Hyperbolic Branching Brownian Motion

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Abbasi, Mohammad Ali | 2014

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 45725 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Esfahani Zade, Mostafa
Abstract:
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 .
Keywords:
Brownian motion ; Hyperbolic Space ; Branching Proccess