Irreducibility and uniqueness of symmetric measure for Markov processes (Q6155665)

Let \(X=(X_t, \mathbb{P}^x)\) be a transient right process with the semigroup \((P_t)\) and the resolvent \((U^\alpha)\) on a state space \(E\). Assume that \(m\) is an excessive measure of \(X\), namely \(m\) is a positive \(\sigma\)-finite measure on \(E\) with \(mP_t\leq m\) for any \(t>0\). For a nearly Borel subset \(A\) of \(E\), \(T_A:=\inf\{t>0:X_t\in A\}\) denotes the first hitting time of \(A\). A universally measurable subset \(A\) of \(E\) is called invariant (resp. \(m\)-invariant) if \(P_t(x,A^c)=0\) for any \(t>0\) and \(x\in A\) (resp. \(m\)-a.e. \(x\in A\)). The process \(X\) is said to satisfy the resolvent absolute continuity to \(m\), or \(m\)-\textbf{RAC}, if \(U^1(x,\cdot)\ll m\) for any \(x\in E\). The main focus of this paper are the various irreducibilities of \(X\). The process \(X\) is called finely (or topologically) irreducible if \(\mathbb{P}^x(T_D < \infty) > 0\) for any non-empty finely open (or open) subset \(D\) of \(E\), and \(x \in E\). Given the excessive measure \(m\), \(X\) is called \(m\)-weakly irreducible (resp. \(m\)-irreducible) if any invariant set (resp. \(m\)-invariant set) \(A\) is \(m\)-trivial, namely either \(m(A)=0\) or \(m(A^c)=0\). It is called \(m\)-finely irreducible if for any finely open subset \(D\), \(\{x\in E: \mathbb{P}^x(T_D<\infty)=0\}\) is \(m\)-trivial. It is worth noting that while fine irreducibility implies topological irreducibility, the reverse is not necessarily true. Additionally, this paper establishes some equivalent conditions for fine irreducibility. Given the excessive measure \(m\), this paper proves the equivalence between \(m\)-weak irreducibility, \(m\)-irreducibility and \(m\)-fine irreducibility. Furthermore, \(X\) is finely irreducible, if and only if it is \(m\)-irreducible (or \(m\)-weakly irreducible, \(m\)-finely irreducible) and satisfies \(m\)-\textbf{RAC}. This holds true for some excessive \(m\) and, equivalently, for every excessive \(m\). It has been shown in previous works [\textit{P. He} and \textit{J. Ying}, Osaka J. Math. 50, No. 2, 417--423 (2013; Zbl 1279.60099); \textit{J. Ying} and \textit{M. Zhao}, Proc. Am. Math. Soc. 138, No. 6, 2181--2185 (2010; Zbl 1197.60078)] that fine irreducibility implies the uniqueness of symmetric measures (or stationary distributions) of \(X\). The present paper further shows that this uniqueness implies \(m\)-fine irreducibility, where \(m\) is such a unique measure. In other words, this uniqueness is a property that lies between fine irreducibility and \(m\)-fine irreducibility.