Potential theory of subordinate killed Brownian motion in a domain

Subordination of a killed Brownian motion in a bounded domain D²Ad via an !/2-stable subordinator gives a process Zt whose infinitesimal generator is m(m(|D)!/2, the fractional power of the negative Dirichlet Laplacian. In this paper we study the properties of the process Zt in a Lipschitz domain D by comparing the process with the rotationally invariant !-stable process killed upon exiting D. We show that these processes have comparable killing measures, prove the intrinsic ultracontractivity of the generator of Zt, prove the intrinsic ultracontractivity of the semigroup of Zt, and, in the case when D is a bounded C1,1 domain, obtain bounds on the Green function and the jumping kernel of Zt.