Subordination of a killed Brownian motion in a bounded domain D ⊂ R d via an α/2-stable subordinator gives a process Z t whose infinitesimal generator is −(− | D ) α/2 , the fractional power of the negative Dirichlet Laplacian. In this paper we study the properties of the process Z t 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 Z t , and, in the case when D is a bounded C 1,1 domain, obtain bounds on the Green function and the jumping kernel of Z t .
We establish a boundary Harnack principle for a large class of subordinate Brownian motion, including mixtures of symmetric stable processes, in bounded κ-fat open set (disconnected analogue of John domains). As an application of the boundary Harnack principle, we identify the Martin boundary and the minimal Martin boundary of bounded κ-fat open sets with respect to these processes with their Euclidean boundary.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.