Hyperbolic Harmonic Functions and Hyperbolic Brownian Motion

Abstract

We study harmonic functions with respect to the Riemannian metric ds2=dx12+⋯+dxn2xn2αn-2in the upper half space R+n={(x1,…,xn)∈Rn:xn>0}. They are called α-hyperbolic harmonic. An important result is that a function f is α-hyperbolic harmonic íf and only if the function g(x)=xn-2-n+α2f(x) is the eigenfunction of the hyperbolic Laplace operator △h=xn2▵-(n-2)xn∂∂xn corresponding to the eigenvalue 14((α+1)2-(n-1)2)=0. This means that in case α= n- 2 , the n- 2 -hyperbolic harmonic functions are harmonic with respect to the hyperbolic metric of the Poincaré upper half-space. We are presenting some connections of α-hyperbolic functions to the generalized hyperbolic Brownian motion. These results are similar as in case of harmonic functions with respect to usual Laplace and Brownian motion.