On stability of single death processes

Abstract

Single birth processes and single death ones are the classical asymmetric Markov processes,on which the investigation is more difficult than that on symmetric ones. In the past three decades, the investigation on single birth processes is fruitful.But the results on single death processes are almost empty. In this paper, we introduce our recent progress on single death processes.We present the explicit representation for the first hitting times and the probabilistic meanings of some numerical characteristics of single death processes.Based on these results, the criteria on ergodicity and strong ergodicity, sufficient or necessary conditions for recurrence and an explicit sufficientcondition for exponential ergodicity as well as presentation of returning probability of single death processes, are obtained respectively.These assertions are also applied to an extended class of branching processes.