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The Backbone Decomposition for Spatially Dependent Supercritical Superprocesses

Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 2123)

Abstract

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 ownn.

Starting with Evans and O’Connell (Can Math Bull 37:187–196, 1994), there exists a cluster of literature, (Engländer and Pinsky, Ann Probab 27:684–730, 1999; Salisbury and Verzani, Probab Theory Relat Fields 115:237–285, 1999; Duquesne and Winkel, Probab Theory Relat Fields 139:313–371, 2007; Berestycki, Kyprianou and Murillo-Salas, Stoch Proc Appl 121:1315–1331, 2011; Kyprianou and Ren, Stat Probab Lett 82:139–144, 2012),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.

Keywords

  • Backbone Decomposition
  • Superprocesses
  • Galton-Watson Process
  • Branching Mechanism
  • Continuous State Branching Process

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Notes

  1. 1.

    The assumptions on \(\mathcal{P}\) may in principle be relaxed. The main reason for this imposition here comes in the proof of Lemma 5 where a comparison principle is used for diffusions.

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Acknowledgements

We would like to thank Maren Eckhoff for a number of helpful comments on earlier versions of this paper. Part of this research was carried out whilst AEK was on sabbatical at ETH Zürich, hosted by the Forschungsinstitute für Mathematik, for whose hospitality he is grateful. The research of YXR is supported in part by the NNSF of China (Grant Nos. 11271030 and 11128101).

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Kyprianou, A.E., Pérez, JL., Ren, YX. (2014). The Backbone Decomposition for Spatially Dependent Supercritical Superprocesses. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLVI. Lecture Notes in Mathematics(), vol 2123. Springer, Cham. https://doi.org/10.1007/978-3-319-11970-0_2

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