Hyperbolic Harmonic Functions and Hyperbolic Brownian Motion

Abstract

We study harmonic functions with respect to the Riemannian metric $$\begin{aligned} ds^{2}=\frac{dx_{1}^{2}+\cdots +dx_{n}^{2}}{x_{n}^{\frac{2\alpha }{n-2}}} \end{aligned}$$ d s 2 = d x 1 2 + ⋯ + d x n 2 x n 2 α n - 2 in the upper half space $$\mathbb {R}_{+}^{n}=\{\left( x_{1},\ldots ,x_{n}\right) \in \mathbb {R}^{n}:x_{n}>0\}$$ R + n = { x 1 , … , x n ∈ R n : x n > 0 } . They are called $$\alpha $$ α -hyperbolic harmonic. An important result is that a function f is $$\alpha $$ α -hyperbolic harmonic íf and only if the function $$g\left( x\right) =x_{n}^{-\frac{ 2-n+\alpha }{2}}f\left( x\right) $$ g x = x n - 2 - n + α 2 f x is the eigenfunction of the hyperbolic Laplace operator $$\bigtriangleup _{h}=x_{n}^{2}\triangle -\left( n-2\right) x_{n}\frac{\partial }{\partial x_{n}}$$ △ h = x n 2 ▵ - n - 2 x n ∂ ∂ x n corresponding to the eigenvalue $$\ \frac{1}{4}\left( \left( \alpha +1\right) ^{2}-\left( n-1\right) ^{2}\right) =0$$ 1 4 α + 1 2 - n - 1 2 = 0 . This means that in case $$\alpha =n-2$$ α = n - 2 , the $$n-2$$ n - 2 -hyperbolic harmonic functions are harmonic with respect to the hyperbolic metric of the Poincaré upper half-space. We are presenting some connections of $$\alpha $$ 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.

Brownian Motion, Martingales and Itô Formula in Clifford Analysis

“…Another group of works [3][4][5] has been devoted to studying stochastic calculus in general Grassmann algebras. Additionally, it is important to mention, that a connection between α-hyperbolic harmonics and hyperbolic Brownian motion has been presented in [12]. Nonetheless, a general Clifford setting has not been addressed so far.…”

Brownian motion, martingales and Itô formula in Clifford analysis

“…Another group of works [3,4,5] has been devoted to studying stochastic calculus in general Grassmann algebras. Additionally, it is important to mention, that a connection between α-hyperbolic harmonics and hyperbolic Brownian motion has been presented in [12]. Nonetheless, a general Clifford setting has not been addressed so far.…”

Toward infinite‐dimensional Clifford analysis

“…Hyperbolic harmonic functions and hyperbolic Brownian motion were studied by Erikson and Kaarakka. 6 Alpay and collaborators studied in previous works [7][8][9] different concepts of infinite-dimensional spaces, Grassmann algebras, and quaternionic functionals.…”

“…Recently, in Clifford analysis, different approaches to Brownian motion and stochastic processes have been made. Hyperbolic harmonic functions and hyperbolic Brownian motion were studied by Erikson and Kaarakka 6 . Alpay and collaborators studied in previous works 7–9 different concepts of infinite‐dimensional spaces, Grassmann algebras, and quaternionic functionals.…”