Explosion and implosion of birth-and-death continuous-time random walks
Andrey Pilipenko, Vadym Tkachenko
TL;DR
The paper addresses when birth-and-death continuous-time random walks explode to infinity or implode from infinity, generalizing classical boundary results to non-Markov, semi-Markov settings. It develops explicit, verifiable criteria expressed in terms of a canonical scale function and speed measure that incorporate the waiting-time Laplace transforms, and it connects these criteria to known diffusion theory. The authors provide detailed analyses for several waiting-time regimes, including finite first moments, regularly varying tails, and stable waiting times, with concrete Markov-case reductions. The results yield both necessary and sufficient conditions and robust proof techniques, enabling applications to a wide range of semi-Markov models and contributing a discrete-analytic parallel to diffusion boundary theory.
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, $α$-stable distribution).