Probability Theory and Related Fields

, Volume 84, Issue 4, pp 505–520

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

  • T. -Y. Lee
Article

DOI: 10.1007/BF01198317

Cite this article as:
Lee, T.Y. Probab. Th. Rel. Fields (1990) 84: 505. doi:10.1007/BF01198317

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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Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • T. -Y. Lee
    • 1
  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA

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