bounds on the density , Green function and jumping function of subordinate killed BM

Abstract. Subordination of a killed Brownian motion in a domain D ⊂ R via an α/2-stable subordinator gives rise to a process Zt whose infinitesimal generator is −(− |D), the fractional power of the negative Dirichlet Laplacian. In this paper we establish upper and lower estimates for the density, Green function and jumping function of Zt when D is either a bounded C1,1 domain or an exterior C1,1 domain. Our estimates are sharp in the sense that the upper and lower estimates differ only by a multiplicative constant.