Heat kernel estimates for the Dirichlet fractional Laplacian

Abstract

In this paper, we consider the fractional Laplacian $-(-\Delta {)}^{\alpha /2}$ on an open subset in $R^d$ with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such Dirichlet fractional Laplacian in $C^{1,1}$ open sets. This heat kernel is also the transition density of a rotationally symmetric $\alpha$-stable process killed upon leaving a $C^{1,1}$ open set. Our results are the first sharp two-sided estimates for the Dirichlet heat kernel of a non-local operator on open sets.