Global heat kernel estimates for $\Delta+\Delta^{\alpha/2}$ in half-space-like domains

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 $R^d$.