Explosion and implosion of continuous-time birth-and-death random walks
DOI:
https://doi.org/10.3842/umzh.v78i9-10.9970Keywords:
continuous-time random walk, semi-Markov process, explosion, implosion, exit boundary, entrance boundaryAbstract
UDC 519.21
We provide necessary and sufficient conditions for the explosion and implosion of birth-and-death (non-Markov) continuous-time random walks. In other words, we establish conditions for $\infty$ to be accessible and conditions under which the infinity is an entrance point. We establish the analytic regularity criteria in terms of the appropriate scale function and speed measure expressed via the transition probabilities and the Laplace transform of waiting times. It is shown that the structure of these criteria is close to the structure of the classical conditions for diffusions and Markov birth-and-death processes. Further, we find the explicit conditions of regularity for semi-Markov processes with waiting times that have (a) finite first moments; (b) regularly varying tails (in particular, the $\alpha$-stable distribution).
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