Decompositions of CSBPs via Poissonian Intertwining
Clément Foucart, Olivier Hénard
TL;DR
This work unifies skeletal decompositions of CSBPs within an intertwining framework. It constructs a two-type $\mathbb{R}_+\times\mathbb{Z}_+$-valued process $(X^\lambda,L^\lambda)$ whose first coordinate preserves the CSBP law while the second is Poisson-poissonized with parameter $\lambda X^\lambda_t$, linked to the CSBP generator via a Poisson kernel $K$. By analyzing the position of $\lambda$ relative to $\rho$, the authors recover classical skeleton results for $\lambda\ge \rho$ and reveal a novel birth-immigration mechanism when $\lambda<\rho$; they also prove explosion occurs simultaneously along the skeleton and the CSBP when killing is absent. Furthermore, they establish a scaling limit: as $\lambda\to\infty$, the rescaled skeleton $L^\lambda/\lambda$ converges to the original CSBP in the Skorokhod sense. Collectively, these results provide a unified, operator-based perspective on skeletal decompositions and genealogies in CSBPs, with implications for branching structure and scaling behavior.
Abstract
We revisit certain decompositions of continuous-state branching processes (CSBPs), commonly referred to as skeletal decompositions, through the lens of intertwining of semi-groups. Precisely, we associate to a CSBP $X$ with branching mechanism $ψ$ a family of $\mathbb{R}_+\times \mathbb{Z}_+$-valued branching processes $(X^λ, L^λ)$, indexed by a parameter $λ\in (0, \infty)$, that satisfies an intertwining relationship with $X$ through the Poisson kernel with parameter $λx$. The continuous component $X^λ$ has the same law as $X$, while the discrete component $L^λ$, conditionally on $X^λ_t$, has a Poisson distribution with parameter $λX^λ_t$. The law of $(X^λ, L^λ)$ depends on the position of $λ$ within $[0, \infty) = [0, ρ) \cup [ρ, \infty)$, where $ρ$ is the largest positive root of $ψ$. When $λ\geq ρ$, various well-known results concerning skeleton decompositions are recovered. In the supercritical case ($ρ> 0$), when $λ<ρ$, a novel phenomenon arises: a birth term appears in the skeleton, corresponding to a one-unit proportional immigration from the continuous to the discrete component. Along the way, the class of continuous-time branching processes taking values in $\mathbb{R}_+ \times \mathbb{Z}_+$ is constructed.