Time-fractional Birth and Death Processes

2020/04/29 by Littin, Jorge
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2004.14177

Abstract

In this article, we provide different representations for a time-fractional birth and death process N_α(t), whose transition probabilities are governed by a time-fractional system of differential equations. More specifically, we present two equivalent characterizations for its trajectories: the first one as a time-changed classic birth and death process, whereas the second one is a Markov renewal process. Also, we provide results for the asymptotic behavior of the process conditioned not to be killed. The most important is that the concept of quasi-limiting distribution and quasi-stationary distribution do not coincide, which is a consequence of the long-memory nature of the process. As an application example, we revisit the linear case to show the consequences of our main theorems.

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