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It is well understood that a supercritical superprocess is equal in law to adiscrete Markov branching process whose genealogy is dressed in a Poissonianway with immigration which initiates subcritial superprocesses. The Markovbranching process corresponds to the genealogical description of prolificindividuals, that is individuals who produce eternal genealogical lines ofdecent, and is often referred to as the skeleton or backbone of the originalsuperprocess. The Poissonian dressing along the skeleton may be considered tobe the remaining non-prolific genealogical mass in the superprocess. Suchskeletal decompositions are equally well understood for continuous-statebranching processes (CSBP). In a previous article, [16], we developed an SDEapproach to study the skeletal representation of CSBPs, which provided a commonframework for the skeletal decompositions of supercritical and (sub)criticalCSBPs. It also helped us to understand how the skeleton thins down onto oneinfinite line of descent when conditioning on survival until larger and largertimes, and eventually forever. Here our main motivation is to show therobustness of the SDE approach by expanding it to the spatial setting ofsuperprocesses. The current article only considers supercriticalsuperprocesses, leaving the subcritical case open.

Skeletal stochastic differential equations for superprocesses