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.


60J45 Probabilistic potential theory
58J65 Diffusion processes and stochastic analysis on manifolds
Full Text: DOI arXiv Euclid EuDML


[1] Baldi, P., Casadio Tarabusi, E. and Figà-Talamanca, A.: Stable laws arising from hitting distributions of processes on homogeneous trees and the hyperbolic half-plane. Pacific J. Math. 197 (2001), 257-273. · Zbl 1049.60015 · doi:10.2140/pjm.2001.197.257
[2] Baldi, P., Casadio Tarabusi, E., Figà-Talamanca, A. and Yor, M.: Non-symmetric hitting distributions on the hyperbolic half-plane and subordinated perpetuities. Rev. Mat. Iberoamericana 17 (2001), 587-605. · Zbl 1001.60018 · doi:10.4171/RMI/305
[3] Bougerol, P. and Jeulin, T.: Brownian bridge on hyperbolic spaces and on homogeneous trees. Probab. Theory Related Fields 115 (1999), 95-120. · Zbl 0947.58032 · doi:10.1007/s004400050237
[4] Byczkowski, T. and Ryznar, M.: Hitting distributions of geometric Brownian motion. Studia Math. 173 (2006), 19-38. · Zbl 1088.60085 · doi:10.4064/sm173-1-2
[5] Chung, K. L. and Zhao, Z.: From Brownian motion to Schrödinger’s equation . Fundamental Principles of Mathematical Sciences 312 . Springer-Verlag, Berlin, 1995. · Zbl 0819.60068
[6] Dufresne, D.: The distribution of a perpetuity, with application to risk theory and pension funding. Scand. Actuar. J. (1990), 39-79. · Zbl 0743.62101 · doi:10.1080/03461238.1990.10413872
[7] Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.: Tables of integral transforms. Vol. I and II . McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954. · Zbl 0055.36401
[8] Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.: Higher Transcendental Functions. Vol. II . McGraw-Hill Book Company, New York-Toronto-London, 1953. · Zbl 0052.29502
[9] Folland, G. B.: Fourier analysis and its applications . The Wadsworth and Brooks/Cole Mathematics Series. Wadsworth and Brooks/Cole, Pacific Grove, CA, 1992. · Zbl 0786.42001
[10] Gradstein, I. S. and Ryzhik, I. M.: Table of integrals, series and products . Sixth edition, Academic Press, London 2000.
[11] Guivarc’h, Y., Ji, L. and Taylor, J. C.: Compactifications of symmetric spaces . Progress in Mathematics 156 . Birkhäuser, Boston, 1998.
[12] Helgason, S.: Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions . Mathematical Surveys and Monographs 83 . American Mathematical Society, Providence, RI, 2000. · Zbl 0965.43007
[13] Yor, M.: Sur certaines fonctionnelles du mouvement Brownien réel. J. Appl. Probab. 29 (1992), 202-208. JSTOR: · Zbl 0758.60085 · doi:10.2307/3214805
[14] Yor, M.: On some exponential functionals of Brownian Motion. Adv. in Appl. Probab. 24 (1992), 509-531. JSTOR: · Zbl 0765.60084 · doi:10.2307/1427477
[15] Yor, M. (ed.): Exponential functionals and principal values related to Brownian motion . Biblioteca de la Revista Matemática Iberoamericana, Madrid, 1997. · Zbl 0889.00015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.