Consider any supercritical Galton-Watson process which may become extinct with positive probability. It is a well-understood and intuitively obvious phenomenon that, on the survival set, the process may be pathwise decomposed into a stochastically `thinner' Galton-Watson process, which almost surely survives and which is decorated with immigrants, at every time step, initiating independent copies of the original Galton-Watson process conditioned to become extinct. The thinner process is known as the backbone and characterizes the genealogical lines of descent of prolific individuals in the original process. Here, prolific means individuals who have at least one descendant in every subsequent generation to their own.
Starting with Evans and O'Connell, there exists a cluster of literature describing the analogue of this decomposition (the so-called backbone decomposition) for a variety of different classes of superprocesses and continuous-state branching processes. Note that the latter family of stochastic processes may be seen as the total mass process of superprocesses with non-spatially dependent branching mechanism.
In this article we consolidate the aforementioned collection of results concerning backbone decompositions and describe a result for a general class of supercritical superprocesses with spatially dependent branching mechanisms. Our approach exposes the commonality and robustness of many of the existing arguments in the literature.
Backbone decomposition for continuous-state branching processes with immigration
在Duqesne and Winkel（2007）和Berestycki et al.（2011）的精神下，我们证明了具有一般分支机制和一般迁移机制的超临界连续态分支过程在定律上等价于具有Poissonian dressing迁移的连续时间Galton Watson过程。该结果还表征了极限骨架分解，这可以从Cao和Winkel（2010）关于移民的Galton-Watson树的一致增长的工作中预测。
