On Kelvin transformation
type: Item
instance of: scholarly article
title: On Kelvin transformation
zbMATH Open document ID: 1105.60057
author: Tomasz Żak
author: Krzysztof Bogdan
published in: Journal of Theoretical Probability
publication date: 2006-10-31T00:00:00Z
review text: The first main result of the paper states that an appropriately time-changed inversion of a rotationally invariant symmetric \(\alpha\)-stable process is equal in law to a Doob's \(h\)-transform of the process itself. More precisely, let \(0<\alpha \leq 2\), \(d\geq 1\), and let \((X_t: t\geq 0)\) be the rotationally invariant \(\alpha\)-stable process in \({\mathbb R}^d\), killed when it hits the origin (in case the process hits points). Define the additive functional \(A_t=\int_0^t | X_s| ^{-2\alpha}\), and denote by \(\tau_t\) its inverse. Let \(Tx=x/| x| ^2\) be the inversion in \({\mathbb R}^d\). Then the process \((TX_{\tau_t}: t\geq 0)\) has the same law as the process \((X_t:\, t\geq 0)\) conditioned in Doob's sense by the function \(h(x)=| x| ^{\alpha -d}\). As a consequence of this, the authors prove the second main result, a transformation rule for the Green function \(G_D\) of the process \(X_t\): If \(D\) is an open subset of \({\mathbb R}^d\setminus \{0\}\), then \(G_{TD}(x,y)=| x| ^{\alpha-d} | y| ^{\alpha -d} G_D(Tx,Ty)\).
reviewed by: Zoran Vondraček
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 31C05
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60J75
zbMATH DE Number: 5068738
zbMATH Keywords: inversion
zbMATH Keywords: Kelvin transform
zbMATH Keywords: isotropic stable Lévy process
zbMATH Keywords: Brownian motion
zbMATH Keywords: Doob's conditional process
zbMATH Keywords: Riesz kernel
zbMATH Keywords: Green function
zbMATH Keywords: Schrödinger equation
zbMATH Keywords: Laplace transform
zbMATH Keywords: resolvent
Wikidata QID: Q110598159
MaRDI profile type: Publication
full work available at URL: s10959-006-0003-8
OpenAlex ID: W2026025542
cites work: Q4023465
cites work: Symmetric stable processes in cones
cites work: Q4065773
cites work: Q4888858
cites work: Potential theory. An analytic and probabilistic approach to balayage
cites work: Q5556844
cites work: On the Distribution of First Hits for the Symmetric Stable Processes
cites work: Q3229718
cites work: Representation of \(a\)-harmonic functions in Lipschitz domains
cites work: Censored stable processes
cites work: Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains
cites work: Potential theory of Schrödinger operator based on fractional Laplacian
cites work: Harnack inequality for stable processes on d-sets
cites work: Harnack's inequality for stable Lévy processes
cites work: Q4767979
cites work: Martin boundary and integral representation for harmonic functions of symmetric stable processes
cites work: From Brownian Motion to Schrödinger’s Equation
cites work: Kelvin transform and multi-harmonic polynomials
cites work: Q5536163
cites work: Q4765076
cites work: Conditional brownian motion and the boundary limits of harmonic functions
cites work: Asymptotic behaviour of densities of stable semigroups of measures
cites work: Q4089199
cites work: Some Theorems Concerning Brownian Motion
cites work: Markoff processes and potentials. I, II, III
cites work: Q3149351
cites work: Q4431523
cites work: Q3250932
cites work: Q5666837
cites work: Introduction to the theory of (non-symmetric) Dirichlet forms
cites work: Relative Fatou theorem for \(\alpha\)-harmonic functions in Lipschitz domains
cites work: Martin representation for \(\alpha\)-harmonic functions
cites work: Hitting times and potentials for recurrent stable processes
cites work: Q4328333
cites work: Q4226355
cites work: Q3763296
cites work: Q4937701
cites work: Q5607484
cites work: Q4187050
cites work: A propos de l'inverse du mouvement brownien
cites work: Q3874718
DOI: 10.1007/S10959-006-0003-8
description: scientific article