Irreducibility and uniqueness of symmetric measure for Markov processes

Abstract

In this paper various irreducibilities for Markov processes related to topologies and excessive measures are discussed and their relations are presented. We shall mainly prove that, while the fine irreducibility is sufficient for the symmetric measure (and stationary distribution) to be unique if exists, it is almost necessary, namely $X$ is $m$-irreducible if $m$ is the unique symmetric measure for $X$.