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Advances in Applied Clifford Algebras

Hyperbolic Harmonic Functions and Hyperbolic Brownian Motion Sirkka-Liisa Eriksson∗

and Terhi Kaarakka

Abstract. We study harmonic functions with respect to the Riemannian metric dx21 + · · · + dx2n ds2 = 2α xnn−2 n in the upper half space Rn : xn > 0}. + = {(x1 , . . . , xn ) ∈ R They are called α-hyperbolic harmonic. An important result is that a function f is α-hyperbolic harmonic ´ıf and only if the function − 2−n+α

g (x) = xn 2 f (x) is the eigenfunction of the hyperbolic Laplace operator h = x2n  − (n − 2) xn ∂x∂n corresponding to the eigenvalue   1 (α + 1)2 − (n − 1)2 = 0. This means that in case α = n − 2, the 4 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 α-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. Mathematics Subject Classification. Primary 31EE, 60J45, Secondary 30G35. Keywords. Hyperbolic harmonic, Hyperbolic metric, Hyperbolic function theory, Brwonian motion, Hyperbolic Brownian motion.

1. Introduction One of the major results in stochastics is that the classical potential theory, connected to Laplace equation, and theory of Brownian motion has strong relations found first by Kakutani [18] (see also [19]). We are pointing out similar results also between generalized Brownian motion and harmonic functions in This article is part of the Topical Collection on ISAAC 12 at Aveiro, July 29-August 2, 2019, edited by Swanhild Bernstein, Uwe Kaehler, Irene Sabadini, and Franciscus Sommen. ∗ Corresponding

author. 0123456789().: V,-vol

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S.-L. Eriksson, T. Kaarakka

Adv. Appl. Clifford Algebras

Rn+ = {(x0 , . . . , x) | x1 , . . . , xn ∈ R, xn > 0} with respect to the Riemannian metric ds2α =

dx21 + · · · + dx2n 2α

,

xnn−2

where α ∈ R. The Riemannian metric dsn−2 is the hyperbolic distance of the Poincar´e upper half space. The Laplace–Beltrami operator connected to ds2α is   2α α ∂f α f = xnn−2 f − . xn ∂xn When α = n − 2 it is called the hyperbolic Laplace operator. If a twice continuously differentiable function f : Ω → R satisfies α f = 0, it is called α-hyperbolic harmonic. If α = n−2, then α-hyperbolic harmonic functions are called briefly hyperbolic harmonic. We recall the result. Theorem 1.1. [20, Lemma 2.1] Let Ω be an open set contained in Rn+1 + . A twice continuously differentiable function f : Ω → R is α-hyperbolic harmonic n−2−α

if and only if the function g (x) = xn 2 f (x) is a solution of the equation  1 2 2 (n − 1) − (α + 1) g = 0. n−2 g + 4 The hyperbolic distance dh (x, y) between the points x and y in Rn+1 + may be computed as follows  2 γ1 (t) + · · · + γn 2 (t) dh (x, y) = dt inf γn (t) γ(0)=x,γ(1)=y γ  

2 = ln λ (x, y) + λ (x, y) − 1 , 2

where λ (x, y) = 1 + |x−y| 2xn yn and cosh dh (x, y) = λ (x, y). The geodesics, representing the shortest distance between the po