Hyperbolic Green Function Estimates

Abstract

For a hyperbolic Brownian motion in the hyperbolic space mathbb {H}^{n}, nge 3, we prove a representation of a Green function and a Poisson kernel for bounded and smooth sets in terms of the corresponding objects for an ordinary Euclidean Brownian motion and a conditional gauge functional. Using this representation we prove bounds for the Green functions and Poisson kernels for smooth sets. In particular, we provide a two sided sharp estimate of the Green function of a hyperbolic ball of any radius. By usual isomorphism argument the same estimate holds in any other model of a real hyperbolic space.