Perturbation bounds of Markov semigroups on abstract states spaces

TL;DR

This paper develops a framework using Dobrushin's ergodicity coefficient to analyze stability and perturbation bounds of positive $C_0$-semigroups on abstract state spaces, with implications for quantum and classical systems.

Contribution

It introduces new perturbation bounds for time averages of stable semigroups and establishes the equivalence of uniform and weak ergodicities via the ergodicity coefficient.

Findings

Derived linear relations between stability and fixed point sensitivity.

Proved equivalence of uniform and weak ergodicities in terms of the ergodicity coefficient.

Studied unique ergodicity of semigroups with weighted averages.

Abstract

In order to successfully explore quantum systems which are perturbations of simple models, it is essential to understand the complexity of perturbation bounds. We must ask ourselves: How quantum many-body systems can be artificially engineered to produce the needed behavior. Therefore, it is convenient to make use of abstract framework to better understand classical and quantum systems. Thus, our investigation's purpose is to explore stability and perturbation bounds of positive $C_0$-semigroups on abstract state spaces using the Dobrushin's ergodicity coefficient. Consequently, we obtain a linear relation between the stability of the semigroup and the sensitivity of its fixed point with respect to perturbations of $C_0$-Markov semigroups. Our investigation leads to the discovery of perturbation bounds for the time averages of uniform asymptotically stable semigroups. A noteworthy mention is that we also prove the equivalence of uniform and weak ergodicities of the time averages Markov operators in terms of the ergodicity coefficient, which shines a new light onto this specific topic. Lastly, in terms of weighted averages, unique ergodicity of semigroups is also studied. Emphasis is put on the newly obtained results which is a new discovery in classical and non-commutative settings.