Hitting hyperbolic half-space

Jacek Malecki; Grzegorz Serafin

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arXiv · 1111.0621

Hitting hyperbolic half-space

Jacek Malecki·Grzegorz Serafin

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Abstract

Let X^\mu={X_t^\mu;t>=0}, \mu>0, be the n-dimensional hyperbolic Brownian motion with drift, that is a diffusion on the real hyperbolic space H^n having the Laplace-Beltrami operator with drift as its generator. We prove the reflection principle for X^\mu, which enables us to study the process X^\mu killed when exiting the hyperbolic half-space, that is the set D={x\in H^n: x_1>0}. We provide formulae, uniform estimates and describe asymptotic behavior of the Green function and the Poisson kernel of D for the process X^\mu. Finally, we derive formula for the lambda-Poisson kernel of the set D.

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Jacek Malecki, Grzegorz Serafin. 2011-11-02. Hitting hyperbolic half-space. https://arxiv.org/abs/1111.0621

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