Dirichlet heat kernel estimates for rotationally symmetric Lévy processes

Abstract

In this paper, we consider a large class of purely discontinuous rotationally symmetric Lévy processes. We establish sharp two-sided estimates for the transition densities of such processes killed upon leaving an open set D. When D is a κ-fat open set, the sharp two-sided estimates are given in terms of surviving probabilities and the global transition density of the Lévy process. When D is a C 1 , 1 open set and the Lévy exponent of the process is given by Ψ ( ξ ) = ϕ ( | ξ | 2 ) with ϕ being a complete Bernstein function satisfying a mild growth condition at infinity, our two-sided estimates are explicit in terms of Ψ, the distance function to the boundary of D and the Lévy density of X. This gives an affirmative answer to the conjecture posted in Chen, Kim and Song [Global heat kernel estimates for relativistic stable processes in half-space-like open sets. Potential Anal. 36 (2012) 235–261]. Our results are the first sharp two-sided Dirichlet heat kernel estimates for a large class of symmetric Lévy processes with general Lévy exponents. We also derive an explicit lower bound estimate for symmetric Lévy processes on R d in terms of their Lévy exponents.