Sıralı Banach Uzayları Üzerinde Tanımlı Homojen Markov Zincirlerinin Cesaro Ortalamalarının Pertürbasyon Sınırları

Abstract

Dobrushin’s ergodicity coefficient is one of the effective tools for the investigations of limiting behaviours on the Markov process theory. Several interesting properties of the ergodicity of positive operators defined on some previous Banach spaces previously have been studied with the aid of Dobrushin’s ergodicity coefficient. In this thesis, we firstly mention ergodicity coefficient properties of Markov operators defined on ordered Banach space with a base. Then we deal with uniformly mean ergodic and asymptotically stable of these operators. Additionally, we examine uniform mean ergodicity criterion in terms of the ergodicity coefficient and set the perturbation theory for uniformly asymptotically stable Markov chains on ordered Banach spaces. Subsequently, we study the uniform asymptotical stability for C0-Markov semigroups in terms of the Dobrushin’s ergodicity coefficient. In this way, we obtain a linear relation between the stability of the semigroup and the sensitivity of its fixed point with respect to perturbations of Markov operators. Moreover, we also establish perturbation bounds for the time averages of the uniform asymptotically stable semigroups. Furthermore, we study the equivalence of uniform and weak ergodicities of the time averages Markov semigroups in terms of the ergodicity coefficient. This result allows us to produce some kind of perturbation bounds for such kind of semigroups. In this thesis finally we consider properties of the ergodicity of LR-nets defined on ordered Banach space. The results of this thesis which we have compiled above in general lines open a new perspective in perturbation theory for quantum Markov processes on Banach spaces. Anahtar Kelimeler: Ordered Banach space, homogen Markov operator, Dobrushin’s coefficient (er-godicity coefficient), perturbation bound, Cesaro average, Lotz-Rabiger nets, ordered norm space, uniformly asymptotically stable