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Poisson kernels of half-spaces in real hyperbolic spaces. (English) Zbl 1161.60024

Let \(\mathbb{H}^n : = \{(x_1,\dots,x_{n-1},x_n)\in\mathbb{R}^{n-1}\times\mathbb{R}^n: x_n>0\}\) be (the half-space model of) the \(n\)-dimensional real hyperbolic half-space, with the Riemannian metric, volume element and the Laplace-Beltrami operator. The authors establish two integral representation formulas for the Poisson kernel \(P_a(x,y)\) for Brownian motion of a half-space (of the form \(x_n > a\)) in \(\mathbb{H}^n\).
The first one is derived using the inverse Fourier transform. The second one is suitable for study the asymptotic properties of the Poisson kernel. The asymptotics for \(P_a(x,y)\) are found for the situations: \(a\rightarrow 0^ {+}\); \(x\rightarrow +\infty\); \(|y|\rightarrow +\infty\); and \(x\rightarrow a^ {+}\). For the cases \(n = 3\), \(4\) or \(6\), explicit formulas for the Poisson kernel itself are computed.

MSC:

60J45 Probabilistic potential theory
58J65 Diffusion processes and stochastic analysis on manifolds
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References:

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