Illinois Journal of Mathematics

Dirichlet heat kernel estimates for $\Delta^{\alpha/2}+ \Delta^{\beta/2}$

Zhen-Qing Chen, Panki Kim, and Renming Song

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Abstract

For $d\geq1$ and $0<\beta<\alpha<2$, consider a family of pseudo differential operators $\{\Delta^{\alpha} + a^\beta \Delta^{\beta/2}$; $a \in[0, 1]\}$ on $\mathbb{R}^d$ that evolves continuously from $\Delta^{\alpha/2}$ to $ \Delta^{\alpha/2}+ \Delta^{\beta/2}$. It gives arise to a family of Lévy processes $\{X^a, a\in[0, 1]\}$ on $\mathbb{R}^d$, where each $X^a$ is the independent sum of a symmetric $\alpha$-stable process and a symmetric $\beta$-stable process with weight $a$. For any $C^{1,1}$ open set $D\subset\mathbb{R}^d$, we establish explicit sharp two-sided estimates, which are uniform in $a\in(0, 1]$, for the transition density function of the subprocess $X^{a, D}$ of $X^a$ killed upon leaving the open set~$D$. The infinitesimal generator of $X^{a, D}$ is the nonlocal operator $\Delta^{\alpha} + a^\beta\Delta^{\beta/2}$ with zero exterior condition on $D^c$. As consequences of these sharp heat kernel estimates, we obtain uniform sharp Green function estimates for $X^{a, D}$ and uniform boundary Harnack principle for $X^a$ in $D$ with explicit decay rate.

Article information

Source
Illinois J. Math., Volume 54, Number 4 (2010), 1357-1392.

Dates
First available in Project Euclid: 24 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1348505533

Digital Object Identifier
doi:10.1215/ijm/1348505533

Mathematical Reviews number (MathSciNet)
MR2981852

Zentralblatt MATH identifier
1268.60101

Subjects
Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 47G20: Integro-differential operators [See also 34K30, 35R09, 35R10, 45Jxx, 45Kxx] 60J75: Jump processes
Secondary: 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx}

Citation

Chen, Zhen-Qing; Kim, Panki; Song, Renming. Dirichlet heat kernel estimates for $\Delta^{\alpha/2}+ \Delta^{\beta/2}$. Illinois J. Math. 54 (2010), no. 4, 1357--1392. doi:10.1215/ijm/1348505533. https://projecteuclid.org/euclid.ijm/1348505533


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