Potential theory of hyperbolic Brownian motion in tube domains

Grzegorz Serafin¹

¹ Institute of Mathematics and Computer Science Wrocław University of Technology Wybrzeże Wyspiańskiego 27 50-370 Wrocław, Poland

Colloquium Mathematicum, Tome 135 (2014) no. 1, pp. 27-52.

Résumé

Let $X=\{X(t);\,t\geq 0\}$ be the hyperbolic Brownian motion on the real hyperbolic space $\mathbb H^n=\{x\in \mathbb R^n:x_n>0\}$. We study the Green function and the Poisson kernel of tube domains of the form $D\times (0,\infty )\subset \mathbb H^n$, where $D$ is any Lipschitz domain in $\mathbb R^{n-1}$. We show how to obtain formulas for these functions using analogous objects for the standard Brownian motion in $\mathbb R^{2n}$. We give formulas and uniform estimates for the set $D_a=\{x\in \mathbb H^n:x_1\in (0,a)\}$. The constants in the estimates depend only on the dimension of the space.

Détail

DOI : 10.4064/cm135-1-3

Keywords: geq hyperbolic brownian motion real hyperbolic space mathbb mathbb n study green function poisson kernel tube domains form times infty subset mathbb where lipschitz domain mathbb n obtain formulas these functions using analogous objects standard brownian motion mathbb formulas uniform estimates set mathbb constants estimates depend only dimension space