Estimates of Dirichlet heat kernels for unimodal Levy processes with low intensity of small jumps

Abstract

In this paper, we study transition density functions for pure jump unimodal Levy processes killed upon leaving an open set D. Under some mild assumptions on the Levy density, we establish two-sided Dirichlet heat kernel estimates when the open set D is C1,1. Our result covers the case that the Levy densities of unimodal Levy processes are regularly varying functions whose indices are equal to the Euclidean dimension. This is the first results on two-sided Dirichlet heat kernel estimates for Levy processes such that the lower scaling index of the Levy densities is not necessarily strictly bigger than the Euclidean dimension.