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.
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 ,
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.
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