Hyperbolic Branching Brownian Motion, M.Sc. Thesis Sharif University of Technology ; Esfahani Zade, Mostafa (Supervisor)
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... Cataloging briefHyperbolic Branching Brownian Motion, M.Sc. Thesis Sharif University of Technology ; Esfahani Zade, Mostafa (Supervisor)
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... Find in contentBookmark |
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