Sharp estimates of the Green function of hyperbolic Brownian motion

Bogus, Kamil; Byczkowski, T.; Małecki, Jacek

doi:10.4064/sm228-3-1

Cited by 3 publications

(9 citation statements)

References 20 publications

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“…This set may seem very special, but studying it is motivated by geometry: the hyperplanes x n = a are horocycles and the hyperplanes x 1 = b are geodesics in H n . Moreover, constants in the estimates depend only on the dimension and the parameter µ, and manipulation of parameters a and b lets us recover and improve existing results for sets like D a = {x ∈ H n : x n > a}, H = {x ∈ H n : [3], [8], [17], [22]).…”

Section: Introductionmentioning

confidence: 69%

“…The first assertion may be found in [17]; however, the proof is short so we repeat it for convenience of the reader. The other assertion is proved for the set D a = {x ∈ H n : x n > a} in [3], but the proof given below is much simpler and covers general sets. xn (τ µ U ) a.s. Thus…”

Section: General Resultsmentioning

confidence: 99%

“…It follows from Theorems 4.1 and 4.2 by replacing x 1 and y 1 by x 1 + b/2 and y 1 + b/2, respectively, and letting b → ∞. In fact, the Poisson kernel P Da (x, y) was estimated in [8], and estimates of the λ-Green function for the process without drift (which, by Theorem 3.1, are equivalent to estimates of the Green function for the process with a suitable drift) are the main results of [3].…”

Section: Estimatesmentioning

confidence: 99%

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On potential theory of hyperbolic Brownian motion with drift

Serafin¹

2020

PMS

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Consider the λ-Green function and the λ-Poisson kernel of a Lipschitz domain U ⊂ H n = {x ∈ R n : xn > 0} for hyperbolic Brownian motion with drift. We provide several relationships that facilitate studying those objects and explain somewhat their nature. As an application, we yield uniform estimates for sets of the form S a,b = {x ∈ H n : xn > a, x1 ∈ (0, b)}, a, b > 0, which covers and extends existing results of that kind.

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Section: Introductionmentioning

confidence: 69%

Section: General Resultsmentioning

confidence: 99%

Section: Estimatesmentioning

confidence: 99%

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On potential theory of hyperbolic Brownian motion with drift

Serafin¹

2020

PMS

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“…(3) where the constant C is dependent only on n, δ B (x) denotes the hyperbolic distance from x to ∂B and d H n is the hyperbolic metric and, as was shown in [4],…”

Section: Introductionmentioning

confidence: 99%

“…A comparison like (3) was obtained in [4] but only for half-spaces D a = {x ∈ H n : x n > a}, a > 0. The methods of [4] are completely different than ours, since they rely on the estimates of the Bessel heat kernel for a half-line. On the other hand their result follows from ours since we can approach any half-space D a by a sequence of balls.…”

Section: Introductionmentioning

confidence: 99%

Hyperbolic Green Function Estimates

2020

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For a hyperbolic Brownian motion in the hyperbolic space $\mathbb {H}^{n}, n\ge 3$ ℍ n , n ≥ 3 , we prove a representation of a Green function and a Poisson kernel for bounded and smooth sets in terms of the corresponding objects for an ordinary Euclidean Brownian motion and a conditional gauge functional. Using this representation we prove bounds for the Green functions and Poisson kernels for smooth sets. In particular, we provide a two sided sharp estimate of the Green function of a hyperbolic ball of any radius. By usual isomorphism argument the same estimate holds in any other model of a real hyperbolic space.

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Hyperbolic Harmonic Functions and Hyperbolic Brownian Motion

Eriksson

Kaarakka

2020

Adv. Appl. Clifford Algebras

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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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Sharp estimates of the Green function of hyperbolic Brownian motion

Cited by 3 publications

References 20 publications

On potential theory of hyperbolic Brownian motion with drift

On potential theory of hyperbolic Brownian motion with drift

Hyperbolic Green Function Estimates

Hyperbolic Harmonic Functions and Hyperbolic Brownian Motion

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