The backbone decomposition for spatially dependent supercritical superprocesses

Andreas Kyprianou, J.-L. Pérez, Y X Ren

Research output: Chapter in Book/Report/Conference proceedingChapter

3 Citations (Scopus)
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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.
Original languageEnglish
Title of host publicationSeminaire de Probabilites XLVI
EditorsCatherine Donati-Martin, Antoine LeJay, Alain Rouault
Place of PublicationSwitzerland
PublisherSpringer
Pages33 - 59
Number of pages27
ISBN (Electronic)978-3-319-11970-0
ISBN (Print)978-3-319-11969-4
DOIs
Publication statusPublished - 30 Oct 2014

Publication series

NameLecture Notes in Mathematics
PublisherSpringer
Volume2123
ISSN (Print)0075-8434

Fingerprint

Superprocess
Backbone
Galton-Watson Process
Decompose
Dependent
Field Theory
Branching
Continuous-state Branching Process
Descent
Stochastic Processes
Robustness
Analogue
Line
Class

Cite this

Kyprianou, A., Pérez, J-L., & Ren, Y. X. (2014). The backbone decomposition for spatially dependent supercritical superprocesses. In C. Donati-Martin, A. LeJay, & A. Rouault (Eds.), Seminaire de Probabilites XLVI (pp. 33 - 59). (Lecture Notes in Mathematics; Vol. 2123). Switzerland: Springer. https://doi.org/10.1007/978-3-319-11970-0_2

The backbone decomposition for spatially dependent supercritical superprocesses. / Kyprianou, Andreas; Pérez, J.-L.; Ren, Y X.

Seminaire de Probabilites XLVI. ed. / Catherine Donati-Martin; Antoine LeJay; Alain Rouault. Switzerland : Springer, 2014. p. 33 - 59 (Lecture Notes in Mathematics; Vol. 2123).

Research output: Chapter in Book/Report/Conference proceedingChapter

Kyprianou, A, Pérez, J-L & Ren, YX 2014, The backbone decomposition for spatially dependent supercritical superprocesses. in C Donati-Martin, A LeJay & A Rouault (eds), Seminaire de Probabilites XLVI. Lecture Notes in Mathematics, vol. 2123, Springer, Switzerland, pp. 33 - 59. https://doi.org/10.1007/978-3-319-11970-0_2
Kyprianou A, Pérez J-L, Ren YX. The backbone decomposition for spatially dependent supercritical superprocesses. In Donati-Martin C, LeJay A, Rouault A, editors, Seminaire de Probabilites XLVI. Switzerland: Springer. 2014. p. 33 - 59. (Lecture Notes in Mathematics). https://doi.org/10.1007/978-3-319-11970-0_2
Kyprianou, Andreas ; Pérez, J.-L. ; Ren, Y X. / The backbone decomposition for spatially dependent supercritical superprocesses. Seminaire de Probabilites XLVI. editor / Catherine Donati-Martin ; Antoine LeJay ; Alain Rouault. Switzerland : Springer, 2014. pp. 33 - 59 (Lecture Notes in Mathematics).
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AB - 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.

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