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Some limit theorems for critical branching Bessel processes, and related semilinear differential equations
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  • Published: December 1990

Some limit theorems for critical branching Bessel processes, and related semilinear differential equations

  • T. -Y. Lee1 nAff2 

Probability Theory and Related Fields volume 84, pages 505–520 (1990)Cite this article

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Summary

For a critical binary branching Bessel process starting fromx≫1 and stopped atx=1, we prove some conditional limit laws of the number of particles arriving atx=1 before a scaled large time. Five regions of the dimensional index of a Bessel process: −∞<d<2,d=2, 2<d<4,d=4 and 4<d<∞ are showed to have somewhat different behaviors. Our probabilistic results are proved by analyzing differential equations satisfied by generating functions. A salient theme is a comparison principle technique deliberately used to estimate solutions of\(u_t - \left( {D^2 + \frac{{d - 1}}{x}D} \right)u + u^p = 0\) inR +×R + wherep is greater than 1. The casep=2 corresponds to the process considered.

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Author notes
  1. T. -Y. Lee

    Present address: Department of Mathematics, University of Maryland, 20742, College Park, MD, USA

Authors and Affiliations

  1. Department of Mathematics, Princeton University, 08544, Princeton, NJ, USA

    T. -Y. Lee

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  1. T. -Y. Lee
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Lee, T.Y. Some limit theorems for critical branching Bessel processes, and related semilinear differential equations. Probab. Th. Rel. Fields 84, 505–520 (1990). https://doi.org/10.1007/BF01198317

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  • Received: 05 July 1988

  • Revised: 17 August 1989

  • Issue Date: December 1990

  • DOI: https://doi.org/10.1007/BF01198317

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Keywords

  • Differential Equation
  • Stochastic Process
  • Probability Theory
  • Limit Theorem
  • Large Time
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