Uniform ergodicities and perturbation bounds of Markov chains on ordered Banach spaces

TL;DR

This paper develops criteria for uniform ergodicity and perturbation bounds of Markov chains on ordered Banach spaces, with applications to classical and quantum Markov processes, using Dobrushin's ergodicity coefficient.

Contribution

It introduces new uniform ergodicity criteria and perturbation bounds for Markov operators on ordered Banach spaces, extending to quantum Markov processes.

Findings

Established a uniform mean ergodicity criterion using the ergodicity coefficient.

Developed a perturbation theory for asymptotically stable Markov chains.

Applied results to quantum Markov processes on von Neumann algebras.

Abstract

It is known that Dobrushin's ergodicity coefficient is one of the effective tools in the investigations of limiting behavior of Markov processes. Several interesting properties of the ergodicity coefficient of a positive mapping defined on ordered Banach space with a base have been studied. In this paper, we consider uniformly mean ergodic and asymptotically stable Markov operators on ordered Banach spaces. In terms of the ergodicity coefficient, we prove uniform mean ergodicity criterion in terms of the ergodicity coefficient. Moreover, we develop the perturbation theory for uniformly asymptotically stable Markov chains on ordered Banach spaces. In particularly, main results open new perspectives in the perturbation theory for quantum Markov processes defined on von Neumann algebras. Moreover, by varying the Banach spaces one can obtain several interesting results in both classical and quantum settings as well.