Markovian tricks for non-Markovian trees: contour process, extinction and scaling limits

TL;DR

This paper analyzes non-Markovian population trees using Markovian techniques, providing conditions for extinction, characterizing laws, and exploring scaling limits like the Bessel tree.

Contribution

It introduces a new approach to study non-Markovian trees via a Feller process, offering criteria for extinction and scaling limit results.

Findings

Characterization of the generator of the coding process.

Necessary and sufficient conditions for extinction.

Identification of the Bessel tree as a scaling limit.

Abstract

In this work, we study a family of non-Markovian trees modeling populations where individuals live and reproduce independently with possibly time-dependent birth-rate and lifetime distribution. To this end, we use the coding process introduced by Lambert. We show that, in our situation, this process is no longer a L{\'e}vy process but remains a Feller process and we give a complete characterization of its generator. This allows us to study the model through well-known Markov processes techniques. On one hand, introducing a scale function for such processes allows us to get necessary and sufficient conditions for extinction or non-extinction and to characterize the law of such trees conditioned on these events. On the other hand, using Lyapounov drift techniques , we get another set of, easily checkable, sufficient criteria for extinction or non-extinction and some tail estimates for the tree length. Finally, we also study scaling limits of such random trees and observe that the Bessel tree appears naturally.