Explosion and implosion of birth-and-death continuous-time random walks

Abstract

We provide necessary and sufficient conditions for explosion and implosion of birth-and-death (non-Markov) continuous-time random walks. In other words, we obtain conditions for $\infty$ to be accessible and for it to be an entrance point. We derive the analytical regularity criteria in terms of the appropriate scale function and the speed measure, which involve transition probabilities and the Laplace transform of the waiting times. We show that these criteria closely resemble classical ones for diffusions and Markov birth-and-death processes. We then calculate explicit conditions of regularity for semi-Markov processes with waiting times that have (a) finite first moments; (b) regularly varying tails (in particular, $\alpha$-stable distribution).