Poisson kernels of half-spaces in real hyperbolic spaces

Abstract

We provide an integral formula for the Poisson kernel of half-spaces for Brownian motion in real hyperbolic space $\H^n$. This enables us to find asymptotic properties of the kernel. Our starting point is the formula for its Fourier transform. When $n=3$, 4 or 6 we give an explicit formula for the Poisson kernel itself. In the general case we give various asymptotics and show convergence to the Poisson kernel of $\H^n$.