Stability of the Markov operator and synchronization of Markovian random products

TL;DR

This paper investigates the stability and synchronization properties of Markovian random products on complex metric spaces, introducing a generalized splitting condition that guarantees asymptotic stability and exponential synchronization.

Contribution

It generalizes the classical splitting condition to a broader class of spaces and proves its implications for stability and synchronization of Markov operators.

Findings

Splitting condition implies asymptotic stability

Markov operator exhibits exponential synchronization

Applicable to complex metric spaces like products of intervals and trees

Abstract

We study Markovian random products on a large class of "m-dimensional" connected compact metric spaces (including products of closed intervals and trees). We introduce a splitting condition, generalizing the classical one by Dubins and Freedman, and prove that this condition implies the asymptotic stability of the corresponding Markov operator and (exponentially fast) synchronization.