Abstract
For a hyperbolic Brownian motion in the hyperbolic space \(\mathbb {H}^{n}, n\ge 3\), we prove a representation of a Green function and a Poisson kernel for bounded and smooth sets in terms of the corresponding objects for an ordinary Euclidean Brownian motion and a conditional gauge functional. Using this representation we prove bounds for the Green functions and Poisson kernels for smooth sets. In particular, we provide a two sided sharp estimate of the Green function of a hyperbolic ball of any radius. By usual isomorphism argument the same estimate holds in any other model of a real hyperbolic space.
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Acknowledgments
G. Serafin was supported by the National Science Centre, Poland, grant no. 2015/18/E/ST1/00239. M. Ryznar and T. Żak were partially supported by the National Science Centre, Poland, grant no. 2015/17/B/ST1/01043. We thank the referee for valuable comments and remarks which improved the presentation of the paper.
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Appendix
Appendix
Proposition A.1 1
Let \(a>\frac {n-1}2\), n ≥ 2, and let B be any Euclidean ball in \(\mathbb {R}^{n}\). Then we have
where the comparability constant depends only on a and n, and ρB(x) stands for Euclidean distance between x and ∂B.
Proof
Without loss of generality we may assume B = B(0, 1) and x = (0,..., 0,xn), where xn > 0. It is clear that for \(x_{n}<\frac 12\) the integral is comparable with a constant. For \(x_{n}\geq \frac 12\) we rewrite it as follows
where Ωn− 2 is the surface area of the unit sphere in \(\mathbb {R}^{n-1}\). Next, we substitute \(u=\frac {(1-x_{n})^{2}}{2x_{n}}s\) and get
Since for \(x_{n}\geq \frac 12\) it holds xn ≈ 1 and 4xn/(1 − xn)2 > 8, we obtain the required estimate. □
Corollary A.1 1
Let γ > 0 and let BR, R > 1, be defined as in (18). If x ∈ BR and \(\delta _{B_{R}}(x)>1\), then
where the comparability constant depends only on γ and n.
Proof
We fix R > 1 and denote B = BR. In view of the previous proposition, it is enough to show that ρB(x) ≈ xn whenever δB(x) > 1. The inequality ρB(x) ≤ xn is clear. On the other hand, we may write
The assumption δB(x) > 1 implies x ∈ BR− 1, which gives us
Finally, we get
which ends the proof. □
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Ryznar, M., Serafin, G. & Żak, T. Hyperbolic Green Function Estimates. Potential Anal 54, 535–559 (2021). https://doi.org/10.1007/s11118-020-09837-5
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DOI: https://doi.org/10.1007/s11118-020-09837-5