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Hyperbolic Branching Brownian Motion

Abbasi, Mohammad Ali | 2014

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 45725 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Esfahani Zade, Mostafa
  7. Abstract:
  8. Hyperbolic branching Brownian motion is a branching diffusion process in which individual particles follow independent Brownian paths in the hyperbolic plane H2, 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 H2 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 ∂H2 (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 − 8λ)/2 .
  9. Keywords:
  10. Brownian motion ; Hyperbolic Space ; Branching Proccess

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