Potential theory of hyperbolic Brownian motion in tube domains

2014 ◽  
Vol 135 (1) ◽  
pp. 27-52
Author(s):  
Grzegorz Serafin
Keyword(s):  
Brownian Motion ◽  
Potential Theory ◽  
Tube Domains ◽  
Hyperbolic Brownian Motion

scholarly journals On the Asymptotic Behavior of the Hyperbolic Brownian Motion

2014 ◽  
Vol 154 (6) ◽  
pp. 1550-1568 ◽  
Author(s):  
Valentina Cammarota ◽  
Alessandro De Gregorio ◽  
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):  
Lanjun Lao ◽  
Enzo Orsingher
Keyword(s):  
Brownian Motion ◽  
Hyperbolic Brownian Motion

scholarly journals Potential theory of subordinate killed Brownian motion

2018 ◽  
Vol 371 (6) ◽  
pp. 3917-3969 ◽  
Author(s):  
Panki Kim ◽  
Renming Song ◽  
Zoran Vondraček
Keyword(s):  
Brownian Motion ◽  
Potential Theory

The measure theory of random fractals

1986 ◽  
Vol 100 (3) ◽  
pp. 383-406 ◽  
Author(s):  
S. James Taylor
Keyword(s):  
Brownian Motion ◽  
Potential Theory ◽  
Measure Theory ◽  
Motion Path ◽  
Natural Question ◽  
Random Fractals ◽  
Limited Space ◽  
Definition Of ◽  
Good Problem

In 1951 A. S. Besicovitch, who was my research supervisor, suggested that I look at the problem of determining the dimension of the range of a Brownian motion path. This problem had been communicated to him by C. Loewner, but it was a natural question which had already attracted the attention of Paul Lévy. It was a good problem to give to an ignorant Ph.D. student because it forced him to learn the potential theory of Frostman [33] and Riesz[75] as well as the Wiener [98] definition of mathematical Brownian motion. In fact the solution of that first problem in [81] used only ideas which were already twenty-five years old, though at the time they seemed both new and original to me. My purpose in this paper is to try to trace the development of these techniques as they have been exploited by many authors and used in diverse situations since 1953. As we do this in the limited space available it will be impossible to even outline all aspects of the development, so I make no apology for giving a biased account concentrating on those areas of most interest to me. At the same time I will make conjectures and suggest some problems which are natural and accessible in the hope of stimulating further research.

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

2015 ◽  
Vol 228 (3) ◽  
pp. 197-221 ◽  
Author(s):  
Kamil Bogus ◽  
Tomasz Byczkowski ◽  
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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