Boundary Behaviour of Harmonic Functions in a Half-Space and Brownian Motion

2011 ◽  
pp. 241-258
Author(s):  
Burgess Davis ◽  
Renming Song
Keyword(s):  
Brownian Motion ◽  
Half Space ◽  
Harmonic Functions ◽  
Boundary Behaviour ◽  
And Brownian Motion

scholarly journals Boundary behaviour of harmonic functions in a half-space and brownian motion

1973 ◽  
Vol 23 (4) ◽  
pp. 195-212 ◽  
Author(s):  
D. L. Burkholder ◽  
Richard F. Gundy
Keyword(s):  
Brownian Motion ◽  
Half Space ◽  
Harmonic Functions ◽  
Boundary Behaviour ◽  
And 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.

scholarly journals AREA INTEGRALS AND BOUNDARY BEHAVIOUR OF HARMONIC FUNCTIONS

2004 ◽  
Vol 47 (2) ◽  
pp. 365-373
Author(s):  
Mary Hanley
Keyword(s):  
Half Space ◽  
Harmonic Functions ◽  
Subject Classification ◽  
Boundary Behaviour ◽  
Primary 31B25

AbstractThis paper introduces a family of area-type integrals over cones. These are used to investigate non-tangential boundary behaviour of harmonic functions on a half-space, extending results of Stein and Brossard.AMS 2000 Mathematics subject classification: Primary 31B25

Thermal analysis of higher-order chemical reactive viscoelastic nanofluids flow in porous media via stretching surface

2021 ◽  
pp. 095440622110084
Author(s):  
Mohamed R Eid ◽  
F Mabood
Keyword(s):  
Brownian Motion ◽  
Transfer Rate ◽  
Fluid Velocity ◽  
Mass Transfer Rate ◽  
Higher Order ◽  
Flow In Porous Media ◽  
Viscoelastic Parameter ◽  
Suction Parameter ◽  
And Brownian Motion ◽  
Magnetic Strength

The essence of the present investigation is to reveal the hydrothermal variations of viscoelastic nanofluid flow in a porous medium over a stretchable surface. A higher-order chemical reaction is incorporated with thermophoresis and Brownian motion. Similarity conversions reduce the resulting equations into their dimensionless form and then solved using Runge-Kutta-Fehlberg (RKF) based shooting procedure. The effects of underlying factors on the flow are discussed through various graphs and tables. Computational results for noteworthy skin friction and heat and mass transport are presented and reviewed with sensible judgment. The study reveals that the fluid velocity reduces with incremental values of the viscoelastic parameter [Formula: see text] and magnetic strength. The temperature reduces for the suction parameter with the existence of stretchable but enhances with thermophoresis and Brownian motion effects. Heat transfer rate amplifies for [Formula: see text] but declines for [Formula: see text]. Mass transfer rate increases with the increase in Brownian parameter and Schmidt number. A comparative analysis shows a better agreement with previous results in limiting scenarios.

Magneto convective flow of casson nanofluid due to Stefan blowing in the presence of bio-active mixers

2021 ◽  
pp. 239779142110166
Author(s):  
Venkatesh Puneeth ◽  
Sarpabhushana Manjunatha ◽  
Bijjanal Jayanna Gireesha ◽  
Rama Subba Reddy Gorla
Keyword(s):  
Magnetic Field ◽  
Brownian Motion ◽  
Convective Flow ◽  
Three Dimensional ◽  
Induced Magnetic Field ◽  
Density Profiles ◽  
Casson Nanofluid ◽  
And Brownian Motion ◽  
Similarity Transformations ◽  
The Mathematical Model

The induced magnetic field for three-dimensional bio-convective flow of Casson nanofluid containing gyrotactic microorganisms along a vertical stretching sheet is investigated. The movement of these microorganisms cause bioconvection and they act as bio-active mixers that help in stabilising the nanoparticles in the suspension. The two forces, Thermophoresis and Brownian motion are incorporated in the Mathematical model along with Stefan blowing. The resulting model is transformed to ordinary differential equations using similarity transformations and are solved using [Formula: see text] method. The Velocity, Induced Magnetic field, Temperature, Concentration of Nanoparticles, and Motile density profiles are interpreted graphically. It is observed that the Casson parameter decreases the flow velocity and enhances the temperature, concentration, and motile density profiles and also it is noticed that the blowing enhances the nanofluid profiles whereas, suction diminishes the nanofluid profiles. On the other hand, it is perceived that the rate of heat conduction is enhanced with Thermophoresis and Brownian motion.

Nanofluids Flow Between Two Rotating Cylinders: Effects of Thermophoresis and Brownian Motion

2013 ◽  
Vol 27 (4) ◽  
pp. 748-755 ◽  
Author(s):  
Meisam Habibi Matin ◽  
Omid Mahian ◽  
Somchai Wongwises
Keyword(s):  
Brownian Motion ◽  
Rotating Cylinders ◽  
And Brownian Motion

Continuous Martingales and Brownian Motion

1991 ◽  
Vol 75 (474) ◽  
pp. 498
Author(s):  
J. R. Norris ◽  
D. Revuz ◽  
M. Yor
Keyword(s):  
Brownian Motion ◽  
And Brownian Motion

scholarly journals Geodesic Random Walks, Diffusion Processes and Brownian Motion on Finsler Manifolds

2021 ◽  
Author(s):  
Tianyu Ma ◽  
Vladimir S. Matveev ◽  
Ilya Pavlyukevich
Keyword(s):  
Brownian Motion ◽  
Diffusion Process ◽  
Random Walks ◽  
Diffusion Processes ◽  
Riemannian Metric ◽  
Finsler Manifold ◽  
Finsler Manifolds ◽  
Bounded Geometry ◽  
And Brownian Motion

AbstractWe show that geodesic random walks on a complete Finsler manifold of bounded geometry converge to a diffusion process which is, up to a drift, the Brownian motion corresponding to a Riemannian metric.

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