Harnack's inequality for stable Lévy processes
type: Item
instance of: scholarly article
title: Harnack's inequality for stable Lévy processes
zbMATH Open document ID: 1081.60055
author: Krzysztof Bogdan
author: Paweł Sztonyk
published in: Potential Analysis
publication date: 2005-04-29T00:00:00Z
review text: It is proved that for a Lévy process \(\{X_{t},t>0\}\) on \(\mathbb R^{d}\) with density function \(f_{\nu}\) of the Lévy measure \(\nu\) satisfying the condition \(f_{\nu}(y)\leq M| y| ^{-d-\alpha},\;y\in \mathbb R^{d},\) there exists a constant \(C_1(\alpha,\nu)\) such that, for every nonegative function \(u\), harmonic in the ball \(B(0,1)\), the following Harnack inequality holds: \[ u(x_1)\leq C_1 u(x_2),\quad x_1,x_2\in B(0,1/2), \] if and only if there is a constant \(C_2\) such that \[ \int_{B(y,1/2)} | y-\nu|^{\alpha-d} f_\nu(\nu)d\nu\leq C_2\int_{B(y,1/2)}f_\nu(\nu)d\nu,\quad | y|\geq1. \] No boundedness from below for the density \(f_\nu\) is required. The proof is based on the following estimate of the Green function \(G(x,y)\) of the unit ball \(B(0,1)\) for the process \(X_{t}\): there is a constant \(C\) such that \[ C^{-1}S(\nu)|\nu-x|^{\alpha-d}\leq G(x,y)\leq CS(\nu)| \nu-x|^{\alpha-d}\quad \text{for }| x| <1/2\text{ and }\nu\in B(0,1). \] Here \(S(\nu)=E^{\nu}\tau_{B}\), and \(\tau_{B}\) is the first exit time from \(B\).
reviewed by: Nijolė Kalinauskaitė
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 31C05
Mathematics Subject Classification ID: 60G51
zbMATH DE Number: 2162535
zbMATH Keywords: relative Kato condition
zbMATH Keywords: Green function
zbMATH Keywords: stable process
Wikidata QID: Q110598161
MaRDI profile type: Publication
full work available at URL: s11118-004-0590-x
OpenAlex ID: W2157773531
cites work: Harnack inequalities for non-local operators of variable order
cites work: Harnack inequalities for jump processes
cites work: Q4888858
cites work: Some Theorems on Stable Processes
cites work: Q5556844
cites work: On the Distribution of First Hits for the Symmetric Stable Processes
cites work: Sharp estimates for the Green function in Lipschitz domains
cites work: Probabilistic proof of boundary Harnack principle for \(\alpha\)-harmonic functions
cites work: Potential theory of Schrödinger operator based on fractional Laplacian
cites work: Harnack inequality for symmetric stable processes on fractals
cites work: Harnack inequality for stable processes on d-sets
cites work: From Brownian Motion to Schrödinger’s Equation
cites work: Q5342182
cites work: Asymptotic behaviour of densities of stable semigroups of measures
cites work: First Passage Times for Symmetric Stable Processes in Space
cites work: Pointwise estimates for densities of stable semigroups of measures
cites work: On some relations between the harmonic measure and the Levy measure for a certain class of Markov processes
cites work: Q3149351
cites work: Q4431523
cites work: First passage distributions of processes with independent increments
cites work: On Harnack's theorem for elliptic differential equations
cites work: Harnack's inequality for elliptic equations and the Hölder property of their solutions
cites work: Harnack inequality for some classes of Markov processes
cites work: On harmonic measure for Lévy processes
cites work: Boundary potential theory for stable Lévy processes
cites work: Q5564805
DOI: 10.1007/S11118-004-0590-X
description: scientific article; zbMATH DE number 2162535