2020/01/17 by Mukhamedov, Farrukh, Al-Rawashdeh, Ahmed
#28D05 #47A35 #60J10 #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Probability (math.PR)
paper · doi:10.48550/arxiv.2001.07703
In our earlier paper, a generalized Dobrushin ergodicity coefficient of Markov operators (acting on abstract state spaces) with respect to a projection P, has been introduced and studied. It turned out that the introduced coefficient was more effective than the usual ergodicity coefficient. In the present work, by means of a left consistent Markov projections and the generalized Dobrushin's ergodicity coefficient, we investigate uniform and weak P-ergodicities of non-homogeneous discrete Markov chains (NDMC) on abstract state spaces. It is easy to show that uniform P-ergodicity implies a weak one, but in general the reverse is not true. Therefore, some conditions are provided together with weak P-ergodicity of NDMC which imply its uniform P-ergodicity. Furthermore, necessary and sufficient conditions are found by means of the Doeblin's condition for the weak P-ergodicity of NDMC. The weak P-ergodicity is also investigated in terms of perturbations. Several perturbative results are obtained which allow us to produce nontrivial examples of uniform and weak P-ergodic NDMC. Moreover, some category results are also obtained. We stress that all obtained results have potential applications in the classical and non-commutative probabilities.