Criteria for three classical problems of downwardly skip-free processes

This paper devotes to studying downwardly skip-free processes (also called single death processes), motivated by a known result, that is, the uniqueness of a time-continuous Markov chain is equivalent to the recurrence of the embedding chain on an enlarged state space corresponding to the original process. The uniqueness, extinction, recurrence and ergodicity (namely, positive recurrence) criteria for the process are obtained. The main new ideas in our proofs are the use of potential theory, method of limit approximation, and the strong Markov property of the process. Some examples are included to illustrate these results are quite effective. Particularly, applying our results to an extended class of branching process, this paper settles the question of its uniqueness which Chen unsolved when dealing with the above problems of this process.