The leftmost particle of branching subordinators

TL;DR

This paper introduces branching subordinators, a new class of continuous-time branching particle systems with particles moving as subordinators, and analyzes the sub-linear growth rate of their leftmost particle, contrasting with classical models.

Contribution

It defines branching subordinators and studies their asymptotic behavior, revealing a sub-linear growth rate for the leftmost particle, which is a novel finding in the context of branching processes.

Findings

Leftmost particle grows at rate t^{γ} with γ  (0,1)

Growth rate depends explicitly on model parameters

Contrasts with classical linear growth in branching random walks

Abstract

We define a family of continuous-time branching particle systems on the non-negative real line, called branching subordinators, where particles move as independent subordinators. Each particle can also split (at possibly infinite rate) into several children (possibly infinitely many) whose positions relative to the position of the parent are random. These particle systems are in the continuity of branching L\'evy processes introduced by Bertoin and Mallein [Ann. Probab. 47(3): 1619-1652 (2019)]. We pay a particular attention to the asymptotic behavior of the leftmost particle of branching subordinators. It turns out that, under some assumptions, the rate of growth of the position of the leftmost particle is of order $t^{\gamma}$ where $\gamma \in (0,1)$ depends explicitly on the parameters. This sub-linear growth is significantly different from the classical linear growth observed for regular branching random walks with non-negative displacements.