11 research outputs found
Limit theorems for some Markov operators
Analysis and Stochastic
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
Equilibrium states and invariant measures for random dynamical systems
Random dynamical systems with countably many maps which admit countable
Markov partitions on complete metric spaces such that the resulting Markov
systems are uniformly continuous and contractive are considered. A
non-degeneracy and a consistency conditions for such systems, which admit some
proper Markov partitions of connected spaces, are introduced, and further
sufficient conditions for them are provided. It is shown that every uniformly
continuous Markov system associated with a continuous random dynamical system
is consistent if it has a dominating Markov chain. A necessary and sufficient
condition for the existence of an invariant Borel probability measure for such
a non-degenerate system with a dominating Markov chain and a finite (16) is
given. The condition is also sufficient if the non-degeneracy is weakened with
the consistency condition. A further sufficient condition for the existence of
an invariant measure for such a consistent system which involves only the
properties of the dominating Markov chain is provided. In particular, it
implies that every such a consistent system with a finite Markov partition and
a finite (16) has an invariant Borel probability measure. A bijective map
between these measures and equilibrium states associated with such a system is
established in the non-degenerate case. Some properties of the map and the
measures are given.Comment: The article is published in DCDS-A, but without the 3rd paragraph on
page 4 (the complete removal of the paragraph became the condition for the
publication in the DCDS-A after the reviewer ran out of the citation
suggestions collected in the paragraph
Existence of a unique invariant measure for a class of equicontinuous Markov operators with application to a stochastic model for an autoregulated gene
Analysis and Stochastic
The law of the iterated logarithm for a piecewise deterministic Markov process assured by the properties of the Markov chain given by its post-jump locations
Analysis and Stochastic