Genealogy of flows of continuous-state branching processes via flows of partitions and the Eve property

TL;DR

This paper introduces a novel stochastic flow of partitions to encode the genealogy of continuous-state branching processes, providing new insights into asymptotic behaviors, the Eve property, and unifying existing representations.

Contribution

It develops a new flow of partitions framework for $ ext{CSBP}$ genealogies, characterizes the Eve property, and unifies different genealogical representations.

Findings

Characterizes the Eve property with a necessary and sufficient condition.

Shows the flow of partitions unifies lookdown and subordinator representations.

Provides a tractable object for studying asymptotic behaviors of $ ext{CSBP}$.

Abstract

We encode the genealogy of a continuous-state branching process associated with a branching mechanism $\Psi$ - or $\Psi$-CSBP in short - using a stochastic flow of partitions. This encoding holds for all branching mechanisms and appears as a very tractable object to deal with asymptotic behaviours and convergences. In particular we study the so-called Eve property - the existence of an ancestor from which the entire population descends asymptotically - and give a necessary and sufficient condition on the $\Psi$-CSBP for this property to hold. Finally, we show that the flow of partitions unifies the lookdown representation and the flow of subordinators when the Eve property holds.