Hyperbolic Harmonic Functions and Hyperbolic Brownian Motion

We study harmonic functions with respect to the Riemannian metric ds2=dx12+⋯+dxn2xn2αn-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} ds^{2}=\frac{dx_{1}^{2}+\cdots +dx_{n}^{2}}{x_{n}^{\frac{2\alpha }{n-2}}} \end{aligned}$$\end{document}in the upper half space R+n={x1,…,xn∈Rn:xn>0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}_{+}^{n}=\{\left( x_{1},\ldots ,x_{n}\right) \in \mathbb {R}^{n}:x_{n}>0\}$$\end{document}. They are called α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-hyperbolic harmonic. An important result is that a function f is α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-hyperbolic harmonic íf and only if the function gx=xn-2-n+α2fx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\left( x\right) =x_{n}^{-\frac{ 2-n+\alpha }{2}}f\left( x\right) $$\end{document} is the eigenfunction of the hyperbolic Laplace operator △h=xn2▵-n-2xn∂∂xn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bigtriangleup _{h}=x_{n}^{2}\triangle -\left( n-2\right) x_{n}\frac{\partial }{\partial x_{n}}$$\end{document} corresponding to the eigenvalue 14α+12-n-12=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ \frac{1}{4}\left( \left( \alpha +1\right) ^{2}-\left( n-1\right) ^{2}\right) =0$$\end{document}. This means that in case α=n-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =n-2$$\end{document}, the n-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n-2$$\end{document}-hyperbolic harmonic functions are harmonic with respect to the hyperbolic metric of the Poincaré upper half-space. We are presenting some connections of α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-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.


Hyperbolic Harmonic Functions and Hyperbolic Brownian Motion
Sirkka-Liisa Eriksson * and Terhi Kaarakka Abstract. We study harmonic functions with respect to the Riemannian metric

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.
R n + = {(x 0 , . . . , x) | x 1 , . . . , x n ∈ R, x n > 0} with respect to the Riemannian metric where α ∈ R. The Riemannian metric ds n−2 is the hyperbolic distance of the Poincaré upper half space. The Laplace-Beltrami operator connected to ds 2 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.
The hyperbolic distance d h (x, y) between the points x and y in R n+1 + may be computed as follows where λ (x, y) = 1 + |x−y| 2 2xnyn and cosh d h (x, y) = λ (x, y). The geodesics, representing the shortest distance between the points, are circular arcs perpendicular to the hyperplane x n = 0 (that is, half-circles whose origin is on x n = 0 and straight vertical lines ending on the hyperplane x n = 0).
We recall important properties of hyperbolic distance (see for example [22]).
with the hyperbolic center a = (a 1 , .., a n ) and the radius r h is the same as the Euclidean ball D e (c a (r h ) , r e ) with the center c a (r h ) = (a 1 , . . . , a n cosh r h ) and the Euclidean radius r e = a n sinh r n .
The hyperbolic distance has the following invariance property.  We recall the definition of the n-dimensional Brownian motion. Definition 1.5. Let (Ω, F, P) be a probability space and {B t } t≥0 be an R nvalued stochastic process. Then {B t } t≥0 is called a standard n-dimensional Brownian motion, if the following properties hold: (i) B 0 = 0; (ii) The function t → B t is continuous almost surely; (iii) The process {B t } t≥0 has independent increments, that is B t+s − B s is independent of (B s1 , . . . , B s k ) for all 0 ≤ s 1 ≤ s 2 ≤ · · · ≤ s k ≤ s < s + t < ∞; (iv) The increments are stationary, that is, B t+s − B s has n-normal distribution with the mean 0 and the covariance tI, that is, dx.
An R n -valued continuous-time stochastic process {B u t } t≥0 is called an n-dimensional Brownian motion started at u if the process {B u t − u} t≥0 is a standard n-dimensional Brownian motion. We denote the transition probability P t u by From the definition we obtain directly that the Radon-Nikodym derivative is The pair (X, H) is called a Brelot space, if X is locally connected and does not contain any isolated points and the following properties hold: (a). For any point x ∈ X there exists a harmonic function h ∈ H (U ) for some open neighborhood U of x and h (x) = 0, (that is H non-generate at any point x ∈ X); (b). The harmonic sheaf H satisfies the Brelot convergence property. That is, the limit of any increasing sequence of harmonic functions on a connected open set is harmonic provided it is finite at a point; (c). Regular sets form a basis for the topology on X.
Brelot spaces, introduced by Brelot in 1957, are also harmonic spaces defined by Constantinescu and Cornea [10].
for any continuous f ∈ C (∂U ). An important tool for computing the harmonic measure of the Laplace operator is the fundamental solution (see [17]). Applying [12] or [13], we find the fundamental α-hyperbolic function from the function (λ) is the Legendre function of the second kind, defined in [16, 8.703] by converging for x satisfying |x| < 1. We recall the Euler relation Then s β,ν (0) = 1. The surface measure ω n of the Euclidean unit ball is The fundamental α-hyperbolic harmonic function was found in [12], but it included constants which were not computed. We verify that the function Adv. Appl. Clifford Algebras is the fundamental α-hyperbolic harmonic function. The function H α (x, y) has the different coefficient than the function computed in [13]. We first notice that if y ∈ R n+1 The proof is similar as in [12].
m e the outward-pointing normal of D h (x, R h ) and ω n is the surface measure of the unit ball.
Proof. Lemma 1.2 implies that the outward-pointing normal at y ∈ ∂D h (x, R h ) is given by Hence we deduce that

