Stability Estimates of Markov Semigroups on Abstract States Spaces

In this paper, the stability and perturbation estimates for Markov C-semigroups on abstract state spaces are explored using the Dobrushin ergodicity coefficient. Consequently, a linear relation between the stability of the semigroup and the sensitivity of its fixed point is obtained with respect to the perturbations of the semigroup. This investigation has led to the discovery of perturbation estimates for the time averages of uniform asymptotically stable semigroups. This work also proves the equivalence of uniform and weak ergodicities of time averages of C-Markov semigroups in terms of the ergodicity coefficient, which shines new light on this topic. Finally, in terms of weighted averages, the unique ergodicity of semigroups is also studied. Emphasis is laid on the newly obtained results, which are new discoveries in the classical and non-commutative settings.