Potential Theory of Subordinate Killed Brownian Motion in a Domain∗

Abstract. Subordination of a killed Brownian motion in a bounded domain D ⊂ R via an α/2-stable subordinator gives a process Zt whose infinitesimal generator is −(− |D), 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 α-stable process killed upon exiting D. We show that these processes have comparable killing measures, 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 .