26 research outputs found
The central limit theorem for random dynamical systems
We consider random dynamical systems with randomly chosen jumps. The choice of deterministic dynamical system and jumps depends on a position. The Central Limit Theorem for random dynamical systems is established
Law of large numbers for random dynamical systems
We consider random dynamical systems with randomly chosen jumps. The choice of deterministic dynamical system and jumps depends on a position. We prove the existence of an exponentially attractive invariant measure and the strong law of large numbers
Law of the Iterated Logarithm for some Markov operators
The Law of the Iterated Logarithm for some Markov operators, which converge
exponentially to the invariant measure, is established. The operators
correspond to iterated function systems which, for example, may be used to
generalize the cell cycle model examined by A. Lasota and M.C. Mackey, J. Math.
Biol. (1999).Comment: 23 page
On absolute continuity of invariant measures associated with a piecewise-deterministic Markov processes with random switching between flows
We are concerned with the absolute continuity of stationary distributions
corresponding to some piecewise deterministic Markov process, being typically
encountered in biological models. The process under investigation involves a
deterministic motion punctuated by random jumps, occurring at the jump times of
a Poisson process. The post-jump locations are obtained via random
transformations of the pre-jump states. Between the jumps, the motion is
governed by continuous semiflows , which are switched directly after the jumps.
The main goal of this paper is to provide a set of verifiable conditions
implying that any invariant distribution of the process under consideration
that corresponds to an ergodic invariant measure of the Markov chain given by
its post-jump locations has a density with respect to the Lebesgue measure
The central limit theorem for Markov processes that are exponentially ergodic in the bounded-Lipschitz norm
In this paper, we establish a version of the central limit theorem for
Markov-Feller continuous time processes (with a Polish state space) that are
exponentially ergodic in the bounded-Lipschitz distance and enjoy a continuous
form of the Foster-Lyapunov condition. As an example, we verify the assumptions
of our main result for a specific piecewise-deterministic Markov process, whose
deterministic component evolves according to continuous semiflows, switched
randomly at the jump times of a Poisson process.Comment: 41 page
Exponential ergodicity in the bounded-Lipschitz distance for a subclass of piecewise-deterministic Markov processes with random switching between flows
In this paper, we study a subclass of piecewise-deterministic Markov
processes with a Polish state space, involving deterministic motion punctuated
by random jumps that occur at exponentially distributed time intervals. Over
each of these intervals, the process follows a flow, selected randomly among a
finite set of all possible ones. Our main goal is to provide a set of
verifiable conditions guaranteeing the exponential ergodicity for such
processes (in terms of the bounded Lipschitz distance), which would refer only
to properties of the flows and the transition law of the Markov chain given by
the post-jump locations. Moreover, we establish a simple criterion on the
exponential ergodicity for a particular instance of these processes, applicable
to certain biological models, where the jumps result from the action of an
iterated function system with place-dependent probabilities
Continuous dependence of an invariant measure on the jump rate of a piecewisedeterministic Markov process
We investigate a piecewise-deterministic Markov process, evolving on a Polish metric
space, whose deterministic behaviour between random jumps is governed by some semi-flow, and any
state right after the jump is attained by a randomly selected continuous transformation. It is assumed
that the jumps appear at random moments, which coincide with the jump times of a Poisson process
with intensity . The model of this type, although in a more general version, was examined in our
previous papers, where we have shown, among others, that the Markov process under consideration
possesses a unique invariant probability measure, say
. The aim of this paper is to prove that the
map 7!
is continuous (in the topology of weak convergence of probability measures). The studied
dynamical system is inspired by certain stochastic models for cell division and gene expression