Poisson kernels of half–spaces in real hyperbolic spaces

We provide an integral formula for the Poisson kernel of half-spaces for Brownian motion in real hyperbolic space \mathbb{H}^n . This enables us to find asymptotic properties of the kernel. We also show convergence to the Poisson kernel of the whole space \mathbb{H}^n . For n=3 , 4 or 6 we compute explicit formulas for the Poisson kernel itself.