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Potential theory of hyperbolic Brownian motion in tube domains

Grzegorz Serafin — 2014

Colloquium Mathematicae

Let X = X(t); t ≥ 0 be the hyperbolic Brownian motion on the real hyperbolic space ℍⁿ = x ∈ ℝⁿ:xₙ > 0. We study the Green function and the Poisson kernel of tube domains of the form D × (0,∞)⊂ ℍⁿ, where D is any Lipschitz domain in $ℝ^{n - 1}$ . We show how to obtain formulas for these functions using analogous objects for the standard Brownian motion in $ℝ^{2 n}$ . We give formulas and uniform estimates for the set $D_{a} = x \in ℍ ⁿ : x ₁ \in (0, a)$ . The constants in the estimates depend only on the dimension of the space.

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