Abstract
For a hyperbolic α-stable process in the hyperbolic space \(\mathbb {H}^{d}, d\ge 2\), we prove that the mean exit time from a halfspace \(H(a)=\{x_{d}>a\}\subset \mathbb {H}^{d} \) is equal to \(\mathbb {E}^{x} \tau _{H(a)} = c(\alpha , d) \delta ^{\alpha /2}_{H(a)} (x),\) where δ D (x) is the (hyperbolic) distance of x to D c. Based on this exact result we provide a sharp estimate of the mean exit time from a hyperbolic ball B(x 0,R) of radius R and center x 0: \(\mathbb {E}^{x}\tau _{B(x_{0},R)}\approx (\delta _{B(x_{0},R)}(x) \tanh R)^{\alpha /2}, x\in \mathbb {H}^{d}\). By usual isomorphism argument the same estimate holds in any other model of real hyperbolic space.
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The research was supported by NCN grant 2011/03/B/ST1/00423.
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Ryznar, M., żak, T. Exit Time of a Hyperbolic α-Stable Process from a Halfspace or a Ball. Potential Anal 45, 83–107 (2016). https://doi.org/10.1007/s11118-016-9536-3
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DOI: https://doi.org/10.1007/s11118-016-9536-3