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

It is known that the Dobrushin’s ergodicity coefficient is one of the effective tools to study the behavior of non-homogeneous Markov chains. In the present paper, we define such an ergodicity coefficient of a positive mapping defined on ordered Banach spaces 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 allows 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). L-weak ergodicity of NDMC defined on OBSB is also studied. We establish that the chain satisfies L-weak ergodicity if and only if it satisfies a modified Doeblin’s condition (D₁ -condition). Moreover, some connections between L-weak ergodicity and L-strong ergodicity have been established. Several nontrivial examples of NDMC which satisfy the D₁-condition are provided.