@article{EJP1751,
author = {Zhen-Qing Chen and Panki Kim and Renming Song},
title = {Global heat kernel estimates for $\Delta+\Delta^{\alpha/2}$ in half-space-like domains},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {17},
year = {2012},
keywords = {symmetric \$\alpha\$-stable process, heat kernel, transition density, Green function, exit time, L\'evy system, harmonic function, fractional Laplacian, Laplacian, Brownian motion},
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 $R^d$.},
pages = {no. 32, 1-32},
issn = {1083-6489},
doi = {10.1214/EJP.v17-1751},
url = {http://ejp.ejpecp.org/article/view/1751}}