Green function estimates for relativistic stable processes in half-space-like open sets
type: Item
instance of: scholarly article
title: Green function estimates for relativistic stable processes in half-space-like open sets
zbMATH Open document ID: 1235.60100
author: Zhen-Qing Chen
author: Panki Kim
author: Renming Song
published in: Stochastic Processes and their Applications
publication date: 2011-06-15T00:00:00Z
review text: Let \(X_t^m\) be the relativistic \(\alpha\)-stable process in \(\mathbb R^n\), i.e., the process with characteristic exponent \((|\xi|^2+m^2)^{\frac{\alpha}{2}}-m\), \(m>0\), \(X_t\equiv X_t^0\). Denote by \(X_t^{m,D}\) (resp., \(X_t^D\)) the process killed upon exiting the domain \(\mathcal{D}\). The paper is devoted to sharp two-sided estimates for the Green function related to \(X_t^D\) when \(\mathcal{D}\) is a half-space-like \(C^{1,1}\) open set. For the case when \(\mathcal{D}\) is a \(C^{1,1}\)-open set in \(\mathbb R^n\), sharp estimates for the transition probability \(p_t^{m,D}(t,x,y)\) and the Green function \(G_D^m(x,y)\) were given in [the authors, Ann. Probab. 40, No.~1, 213--244 (2012; Zbl 1235.60101)]. For bounded \(\mathcal{D}\), the estimates for the Green function show that it is comparable to the Green function of the killed stable process in \(\mathcal{D}\). In this paper, the estimates for the Green function are obtained under the assumption that \(\mathcal{D}\) is an unbounded \(C^{1,1}\) domain.
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 47G20
Mathematics Subject Classification ID: 60J75
Mathematics Subject Classification ID: 47D07
Mathematics Subject Classification ID: 60G51
zbMATH DE Number: 5908028
zbMATH Keywords: symmetric \(\alpha \)-stable process
zbMATH Keywords: relativistic stable process
zbMATH Keywords: Green function
zbMATH Keywords: exit time
zbMATH Keywords: Lévy system
zbMATH Keywords: uniform Harnack inequality
zbMATH Keywords: uniform boundary Harnack principle
reviewed by: V. Knopova
MaRDI profile type: Publication
full work available at URL: j.spa.2011.01.004
OpenAlex ID: W2151165919
cites work: On the Distribution of First Hits for the Symmetric Stable Processes
cites work: The boundary Harnack principle for the fractional Laplacian
cites work: Sharp estimates for the Green function in Lipschitz domains
cites work: Bessel potentials, hitting distributions and Green functions
cites work: Q3417697
cites work: Relativistic Schrödinger operators: Asymptotic behavior of the eigenfunctions
cites work: Heat kernel estimates for the Dirichlet fractional Laplacian
cites work: Sharp heat kernel estimates for relativistic stable processes in open sets
cites work: Heat kernel estimates for stable-like processes on \(d\)-sets.
cites work: Heat kernel estimates for jump processes of mixed types on metric measure spaces
cites work: Drift transforms and Green function estimates for discontinuous processes
cites work: Global heat kernel estimates for fractional Laplacians in unbounded open sets
cites work: Hartree-Fock theory for pseudorelativistic atoms
cites work: Relativistic stability of matter. I
cites work: Stability of relativistic matter with magnetic fields for nuclear charges up to the critical value
cites work: Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators
cites work: Two-sided optimal bounds for Green functions of half-spaces for relativistic \(\alpha\)-stable process
cites work: Spectral theory of the operator \((p^2+m^2)^{1/2}-Ze^2/r\)
cites work: Relative Fatou's theorem for (\(-\Delta)^{\alpha/2}\)-harmonic functions in bounded \(\kappa\)-fat open sets
cites work: Generalized 3G theorem and application to relativistic stable process on non-smooth open sets
cites work: Potential theory of truncated stable processes
cites work: Boundary Harnack principle for subordinate Brownian motions
cites work: Intrinsic ultracontractivity of the Feynman-Kac semigroup for relativistic stable processes
cites work: The stability and instability of relativistic matter
cites work: Estimates of Green function for relativistic \(\alpha\)-stable process
cites work: Q3999823
cites work: Estimates on the Dirichlet heat kernel of domains above the graphs of bounded C1,1 functions
cites work: Boundary Harnack principle for symmetric stable processes
cites work: The large-Z Behavior of pseudorelativistic atoms
cites work: On harmonic measure for Lévy processes
cites work: Spectral analysis of pseudodifferential operators
DOI: 10.1016/J.SPA.2011.01.004
description: scientific article; zbMATH DE number 5908028