Uniform Ergodicity of Lotz–Räbiger Nets of Markov Operators on Abstract State Spaces

Abstract

It is known that Dobrushin’s ergodicity coefficient is one of the powerful tools in the investigation of limiting behavior of Markov chains. Several interesting properties of the ergodicity coefficient of a positive mapping defined on an abstract state space have been studied. In this paper, we consider uniform ergodicity of Lotz–Rabiger nets of Markov operators on abstract state spaces. We prove a uniform mean ergodicity criterion in terms of the ergodicity coefficient. This result allows us to investigate perturbations of uniformly mean ergodic operators. Moreover, our main results open new perspectives in quantum Markov processes defined on von Neumann algebras.