Using Lemma 1.2 we infer that
since sinh R h goes to 0 and the rest of the formula is just the average value of the continuous function over D h (x, R h ). Similarly, we compute that completing the proof.
where dy (α) = y Proof. Denote D h (a, ρ) = D and pick a hyperbolic ball such that Since H α is α-hyperbolic harmonic, applying the Green's identity − v ∂u ∂m (α) dσ ( Adv. Appl. Clifford Algebras with respect to the Riemannian metric ds 2 α (see [1]) we obtain and sinh R h goes to 0 we infer that the rest of the formula is just the average value of the continuous function over D h (x, R h ) and we conclude Applying the previous result we obtain the desired result.
and therefore − α H α (x, y) = δ x in the distribution sense. We conclude the following result.

Theorem 2.7. The function H α (x, y) is the fundamental α-hyperbolic harmonic function.
The fundamental solution may also be computed using hyperbolic Brownian motion explained in the next section.
The generator of this diffusion is given by We also immediately notice that the hyperbolic Brownian motion is n−2-hyperbolic Brownian motion. Using Itô calculus, we obtain the solution.
The solution X (μ) (t) of the previous system of stochastic differential equations is generated by the geometric Brownian motion X (μ) where B 1 , . . . , B n are standard Brownian motions.
α-hyperbolic Brownian motion has the following characterization (see [23]).  Using the previous result, the transition density function of the generalized hyperbolic Brownian motion was computed by Ma lecki and Serafin in [23].
Our main result is the following connection between the potential kernel and the fundamental solution.
where u = cosh r h and Using the Euler relation and (2.2) we obtain we conclude We recall the definition of the first exit time from a set D (the first hit time for the complement of D). Definition 3.6. If D is a nonempty, open, connected bounded subset of R n , then we denote We recall the Dynkin formula and its direct corollary.
Moreover, if h is α-hyperbolic harmonic in Ω andD ⊂ Ω, then In case of harmonic functions with respect to Laplace operator, Kakutani [18] (see also [19]) noticed that the harmonic measure has a probabilistic interpretation. If K is a nice compact subset of ∂U , then the harmonic measure μ U x (K) is also equal to the probability that a Brownian motion started at x reaches K before hitting ∂U \K. Future problem is to find out if the similar result holds for α-hyperbolic Brownian motion. The difficulty is the computation of the harmonic measure of α-hyperbolic harmonic functions.
Funding Open access funding provided by University of Helsinki including Helsinki University Central Hospital.
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