Two-sided Green function estimates for killed subordinate Brownian motions
type: Item
instance of: scholarly article
title: Two-sided Green function estimates for killed subordinate Brownian motions
zbMATH Open document ID: 1248.60091
author: Panki Kim
author: Zoran Vondraček
author: Renming Song
publication date: 2012-06-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1007.5455
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 60J75
Mathematics Subject Classification ID: 60G51
zbMATH DE Number: 6042726
zbMATH Keywords: Green functions
zbMATH Keywords: Lévy process
zbMATH Keywords: \(\kappa \)-fat open set
zbMATH Keywords: subordinator
zbMATH Keywords: subordinate Brownian motion
zbMATH Keywords: boundary Harnack principle
MaRDI profile type: Publication
OpenAlex ID: W2131987832
arXiv ID: 1007.5455
DOI: 10.1112/PLMS/PDR050
published in: Proceedings of the London Mathematical Society. Third Series
review text: A subordinate Brownian motion is a Lévy process that can be obtained by replacing the time of the Brownian motion by an independent subordinator. The infinitesimal generator of a subordinate Brownian motion is \(-\phi (- \Delta )\), where \(\phi \) is the Laplace exponent of the subordinator.NEWLINENEWLINEIn this paper, the authors consider a large class of subordinate Brownian motions without diffusion component. The most important improvement with respect to their previous paper [Stochastic Processes Appl. 119, No. 5, 1601--1631 (2009; Zbl 1166.60046)] is that \(\phi \) is only supposed to be comparable (at infinity) to a regularly varying function. Thus, the estimates below hold for a lot more processes such as (sums of independent) symmetric stable processes, (sums of independent) relativistic stable processes. Also some redundant additional assumptions are eliminated.NEWLINENEWLINEThe main result gives the following sharp estimates for the Green function \(G_D \) of \(X\) in a bounded, \(\kappa \)-fat open set \(D \subset \mathbb R^d\): NEWLINE\[NEWLINE\begin{multlined} C_1 ^ {-1} \left (1 \wedge \frac{\phi (|x-y|^ {-2})}{\sqrt{\phi (\delta _D (x) ^ {-2} ) \phi (\delta _D (y) ^ {-2} ) }} \right ) \frac{1}{|x-y|^ {d} \phi (|x-y|^ {-2})}\\ \leq G_D (x, y) \leq C_1 \left (1 \wedge \frac{\phi (|x-y|^ {-2})}{\sqrt{\phi (\delta _D (x) ^ {-2} ) \phi (\delta _D (y) ^ {-2} ) }} \right ) \frac{1}{|x-y|^ {d} \phi (|x-y|^ {-2})}.\end{multlined}NEWLINE\]NEWLINE When \(D\) is moreover \(C^ {1, 1}\), the authors obtain an explicit form of the estimates in terms of the distance to the boundary. As a consequence, they obtain a boundary Harnack principle in \(C^ {1, 1}\) open sets with explicit rate of decay.
reviewed by: Liliana Popa
description: scientific article; zbMATH DE number 6042726