On Maximal Functions for Harmonic Functions with Respect to Brownian Motion

1994 ◽  
Vol 38 (1) ◽  
pp. 14-26
Author(s):  
I. A. Dzhwarsheishvili
Keyword(s):  
Brownian Motion ◽  
Harmonic Functions ◽  
Maximal Functions

Geometric Cardinal Invariants, Maximal Functions and a Measure Theoretic Pigeonhole Principle

2005 ◽  
Vol 11 (4) ◽  
pp. 517-525
Author(s):  
Juris Steprāns
Keyword(s):  
Harmonic Analysis ◽  
Set Theory ◽  
Harmonic Functions ◽  
Maximal Functions ◽  
Pigeonhole Principle ◽  
Cardinal Invariants ◽  
Geometric Objects

AbstractIt is shown to be consistent with set theory that every set of reals of size ℵ1 is null yet there are ℵ1 planes in Euclidean 3-space whose union is not null. Similar results will be obtained for other geometric objects. The proof relies on results from harmonic analysis about the boundedness of certain harmonic functions and a measure theoretic pigeonhole principle.

scholarly journals Brownian Motion and Harmonic Analysis on Sierpinski Carpets

1999 ◽  
Vol 51 (4) ◽  
pp. 673-744 ◽  
Author(s):  
Martin T. Barlow ◽  
Richard F. Bass
Keyword(s):  
Brownian Motion ◽  
Heat Equation ◽  
Harmonic Analysis ◽  
Heat Kernel ◽  
Lower Bounds ◽  
Harnack Inequality ◽  
Harmonic Functions ◽  
Upper And Lower Bounds ◽  
Isotropic Diffusion ◽  
Sierpinski Carpets

AbstractWe consider a class of fractal subsets of d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion X and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting.

scholarly journals Harmonic functions on Walsh’s Brownian motion

2014 ◽  
Vol 124 (6) ◽  
pp. 2228-2248 ◽  
Author(s):  
Patrick J. Fitzsimmons ◽  
Kristin E. Kuter
Keyword(s):  
Brownian Motion ◽  
Harmonic Functions

scholarly journals Brownian Motion and Harmonic Functions on Sol(p,q)†

2011 ◽  
Vol 2012 (22) ◽  
pp. 5182-5218 ◽  
Author(s):  
Sara Brofferio ◽  
Maura Salvatori ◽  
Wolfgang Woess
Keyword(s):  
Brownian Motion ◽  
Harmonic Functions

scholarly journals Brownian motion on stationary random manifolds

2016 ◽  
Vol 16 (02) ◽  
pp. 1660001
Author(s):  
Pablo Lessa
Keyword(s):  
Brownian Motion ◽  
Harmonic Functions ◽  
Dimensional Space ◽  
Volume Growth ◽  
Upper And Lower Bounds ◽  
Entropy Theory ◽  
Liouville Property ◽  
Average Volume ◽  
Infinite Dimensional ◽  
Bounded Harmonic Functions

We introduce the notion of a stationary random manifold and develop the basic entropy theory for it. Examples include manifolds admitting a compact quotient under isometries and generic leaves of a compact foliation. We prove that the entropy of an ergodic stationary random manifold is zero if and only if the manifold satisfies the Liouville property almost surely, and is positive if and only if it admits an infinite dimensional space of bounded harmonic functions almost surely. Upper and lower bounds for the entropy are provided in terms of the linear drift of Brownian motion and average volume growth of the manifold. Other almost sure properties of these random manifolds are also studied.

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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