Fractional telegraph-type equations and hyperbolic Brownian motion

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Vol 89 ◽  
pp. 131-137 ◽  
Author(s):  
Mirko D’Ovidio ◽  
Enzo Orsingher ◽  
Bruno Toaldo
Keyword(s):  
Brownian Motion ◽  
Hyperbolic Brownian Motion

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Vol 154 (6) ◽  
pp. 1550-1568 ◽  
Author(s):  
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Claudio Macci
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Hyperbolic Brownian Motion

Hyperbolic and fractional hyperbolic Brownian motion

Stochastics ◽  
2007 ◽  
Vol 79 (6) ◽  
pp. 505-522 ◽  
Author(s):  
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Enzo Orsingher
Keyword(s):  
Brownian Motion ◽  
Hyperbolic Brownian Motion

scholarly journals Sharp estimates of the Green function of hyperbolic Brownian motion

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Vol 228 (3) ◽  
pp. 197-221 ◽  
Author(s):  
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Jacek Małecki
Keyword(s):  
Brownian Motion ◽  
Green Function ◽  
Sharp Estimates ◽  
The Green Function ◽  
Hyperbolic Brownian Motion

scholarly journals Hyperbolic Harmonic Functions and Hyperbolic Brownian Motion

2020 ◽  
Vol 30 (5) ◽  
Author(s):  
Sirkka-Liisa Eriksson ◽  
Terhi Kaarakka
Keyword(s):  
Brownian Motion ◽  
Half Space ◽  
Laplace Operator ◽  
Harmonic Functions ◽  
Riemannian Metric ◽  
Hyperbolic Metric ◽  
Hyperbolic Functions ◽  
The Hyperbolic Metric ◽  
Hyperbolic Brownian Motion ◽  
Hyperbolic Harmonic Functions

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.

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