The narrow recurrence of continuous-time Markov chains

Abstract

Let (Ω, F, P) be a probability space and X be a Polish space equipped with its Borel σ-algebra B. We consider a transition function probability {Pt, t ∈ R +} of a continuous Markov chain on (Ω, F, P) with values in X. This transition function defines a semi group acting on P r(X), the set of all probability measures on X, which is also a Polish space endowed with the narrow topology. In this paper, we introduce and study the notion of the narrow recurrence of transition functions of continuous Markov chains and provide some properties, which can be considered as an initiation of applications of properties of topological dynamics on stochastic process theory and random dynamical systems.