Abstract
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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