Global Heat Kernel Estimates for $\Delta+\Delta^{\alpha/2}$ in Half-space-like domains

TL;DR

This paper derives precise two-sided estimates for the heat kernels of a combined Laplacian and fractional Laplacian operator in half-space-like domains, uniformly in the parameter, using probabilistic methods.

Contribution

It provides sharp, uniform heat kernel and Green function estimates for the operator  + a^lpha ^{lpha/2} in half-space-like domains, extending previous results to large time and unbounded domains.

Findings

Established sharp two-sided heat kernel estimates for all time.

Derived uniform Green function estimates independent of parameter a.

Extended heat kernel estimates to large time in unbounded domains.

Abstract

Suppose that $d\ge 1$ and $\alpha\in (0, 2)$. In this paper, by using probabilistic methods, we establish 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 in ${\mathbb R}^d$ 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 it yields the Dirichlet heat kernel estimates for Brownian motion in half-space-like domains by taking $a\to 0$. Integrating the heat kernel estimates in time $t$, 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 ${\mathbb R}^d$.