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

TL;DR

This paper establishes precise conditions for explosion and implosion in birth-and-death continuous-time random walks, extending classical diffusion and Markov process criteria to semi-Markov processes with various waiting time distributions.

Contribution

It provides necessary and sufficient analytical criteria for explosion and implosion in non-Markov birth-and-death processes, including explicit conditions for semi-Markov processes with different waiting time distributions.

Findings

Derived regularity criteria involving scale functions and speed measures.

Established conditions for explosion and implosion in semi-Markov processes.

Connected criteria to classical diffusion and Markov birth-and-death processes.

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).