The leftmost particle of branching subordinators

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.