Branching diffusions on $H^d$ with variable fission: The Hausdorff dimension of the limiting set

For a hyperbolic Brownian motion on the Poincaré half-plane H 2 , starting from a point of hyperbolic coordinates z = (η, α) inside a hyperbolic disc U of radiusη, we obtain the probability of hitting the boundary ∂U at the point (η,ᾱ). Forη → ∞ we derive the asymptotic Cauchy hitting distribution on ∂H 2 and for small values of η andη we obtain the classical Euclidean Poisson kernel. The exit probabilities Pz{Tη 1 < Tη 2 } from a hyperbolic annulus in H 2 of radii η1 and η2 are derived and the transient behaviour of hyperbolic Brownian motion is considered. Similar probabilities are calculated also for a Brownian motion on the surface of the three dimensional sphere.For the hyperbolic half-space H n we obtain the Poisson kernel of a ball in terms of a series involving Gegenbauer polynomials and hypergeometric functions. For small domains in H n we obtain the ndimensional Euclidean Poisson kernel. The exit probabilities from an annulus are derived also in the n-dimensional case.

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Branching diffusions on $H^d$ with variable fission: The Hausdorff dimension of the limiting set

Cited by 2 publications

References 8 publications

Асимптотическое Поведение Ветвящейся Диффузии На Гиперболическом Пространстве

Асимптотическое Поведение Ветвящейся Диффузии На Гиперболическом Пространстве

Hitting spheres on hyperbolic spaces

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