Let \((p_{ij}(t))\) be a sub-Markov transition probability matrix on a countable state space \(E\) with the totally stable and conservative \(Q\)- matrix \(Q=(q_{ij})\). The author studies three classical problems: The uniqueness of the \(Q\)-semigroup, recurrence and positive recurrence of the corresponding Markov chain.
For a flavor of obtained results, here is the uniqueness theorem: If there exists a sequence \(\{E_ n\}^ \infty_ 1\), \(E_ n\uparrow E\), and a non-negative function \(\phi\) such that \(\sup_{i\in E_ n} q_ i<\infty\), \(\lim_{n\to\infty}\) \(\inf_{i\not\in E_ n}\phi_ i=\infty\), and \[ \sum_ j q_{ij}(\phi_ j-\phi_ i)\leq c\phi_ i,\quad i\in E, \] for some \(c\in{\mathbf R}\), then the \(Q\)-process is unique. - The main tool in proving results is the concept of the minimal, non- negative solution to a non-negative linear equation. Examples are given illustrating applicability of criteria.