Hitting hyperbolic half-space

2011/11/02 by Jacek Malecki, Malecki, Jacek, Grzegorz Serafin +1
Mathematics · #60J60 #60J65 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60J60 #msc:60J65

paper · pdf · doi:10.48550/arxiv.1111.0621

20 pages

arxiv created 2011/11/02 · arxiv updated 2011/11/03

Abstract

Let X^μ=X_t^μ;t>=0, μ>0, be the n-dimensional hyperbolic Brownian motion with drift, that is a diffusion on the real hyperbolic space Hⁿ having the Laplace-Beltrami operator with drift as its generator. We prove the reflection principle for X^μ, which enables us to study the process X^μkilled when exiting the hyperbolic half-space, that is the set D=x∈ Hⁿ: x₁>0. We provide formulae, uniform estimates and describe asymptotic behavior of the Green function and the Poisson kernel of D for the process X^μ. Finally, we derive formula for the lambda-Poisson kernel of the set D.

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