A small-time coupling between $\Lambda$-coalescents and branching processes

TL;DR

This paper establishes a new explicit coupling between Lambda-coalescents and continuous-state branching processes, linking their extinction and coming down from infinity, and providing insights into their speed and power-law behaviors.

Contribution

It introduces a novel explicit coupling based on a particle representation that connects coalescent and branching process genealogies, answering a key open question.

Findings

Coalescent comes down from infinity iff the branching process becomes extinct.

The coupling relates the speed of coming down from infinity to properties of Levy processes.

Power-law behavior of N^Lambda(t) is connected to classical Levy process indices.

Abstract

We describe a new general connection between $\Lambda$-coalescents and genealogies of continuous-state branching processes. This connection is based on the construction of an explicit coupling using a particle representation inspired by the lookdown process of Donnelly and Kurtz. This coupling has the property that the coalescent comes down from infinity if and only if the branching process becomes extinct, thereby answering a question of Bertoin and Le Gall. The coupling also offers new perspective on the speed of coming down from infinity and allows us to relate power-law behavior for $N^{\Lambda}(t)$ to the classical upper and lower indices arising in the study of pathwise properties of L\'{e}vy processes.