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2012 Global heat kernel estimates for $\Delta+\Delta^{\alpha/2}$ in half-space-like domains
Zhen-Qing Chen, Panki Kim, Renming Song
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Electron. J. Probab. 17: 1-32 (2012). DOI: 10.1214/EJP.v17-1751

Abstract

Suppose that $d\ge 1$ and $\alpha\in (0, 2)$. In this paper, we establish by using probabilistic methods sharp two-sided pointwise estimates for the Dirichlet heat kernels of $\{\Delta+ a^\alpha \Delta^{\alpha/2}; \ a\in (0, 1]\}$ on half-space-like $C^{1, 1}$ domains for all time $t>0$. The large time estimates for half-space-like domains are very different from those for bounded domains. Our estimates are uniform in $a \in (0, 1]$ in the sense that the constants in the estimates are independent of $a\in (0, 1]$. Thus they yield the Dirichlet heat kernel estimates for Brownian motion in half-space-like domains by taking $a\to 0$. Integrating the heat kernel estimates with respect to the time variable $t$, we obtain uniform sharp two-sided estimates for the Green functions of $\{\Delta+ a^\alpha \Delta^{\alpha/2}; \ a\in (0, 1]\}$ in half-space-like $C^{1, 1}$ domains in $R^d$.

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Zhen-Qing Chen. Panki Kim. Renming Song. "Global heat kernel estimates for $\Delta+\Delta^{\alpha/2}$ in half-space-like domains." Electron. J. Probab. 17 1 - 32, 2012. https://doi.org/10.1214/EJP.v17-1751

Information

Accepted: 25 April 2012; Published: 2012
First available in Project Euclid: 4 June 2016

zbMATH: 1247.60115
MathSciNet: MR2915668
Digital Object Identifier: 10.1214/EJP.v17-1751

Subjects:
Primary: 47G20 , 60J35 , 60J75
Secondary: 47D07

Keywords: Brownian motion , Exit time , fractional Laplacian , Green function , Harmonic function , heat kernel , L\'evy system , Laplacian , Symmetric $\alpha$-stable process , Transition density

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