Displaying similar documents to “Limiting behaviors of the Brownian motions on hyperbolic spaces”

Sharp estimates of the Green function of hyperbolic Brownian motion

Kamil Bogus, Tomasz Byczkowski, Jacek Małecki (2015)

Studia Mathematica

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The main objective of the work is to provide sharp two-sided estimates of the λ-Green function, λ ≥ 0, of the hyperbolic Brownian motion of a half-space. We rely on the recent results obtained by K. Bogus and J. Małecki (2015), regarding precise estimates of the Bessel heat kernel for half-lines. We also substantially use the results of H. Matsumoto and M. Yor (2005) on distributions of exponential functionals of Brownian motion.

Hitting distributions of geometric Brownian motion

T. Byczkowski, M. Ryznar (2006)

Studia Mathematica

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Let τ be the first hitting time of the point 1 by the geometric Brownian motion X(t) = x exp(B(t) - 2μt) with drift μ ≥ 0 starting from x > 1. Here B(t) is the Brownian motion starting from 0 with EB²(t) = 2t. We provide an integral formula for the density function of the stopped exponential functional A ( τ ) = ∫ 0 τ X ² ( t ) d t and determine its asymptotic behaviour at infinity. Although we basically rely on methods developed in [BGS], the present paper covers the case of arbitrary drifts μ ≥ 0 and provides...

Hitting half-spaces or spheres by Ornstein-Uhlenbeck type diffusions

Tomasz Byczkowski, Jakub Chorowski, Piotr Graczyk, Jacek Małecki (2012)

Colloquium Mathematicae

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The purpose of the paper is to provide a general method for computing the hitting distributions of some regular subsets D for Ornstein-Uhlenbeck type operators of the form 1/2Δ + F·∇, with F bounded and orthogonal to the boundary of D. As an important application we obtain integral representations of the Poisson kernel for a half-space and balls for hyperbolic Brownian motion and for the classical Ornstein-Uhlenbeck process. The method developed in this paper is based on stochastic calculus...