Ergodic properties of nonhomogeneous Markov chains defined on ordered Banach spaces with a base

Abstract

It is known that the Dobrushin's ergodicity coefficient is one of the effective tools to study a behavior of non-homogeneous Markov chains. In the present paper, we define such an ergodicity coefficient of a positive mapping defined on ordered Banach space with a base (OBSB), and study its properties. In terms of this coefficient we prove the equivalence uniform and weak ergodicities of homogeneous Markov chains. This extends earlier results obtained in case of von Neumann algebras. Such a result allowed to establish a category theorem for uniformly ergodic Markov operators. We find necessary and sufficient conditions for the weak ergodicity of nonhomogeneous discrete Markov chains (NDMC). It is also studied $L$-weak ergodicity of NDMC defined on OBSB. We establish that the chain satisfies $L$-weak ergodicity if and only if it satisfies a modified Doeblin's condition ($\frak{D}_1$-condition). Moreover, several nontrivial examples of NDMC which satisfy the $\frak{D}_1$-condition are provided.