Coalescences in Continuous-State Branching Processes

Clément Foucart (1) , Chunhua Ma (2, 3) , Bastien Mallein (1)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 Nankai University. School of Mathematical Sciences and LPMC
3 NKU - Nankai University

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.

Mots clés: Branching processes; coalescent processes; continuous-state branching processes; flow of subordinators; genealogy; duality

Domaines: Probabilités [math.PR]

Clément Foucart, Chunhua Ma, Bastien Mallein. Coalescences in Continuous-State Branching Processes. Electronic Journal of Probability, 2019, 24 (103), ⟨10.1214/19-EJP358⟩. ⟨hal-01944043v2⟩