Electronic Journal of Probability

Limit theorems for conditioned multitype Dawson-Watanabe processes and Feller diffusions

Nicolas Champagnat and Sylvie Roelly

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A multitype Dawson-Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every finite time interval, its distribution is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. Several results on the long time behavior of the conditioned mass process-the conditioned multitype Feller branching diffusion-are then proved. The general case is first considered, where the mutation matrix which models the interaction between the types, is irreducible. Several two-type models with decomposable mutation matrices are analyzed too.

Article information

Electron. J. Probab., Volume 13 (2008), paper no. 25, 777-810.

Accepted: 6 May 2008
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G57: Random measures

multitype measure-valued branching processes conditionedDawson-Watanabe process critical and subcritical Dawson-Watanabeprocess conditioned Feller diffusion remote survival long time behavior

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Champagnat, Nicolas; Roelly, Sylvie. Limit theorems for conditioned multitype Dawson-Watanabe processes and Feller diffusions. Electron. J. Probab. 13 (2008), paper no. 25, 777--810. doi:10.1214/EJP.v13-504. https://projecteuclid.org/euclid.ejp/1464819098

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  • Athreya, Krishna B.; Ney, Peter E. Branching processes.Die Grundlehren der mathematischen Wissenschaften, Band 196.Springer-Verlag, New York-Heidelberg, 1972. xi+287 pp.
  • Dawson, D.A. Limit theorems for interaction free geostochastic systems. Col. Math. Soc. Bolyai, 22–47 (1978).
  • Dawson, Donald A. Measure-valued Markov processes. École d'Été de Probabilités de Saint-Flour XXI–-1991, 1–260, Lecture Notes in Math., 1541, Springer, Berlin, 1993.
  • Dawson, D. A.; Gorostiza, L. G.; Wakolbinger, A. Hierarchical equilibria of branching populations. Electron. J. Probab. 9 (2004), no. 12, 316–381 (electronic).
  • Dynkin, E.B. An introduction to Branching Measure-valued Processes, CRM Monographs 6, Amer. Math. Soc. Providence (1994).
  • El Karoui, Nicole; Roelly, Sylvie. Propriétés de martingales, explosion et représentation de Lévy-Khintchine d'une classe de processus de branchement à valeurs mesures.(French) [Martingale properties, explosion and Levy-Khinchin representation of a class of measure-valued branching processes] Stochastic Process. Appl. 38 (1991), no. 2, 239–266.
  • Etheridge, Alison M. An introduction to superprocesses.University Lecture Series, 20. American Mathematical Society, Providence, RI, 2000. xii+187 pp. ISBN: 0-8218-2706-5
  • Etheridge, A. and Williams, D.R.E. A decomposition of the (1+beta)-superprocess conditioned on survival. Proceed. of the Royal Soc. of Edinburgh (2004)
  • Evans, Steven N. Two representations of a conditioned superprocess. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), no. 5, 959–971.
  • Evans, Steven N.; Perkins, Edwin. Measure-valued Markov branching processes conditioned on nonextinction. Israel J. Math. 71 (1990), no. 3, 329–337.
  • Foster, J.; Ney, P. Decomposable critical multi-type branching processes. Sankhyā Ser. A 38 (1976), no. 1, 28–37.
  • Foster, J. and Ney, P. Limit laws for decomposable critical branching processes. Z. Wahrsch. Verw. Gebiete 46, 13–43 (1978/1979).
  • Frobenius, G. Uber Matrizen aus positiven Elementen Sitz. Ber. der Preussischen Akademie der Wissenschaft 456–477 (1912).
  • Gantmacher, Felix R. Matrizentheorie.(German) [Matrix theory] With an appendix by V. B. Lidskij.With a preface by D. P. Želobenko.Translated from the second Russian edition by Helmut Boseck, Dietmar Soyka and Klaus Stengert.Springer-Verlag, Berlin, 1986. 654 pp. ISBN: 3-540-16582-7
  • Gorostiza, Luis G.; López-Mimbela, Jose A. The multitype measure branching process. Adv. in Appl. Probab. 22 (1990), no. 1, 49–67.
  • Gorostiza, Luis G.; Roelly, Sylvie. Some properties of the multitype measure branching process. Stochastic Process. Appl. 37 (1991), no. 2, 259–274.
  • Jivrina, Miloslav. Branching processes with measure-valued states. 1964 Trans. Third Prague Conf. Information Theory, Statist. Decision Functions, Random Processes (Liblice, 1962) pp. 333–357 Publ. House Czech. Acad. Sci., Prague
  • Kawazu, Kiyoshi; Watanabe, Shinzo. Branching processes with immigration and related limit theorems. Teor. Verojatnost. i Primenen. 16 1971 34–51.
  • Lambert, Amaury. Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Probab. 12 (2007), no. 14, 420–446 (electronic).
  • Lamperti, John; Ney, Peter. Conditioned branching processes and their limiting diffusions. Teor. Verojatnost. i Primenen. 13 1968 126–137.
  • Li, Zeng-Hu. Asymptotic behaviour of continuous time and state branching processes. J. Austral. Math. Soc. Ser. A 68 (2000), no. 1, 68–84.
  • Meyer, Paul-André. Fonctionelles multiplicatives et additives de Markov.(French) Ann. Inst. Fourier (Grenoble) 12 1962 125–230.
  • Ogura, Y. Asymptotic behavior of multitype Galton-Watson processes. J. Math. Kyoto Univ. 15, 251–302 (1975).
  • Overbeck, L. Conditioned super-Brownian motion. Probab. Theory Related Fields 96 (1993), no. 4, 545–570.
  • Perkins, Edwin. Dawson-Watanabe superprocesses and measure-valued diffusions. Lectures on probability theory and statistics (Saint-Flour, 1999), 125–324, Lecture Notes in Math., 1781, Springer, Berlin, 2002.
  • Perko, Lawrence. Differential equations and dynamical systems.Texts in Applied Mathematics, 7. Springer-Verlag, New York, 1991. xii+403 pp. ISBN: 0-387-97443-1
  • Perron, O. Uber Matrizen. Math. Annalen 64, 248–263 (1907)
  • Roelly-Coppoletta, Sylvie; Rouault, Alain. Processus de Dawson-Watanabe conditionné par le futur lointain.(French) [A Dawson-Watanabe process conditioned by the remote future] C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 14, 867–872.
  • Seneta, E. Non-negative matrices and Markov chains.Revised reprint of the second (1981) edition [Springer-Verlag, New York; ].Springer Series in Statistics. Springer, New York, 2006. xvi+287 pp. ISBN: 978-0387-29765-1; 0-387-29765-0
  • Sugitani, Sadao. On the limit distributions of decomposable Galton-Watson processes with the Perron-Frobenius root $1$. Osaka J. Math. 18 (1981), no. 1, 175–224.
  • Vatutin, V.A. and Sagitov, S.M. A decomposable critical branching processes with two types of particles. Proc. Steklov Inst. of Math. 4, 1–19 (1988)
  • Watanabe, Shinzo. A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ. 8 1968 141–167.
  • Zubkov, A. M. The limit behavior of decomposable critical branching processes with two types of particles.(Russian) Teor. Veroyatnost. i Primenen. 27 (1982), no. 2, 228–238.