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.
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Notes
- 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.
References
J. Berestycki, A.E. Kyprianou, A. Murillo-Salas, The prolific backbone decomposition for supercritical superdiffusions. Stoch. Proc. Appl. 121, 1315–1331 (2011)
J. Bertoin, J. Fontbona, S. Martínez, On prolific individuals in a continuous-state branching process. J. Appl. Probab. 45, 714–726 (2008)
T. Duquesne, M. Winkel, Growth of Lévy trees. Probab. Theory Relat. Fields 139, 313–371 (2007)
E.B. Dynkin, Markov Processes I (English translation) (Springer, Berlin, 1965)
E.B. Dynkin, A probabilistic approach to one class of non-linear differential equations. Probab. Theory Relat. Fields 89, 89–115 (1991)
E.B. Dynkin, Superprocesses and partial differential equations. Ann. Probab. 21, 1185–1262 (1993)
E.B. Dynkin, An Introduction to Branching Measure-Valued Processes. CRM Monograph Series, vol. 6 (American Mathematical Society, Providence, 1994), 134 pp.
E.B. Dynkin, Branching exit Markov systems and superprocesses. Ann. Probab. 29, 1833–1858 (2001)
E.B. Dynkin, Diffusions, Superprocesses and Partial Differential Equations. American Mathematical Society, Colloquium Publications, vol. 50 (Providence, Rhode Island, 2002)
E.B. Dynkin, S.E. Kuznetsov, \(\mathbb{N}\)-measures for branching exit Markov systems and their applications to differential equations. Probab. Theory Relat. Fields 130, 135–150 (2004)
J. Engländer, R.G. Pinsky, On the construction and support properties of measure-valued diffusions on \(D \subseteq R^{d}\) with spatially dependent branching. Ann. Probab. 27, 684–730 (1999)
A.M. Etheridge, An Introduction to Superprocesses. University Lecture Series, vol. 20 (American Mathematical Society, Providence, 2000), 187 pp.
A. Etheridge, D.R.E. Williams, A decomposition of the (1 +β)-superprocess conditioned on survival. Proc. Roy. Soc. Edin. 133A, 829–847 (2003)
S.N. Evans, Two representations of a superprocess. Proc. Roy. Soc. Edin. 123A, 959–971 (1993)
S.N. Evans, N. O’Connell, Weighted occupation time for branching particle systems and a representation for the supercritical superprocess. Can. Math. Bull. 37, 187–196 (1994)
P.J. Fitzsimmons, Construction and regularity of measure-valued Markov branching processes. Isr. J. Math. 63, 337–361 (1988)
R. Hardy, S.C. Harris, A spine approach to branching diffusions with applications to L p-convergence of martingales. Séminaire de Probab. XLII, 281–330 (2009)
T. Harris, The Theory of Branching Processes (Springer, Berlin; Prentice-Hall, Englewood Cliffs, 1964)
K. Ito, S. Watanabe, Transformation of Markov processes by multiplicative functionals. Ann. Inst. Fourier Grenobl 15, 13–30 (1965)
A.E. Kyprianou, Y-X. Ren, Backbone decomposition for continuous-state branching processes with immigration. Stat. Probab. Lett. 82, 139–144 (2012)
J-F. Le Gall, Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics (ETH Zürich, Birkhäuser, 1999)
Z. Li, Measure-Valued Branching Markov Processes. Probability and Its Applications (New York) (Springer, Heidelberg, 2011), 350 pp.
T. Salisbury, J. Verzani, On the conditioned exit measures of super Brownian motion. Probab. Theory Relat. Fields 115, 237–285 (1999)
S. Roelly-Coppoletta, A. Rouault, Processus de Dawson-Watanabe conditioné par le futur lointain. C.R. Acad. Sci. Paris Série I 309, 867–872 (1989)
M.J. Sharpe, General Theory of Markov Processes (Academic Press, San Diego, 1988)
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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