Skeletal stochastic differential equations for continuous-state branching process

It is well understood that a supercritical continuous-state branching process (CSBP) is equal in law to a discrete continuous-time Galton Watson process (the skeleton of prolific individuals) whose edges are dressed in a Poissonian way with immigration which initiates subcritial CSBPs (non-prolific mass). Equally well understood in the setting of CSBPs and super-processes is the notion of a spine or immortal particle dressed in a Poissonian way with immigration which initiates copies of the original CSBP, which emerges when conditioning the process to survive eternally.
In this article, we revisit these notions for CSBPs and put them in a common framework using the language of (coupled) SDEs. In this way, we are able to deal simultaneously with all types of CSBPs (supercritical, critical and subcritical) as well as understanding how the backbone representation becomes, in the sense of weak convergence, a spinal decomposition when condi- tioning on survival. Our principal motivation is to prepare the way to expand the SDE approach to the spatial setting of super-processes, where recent results have increasingly sought the use of skeletal decompositions to transfer results from the branching particle setting to the setting of measure valued processes.