  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.
