Strict irreducibility of Markov chains and ergodicity of skew products

TL;DR

This paper investigates the conditions under which ergodicity of a family of measure-preserving transformations implies ergodicity of the associated skew product, introducing strict irreducibility as a key concept.

Contribution

It characterizes when ergodicity of a Markov chain family implies ergodicity of the skew product, extending Bufetov's condition to general state spaces and proving its necessity.

Findings

Introduces strict irreducibility for Markov kernels.

Extends ergodicity conditions from finite to general state spaces.

Provides explicit limit descriptions in ergodic theorems for random transformations.

Abstract

We consider a family of measure preserving transformations, which act on a common probability space and are chosen at random by a stationary ergodic Markov chain. This setting defines an instance of a random dynamical system (RDS), which may be described in terms of a step skew product. In many contexts it is desirable to know whether ergodicity of the family implies ergodicity of the skew product. Introducing the notion of strict irreducibility for Markov kernels we shall characterize the class of Markov chains for which the aforementioned implication holds true. We thereby extend a sufficient condition of Bufetov for the case of finite state Markov chains to general state spaces and show that it is in fact also necessary. As an application we obtain an explicit description of the limit in ergodic theorems for a suitable class of random transformations.