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Birth and death processes on certain random trees: classification and stationary laws
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  • Published: 26 November 2003

Birth and death processes on certain random trees: classification and stationary laws

  • Guy Fayolle1,
  • Maxim Krikun2 &
  • Jean-Marc Lasgouttes3 

Probability Theory and Related Fields volume 128, pages 386–418 (2004)Cite this article

Abstract.

The main substance of the paper concerns the growth rate and the classification (ergodicity, transience) of a family of random trees. In the basic model, new edges appear according to a Poisson process of parameter λ and leaves can be deleted at a rate μ. The main results lay the stress on the famous number e. A complete classification of the process is given in terms of the intensity factor ρ=λ/μ: it is ergodic if ρ≤e −1, and transient if ρ>e −1. There is a phase transition phenomenon: the usual region of null recurrence (in the parameter space) here does not exist. This fact is rare for countable Markov chains with exponentially distributed jumps. Some basic stationary laws are computed, e.g. the number of vertices and the height. Various bounds, limit laws and ergodic-like theorems are obtained, both for the transient and ergodic regimes. In particular, when the system is transient, the height of the tree grows linearly as the time t→∞, at a rate which is explicitly computed. Some of the results are extended to the so-called multiclass model.

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Author information

Authors and Affiliations

  1. INRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153, Le Chesnay Cedex, France

    Guy Fayolle

  2. LLRS, Faculty of Mathematics and Mechanics, Moscow State University, 119899, Moscow, Russia

    Maxim Krikun

  3. INRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153, Le Chesnay Cedex, France

    Jean-Marc Lasgouttes

Authors
  1. Guy Fayolle
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  2. Maxim Krikun
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  3. Jean-Marc Lasgouttes
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Corresponding author

Correspondence to Guy Fayolle.

Additional information

J.-M. Lasgouttes worked partly on the present study while spending a sabbatical at EURANDOM in Eindhoven.

Mathematics Subject Classification (2000): 60B05, 60J80, 34L30

Revised: 2003

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Fayolle, G., Krikun, M. & Lasgouttes, JM. Birth and death processes on certain random trees: classification and stationary laws. Probab. Theory Relat. Fields 128, 386–418 (2004). https://doi.org/10.1007/s00440-003-0311-1

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  • Received: 30 May 2002

  • Published: 26 November 2003

  • Issue Date: March 2004

  • DOI: https://doi.org/10.1007/s00440-003-0311-1

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Keywords

  • Random trees
  • Ergodicity
  • Transience
  • Nonlinear differential equations
  • Phase transition
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