On the coming down from infinity of continuous-state branching processes with drift-interaction
Abstract
We study the phenomenon of coming down from infinity - that is, when the process starts from infinity and never returns to it - for continuous-state branching processes with generalized drift. We provide sufficient conditions on the drift term and the branching mechanism to ensure both non-explosion and coming down from infinity, without requiring the associated jump measure to have a finite first moment. Assuming the process comes down from infinity and the drift satisfies a one-sided Lipschitz condition, we show that, as the initial values tend to infinity, the process converges locally uniformly almost surely to the strong solution of a stochastic differential equation. The main techniques employed are comparison principles for solutions of stochastic equations and the method of Lyapunov functions, the latter being briefly reviewed in a broader setting.
Cite
@article{arxiv.2510.05958,
title = {On the coming down from infinity of continuous-state branching processes with drift-interaction},
author = {Félix Rebotier},
journal= {arXiv preprint arXiv:2510.05958},
year = {2025}
}