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 behaviour 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 individuals according to an independent Poisson point process 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.
Highlights
Consider a continuous-state branching population constructed as a flow of nested subordinators
We briefly review some methods that have been developed recently in order to study the genealogy backwards in time of branching processes
We show how continuous-state branching processes can be constructed as flows of subordinators
Summary
Random population models can be divided in two classes, those with constant finite size and those whose size is varying randomly. One approach consists in conditioning the process to be non-extinct at a given time, sampling two or more individuals uniformly in the population and study the time of coalescence of their ancestral lineages. We introduce new elementary non-exchangeable Markovian coalescents as simple dual objects of immortal continuous-time Galton-Watson processes These processes are taking values in the set of partitions of N whose blocks are formed with consecutive integers.
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