Invariant distribution of \(Q\)-process. I (Q2749019)

This paper addresses the open problem of \textit{D. Williams} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 3, 227-246 (1964; Zbl 0143.19804)], i.e., assumes that \(E\) is a countable set, \(Q=(q_{ij})\) (\(i,j\in E\)) is a matrix on \(E\times E\) such that NEWLINE\[NEWLINE q_{ij}\geq 0 \quad (i\neq j), \quad\quad \displaystyle\sum_{k\neq i} q_{ik}=-q_{ii}\leq +\infty , \quad \forall i\in E. NEWLINE\]NEWLINE Moreover, \(m\) is a strictly positive probability distribution on \(E\) such that NEWLINE\[NEWLINE \displaystyle\sum_{i\neq j} m_iq_{ij}=-m_jq_{jj}, \quad \forall j\in E. NEWLINE\]NEWLINE Then under what conditions there exists a \(Q\)-function \(P\) such that \(m\) is an invariant distribution of \(P\)? The authors prove the existence of a \(Q\)-function under the assumption that \(Q\) is globally stable, i.e., \(q_i<\infty \) for all \(i\in E\).