Potential theory of hyperbolic Brownian motion in tube domains

Volume 135 / 2014

Grzegorz Serafin Colloquium Mathematicum 135 (2014), 27-52 MSC: Primary 60J65; Secondary 60J60. DOI: 10.4064/cm135-1-3


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.


  • Grzegorz SerafinInstitute of Mathematics and Computer Science
    Wrocław University of Technology
    Wybrzeże Wyspiańskiego 27
    50-370 Wrocław, Poland

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