Heat kernel estimates for the Dirichlet fractional Laplacian

Abstract

We consider the fractional Laplacian - (-Δ) α/2 on an open subset in ℝ d with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such a Dirichlet fractional Laplacian in C 1,1, open sets. This heat kernel is also the transition density of a rotationally symmetric α-stable process killed upon leaving a C 1,1, open set. Our results are the first sharp twosided estimates for the Dirichlet heat kernel of a non-local operator on open sets. © European Mathematical Society 2010.