# Coalescences in Continuous-State Branching Processes

## TL;DR

This paper explores the ancestral lineages in continuous-state branching processes, introducing new coalescent models that describe genealogical structures and their long-term behaviors.

## Contribution

It introduces the concept of consecutive coalescents and characterizes their law via finite measures, linking genealogies to inhomogeneous Galton-Watson processes.

## Key findings

- Characterization of ancestral lineage processes as Siegmund duals.
- Development of non-exchangeable Markovian coalescent processes.
- Description of the long-term behavior and generators of these processes.

## Abstract

Consider a continuous-state branching population constructed as a flow of nested subordinators. Inverting the subordinators and reversing time give rise to a flow of coalescing Markov processes (with negative jumps) which correspond to the ancestral lineages of individuals in the current generation. The process of the ancestral lineage of a fixed individual is the Siegmund dual process of the continuous-state branching process. We study its semi-group, its long-term behavior and its generator. In order to follow the coalescences in the ancestral lineages and to describe the backward genealogy of the population, we define non-exchangeable Markovian coalescent processes obtained by sampling independent Poisson arrival times over the flow. These coalescent processes are called consecutive coalescents, as only consecutive blocks can merge. They are characterized in law by finite measures on $\mathbb{N}$ which can be thought as the offspring distributions of some inhomogeneous immortal Galton-Watson processes forward in time.