain a one-to-one correspondence between invariant measures of such a chain and invariant measures of the PDMP. Finally, we illustrate the applicability of our results for a model of prokaryotic gene expression.