Decompositions of CSBPs via Poissonian Intertwining

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 $\psi$ a family of ${\mathbb{R}}_{+}\times {\mathbb{Z}}_{+}$-valued branching processes $(X^{\lambda },L^{\lambda })$, indexed by a parameter $\lambda \in (0,\infty )$, that satisfies an intertwining relationship with $X$ through the Poisson kernel with parameter $\lambda x$. The continuous component $X^{\lambda }$ has the same law as $X$, while the discrete component $L^{\lambda }$, conditionally on $X_t^{\lambda }$, has a Poisson distribution with parameter $\lambda X_t^{\lambda }$. The law of $(X^{\lambda },L^{\lambda })$ depends on the position of $\lambda$ within $[0,\infty )=[0,\rho )\cup [\rho ,\infty )$, where $\rho$ is the largest positive root of $\psi$. When $\lambda \ge \rho$, various well-known results concerning skeleton decompositions are recovered. In the supercritical case ($\rho >0$), when $\lambda <\rho$, 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.

arXiv comment31 pages, 2 figures