Hyperbolic Harmonic Functions and Hyperbolic Brownian Motion

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http://hdl.handle.net/10138/321413

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Eriksson , S-L & Kaarakka , T 2020 , ' Hyperbolic Harmonic Functions and Hyperbolic Brownian Motion ' , Advances in Applied Clifford Algebras , vol. 30 , no. 5 , 72 . https://doi.org/10.1007/s00006-020-01099-z

Title: Hyperbolic Harmonic Functions and Hyperbolic Brownian Motion
Author: Eriksson, Sirkka-Liisa; Kaarakka, Terhi
Contributor: University of Helsinki, Department of Mathematics and Statistics
Date: 2020-10-31
Language: eng
Number of pages: 13
Belongs to series: Advances in Applied Clifford Algebras
ISSN: 0188-7009
URI: http://hdl.handle.net/10138/321413
Abstract: We study harmonic functions with respect to the Riemannian metric ds(2) = dx(1)(2) + ... + dx(n)(2)/x(n) (2 alpha/n-2) n in the upper half space R-+(n) = {(x(1), ... , x(n)) is an element of R-n : x(n) > 0}. They are called alpha-hyperbolic harmonic. An important result is that a function f is alpha-hyperbolic harmonic ' if and only if the function g (x) = x- 2-n+alpha/2 n f (x) is the eigenfunction of the hyperbolic Laplace operator Delta(h) = x(n)(2) Delta - (n - 2) x(n) partial derivative/partial derivative x(n) corresponding to the eigenvalue 1/4 (alpha + 1)(2) - (n - 1)(2) = 0. This means that in case a = n - 2, the n - 2-hyperbolic harmonic functions are harmonic with respect to the hyperbolic metric of the Poincar ' e upper half-space. We are presenting some connections of alpha-hyperbolic functions to the generalized hyperbolic Brownian motion. These results are similar as in case of harmonic functions with respect to usual Laplace and Brownian motion.
Subject: Hyperbolic harmonic
Hyperbolic metric
Hyperbolic function theory
Brwonian motion
Hyperbolic Brownian motion
HITTING DISTRIBUTIONS
GREEN-FUNCTION
KERNELS
111 Mathematics
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