Year: 2013
Type: article
Abstract: For a hyperbolic Brownian motion on the Poincaré half-plane ${\mathbf H}^2$, starting from a point $z=(\eta, \alpha)$ inside a hyperbolic disc $U$ of radius $\bar{\eta}$, we obtain the probability of hitting the boundary $\partial U$ at the point $(\bar \eta,\bar \alpha)$. For $\bar{\eta} \to \infty$ we derive the asymptotic Cauchy hitting distribution on $\partial{\mathbf H}^2$. In particular, it follows that the hyperbolic Brownian motion starting at $(x,y) \in {\mathbf H}^2$ “hits” the boundary of ${\mathbf H}^2$ at a point which is Cauchy distributed with scale parameter $y^{\prime}=y/(x^2+y^2)$ and position parameter $x^{\prime}=x/(x^2+y^2)$. For small values of $\eta$ and $\bar \eta$ we obtain the classical Euclidean Poisson kernel. The exit probabilities from a hyperbolic annulus in ${\mathbf H}^2$ of radii $\eta_1$ and $\eta_2$ are derived and the transient behavior 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 ${\mathbf H}^n$ we obtain, with a proof based on the method of separation of variables, the Poisson kernel of a ball. For small domains in ${\mathbf H}^n$ we obtain the $n$-dimensional Euclidean Poisson kernel. The exit probabilities from an annulus are derived also in the $n$-dimensional case.
Authors: V. Cammarota, E. Orsingher
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2951816446