Existence of a unique invariant measure for a class of equicontinuous Markov operators with application to a stochastic model for an autoregulated gene

Kubieniec

2018

Dynamical Systems

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 sequence of jump times, the conditional probability that the next jump, say τ n+1 , will occur before time t has the formwhere {ξ(t)} t≥0 is a stochastic process with values in I which indicates the semiflow that currently determines the evolution of the system. We shall focus on the limit behaviour of the Markov chain {(Y n , ξ n )} n∈N0 given by the post-jump locations of the PDMP {(Y (t), ξ(t))} t≥0 , that is, defined by (Y n , ξ n ) = (Y (τ n ), ξ(τ n )) for n ∈ N 0 . Such a discrete-time dynamical system (in the context presented here) includes as a special case, for instance, a simple cell cycle model examined by Lasota and Mackey [21], and, furthermore, may prove useful for improvement of the model in [8].Our first goal is to establish a criterion for exponential ergodicity of the transition operator associated with the chain {(Y n , ξ n )} n∈N0 , analogously as in [12, Theorem 4.1]. More precisely, letting (·)P stand for the Markov operator acting on Borel measures in such a way that µ n+1 = µ n P for n ∈ N 0 , where µ n is the distibution of (Y n , ξ n ), we provide sufficient conditions under which there exists exactly one probability measure µ * that is invariant for P , i.e. µ * = µ * P . It turns out that such a measure is exponentially attracting in the so-called dual bounded Lipschitz distance, which is induced by the Fortet-Mourier norm ([5, 23]), denoted by ||·|| F M . We mean by this that there exists a constant β ∈ [0, 1) such that ||µP n − µ * || F M ≤ C(µ)β n for all n ∈ N and µ ∈ M ρc,1 prob ,

mentioning

confidence: 99%

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

Kubieniec

2018

Dynamical Systems

“…For this reason we refer to an abstract model, which occurs mainly in gene expression analysis (cf. [8,11,17]). Such a model has already been investigated in terms of its exponential ergodicity and the strong law of large numbers in [3,11].…”

Section: An Abstract Markov Model For Gene Expressionmentioning

confidence: 99%

A useful version of the central limit theorem for a general class of Markov chains

Journal of Mathematical Analysis and Applications

Horbacz

Wojewódka–Ściążko

2020

Self Cite

In the paper we propose certain conditions, relatively easy to verify, which ensure the central limit theorem for some general class of Markov chains. To justify the usefulness of our criterion, we further verify it for a particular discrete-time Markov dynamical system. From the application point of view, the examined system provides a useful tool in analysing the stochastic dynamics of gene expression in prokaryotes.Obviously, V is a Lyapunov function on X 2 , which, due to (B1), satisfies the inequality U V (x, y) ≤ aV (x, y) + 2b for any (x, y) ∈ X 2 . Consequently, referring to Lemma 1.3, we can choose λ ∈ (0, 1) and c λ > 0 so that(2.9)Define T : (X 2 ) N 0 → (X 2 ) N 0 by T ((x n , y n ) n∈N 0 ) = (x n+1 , y n+1 ) n∈N 0 , and letwhere (F k ) k∈N 0 stands for the natural filtration of (φk ) k∈N 0 . Now, put Λ := max{λ, γ}. Using the fact that ρ N K ≤ ρ N J + ρ K • T ρ N J , and, further, applying sequentially the strong Markov property, condition (B5) and inequality (2.9) with m = N , we obtain

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Horbacz

Wojewódka–Ściążko

2020

Asymptotic Analysis

Self Cite

We propose certain conditions which are sufficient for the functional law of the iterated logarithm (the Strassen invariance principle) for some general class of nonstationary Markov-Feller chains. This class may be briefly specified by the following two properties: firstly, the transition operator of the chain under consideration enjoys a nonlinear Lyapunov-type condition, and secondly, there exists an appropriate Markovian coupling whose transition probability function can be decomposed into two parts, one of which is contractive and dominant in some sense. The construction of such a coupling derives from the paper of M. Hairer (Probab. Theory Related Fields, 124(3):345-380, 2002). Our criterion may serve as a useful tool in verifying the functional law of the iterated logarithm for certain random dynamical systems, developed eg. in molecular biology. In the final part of the paper we present an example application of our main theorem to the mathematical model describing stochastic dynamics of gene expression.