On potential theory of hyperbolic Brownian motion with drift

Abstract

Consider the λ-Green function and the λ-Poisson kernel of a Lipschitz domain U ⊂ H n = {x ∈ R n : xn > 0} for hyperbolic Brownian motion with drift. We provide several relationships that facilitate studying those objects and explain somewhat their nature. As an application, we yield uniform estimates for sets of the form S a,b = {x ∈ H n : xn > a, x1 ∈ (0, b)}, a, b > 0, which covers and extends existing results of that kind.

Hyperbolic Green Function Estimates

“…Hence, the assertion is clear for z n ≤ x n M . For z ∈ ∂B R such that x n /M < z n < cosh R we have, by (22), z n > x n /M > Me −R and consequently…”