Skeletal stochastic differential equations for superprocesses

Abstract

It is well understood that a supercritical superprocess is equal in law to a discrete Markov branching process whose genealogy is dressed in a Poissonian way with immigration which initiates subcritical superprocesses. The Markov branching process corresponds to the genealogical description of prolific individuals, that is, individuals who produce eternal genealogical lines of descent, and is often referred to as the skeleton or backbone of the original superprocess. The Poissonian dressing along the skeleton may be considered to be the remaining non-prolific genealogical mass in the superprocess. Such skeletal decompositions are equally well understood for continuous-state branching processes (CSBP).

In a previous article [16] we developed an SDE approach to study the skeletal representation of CSBPs, which provided a common framework for the skeletal decompositions of supercritical and (sub)critical CSBPs. It also helped us to understand how the skeleton thins down onto one infinite line of descent when conditioning on survival until larger and larger times, and eventually forever.

Here our main motivation is to show the robustness of the SDE approach by expanding it to the spatial setting of superprocesses. The current article only considers supercritical superprocesses, leaving the subcritical case open.