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Hyperbolic branching Brownian motion
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  • Published: June 1997

Hyperbolic branching Brownian motion

  • Steven P. Lalley1 &
  • Tom Sellke1 

Probability Theory and Related Fields volume 108, pages 171–192 (1997)Cite this article

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  • 20 Citations

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Summary.

Hyperbolic branching Brownian motion is a branching diffusion process in which individual particles follow independent Brownian paths in the hyperbolic plane ? 2, and undergo binary fission(s) at rate λ > 0. It is shown that there is a phase transition in λ: For λ≦ 1/8 the number of particles in any compact region of ? 2 is eventually 0, w.p.1, but for λ > 1/8 the number of particles in any open set grows to ∞ w.p.1. In the subcritical case (λ≦ 1/8) the set Λ of all limit points in ∂? 2 (the boundary circle at ∞) of particle trails is a Cantor set, while in the supercritical case (λ > 1/8) the set Λ has full Lebesgue measure. For λ≦ 1/8 it is shown that w.p.1 the Hausdorff dimension of Λ is δ = (1−√1−8 λ)/2.

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Authors and Affiliations

  1. Department of Statistics, Mathematical Sciences Building, Purdue University, West Lafayette, IN 47907, USA email: lalley@stat.purdue.edu; tsellke@stat.purdue.edu, , , , , , US

    Steven P. Lalley & Tom Sellke

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  1. Steven P. Lalley
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  2. Tom Sellke
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Received: 2 November 1995 / In revised form: 22 October 1996

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Lalley, S., Sellke, T. Hyperbolic branching Brownian motion. Probab Theory Relat Fields 108, 171–192 (1997). https://doi.org/10.1007/s004400050106

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  • Issue Date: June 1997

  • DOI: https://doi.org/10.1007/s004400050106

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  • Mathematics Subject Classification (1991): 60K35 (primary)
  • 60J80 (secondary)
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