Illinois Journal of Mathematics

Sharp Green function estimates for $\Delta+ \Delta^{\alpha /2}$ in $C^{1,1}$ open sets and their applications

Zhen-Qing Chen, Panki Kim, Renming Song, and Zoran Vondraček

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Abstract

We consider a family of pseudo differential operators $\{\Delta+ a^\alpha\Delta^{\alpha/2}$; $a\in[0, 1]\}$ on ${\mathbb R}^d$ that evolves continuously from $\Delta$ to $\Delta+ \Delta^{\alpha/2}$, where $d\geq1$ and $\alpha\in(0, 2)$. It gives rise to a family of Lévy processes $\{X^a, a\in[0, 1]\}$, where $X^a$ is the sum of a Brownian motion and an independent symmetric $\alpha$-stable process with weight $a$. Using a recently obtained uniform boundary Harnack principle with explicit decay rate, we establish sharp bounds for the Green function of the process $X^a$ killed upon exiting a bounded $C^{1,1}$ open set $D\subset{\mathbb R}^d$. Our estimates are uniform in $a\in(0, 1]$ and taking $a\to0$ recovers the Green function estimates for Brownian motion in $D$. As a consequence of the Green function estimates for $X^a$ in $D$, we identify both the Martin boundary and the minimal Martin boundary of $D$ with respect to $X^a$ with its Euclidean boundary. Finally, sharp Green function estimates are derived for certain Lévy processes which can be obtained as perturbations of $X^a$.

Article information

Source
Illinois J. Math., Volume 54, Number 3 (2010), 981-1024.

Dates
First available in Project Euclid: 3 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1336049983

Digital Object Identifier
doi:10.1215/ijm/1336049983

Mathematical Reviews number (MathSciNet)
MR2928344

Zentralblatt MATH identifier
1257.31002

Subjects
Primary: 31A20: Boundary behavior (theorems of Fatou type, etc.) 31B25: Boundary behavior 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]
Secondary: 47G20: Integro-differential operators [See also 34K30, 35R09, 35R10, 45Jxx, 45Kxx] 60J75: Jump processes 31B05: Harmonic, subharmonic, superharmonic functions

Citation

Chen, Zhen-Qing; Kim, Panki; Song, Renming; Vondraček, Zoran. Sharp Green function estimates for $\Delta+ \Delta^{\alpha /2}$ in $C^{1,1}$ open sets and their applications. Illinois J. Math. 54 (2010), no. 3, 981--1024. doi:10.1215/ijm/1336049983. https://projecteuclid.org/euclid.ijm/1336049983


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