Dynamical Systems and Markov Chains

Deterministic chaos for Markov chains

We find that Markov chains with finite state space are Poincare chaotic. Moreover, finite realizations of the chains are arcs of each unpredictable orbit for sure. An illustrating example with a proper numerical simulation is provided.

On the Vere-Jones classification and existence of maximal measures for countable topological Markov chains

We consider topological Markov chains (also called Markov shifts) on countable graphs. We show that a transient graph can be extended to a recurrent graph of equal entropy which is either positive recurrent of null recurrent, and we give an example of each type. We extend the notion of local entropy to topological Markov chains and prove that a transitive Markov chain admits a measure of maximal entropy (or maximal measure) whenever its local entropy is less than its (global) entropy.

Coupling methods for random topological Markov chains

We apply coupling techniques in order to prove that the transfer operators associated with random topological Markov chains and non-stationary shift spaces with the big images and preimages-property have a spectral gap.

Exponential stability of nonhomogeneous matrix-valued Markovian chains

In this paper, we characterize the stability of matrix-valued Markovian chains by periodic data.

On regularity of functions of Markov chains

We consider processes which are functions of finite-state Markov chains. It is well known that such processes are rarely Markov. However, such processes are often regular in the following sense: the distant past values of the process have diminishing influence on the distribution of the present value. In the present paper, we present novel sufficient conditions for regularity of functions of Markov chains.

将亚稳态连续空间马尔可夫链约化为有限集上的马尔可夫链

Reducing metastable continuous-space Markov chains to Markov chains on a finite set

本文研究具有有限亚稳态的连续空间马尔可夫链，证明其核有 N 个接近 1 的分离特征值，并可用 N 状态有限马尔可夫链在时间上一致逼近，转移概率与首达概率指数接近。