Potential theory for subordinate killed Brownian motion in some unbounded domains

In this dissertation, we study positive harmonic functions of subordinate killed Brownian motion in two types of unbounded domains: domains above the graphs of bounded {dollar}C^{lcub}1,1{rcub}{dollar} functions and exterior domains with{dollar}C^{lcub}1,1{rcub}{dollar} boundaries. With the help of two-sided estimates of the densityfunction of the subordinate killed Brownian motion in each domain, we are able to identify the Martin boundary and the minimal Martin boundary, and then obtain the canonical representation of the positive harmonic functions in terms of the Martin boundary and the Martin kernel. Moreover, each positive harmonic function consists of two nonnegative harmonic functions: one is purely excessive and the other is invariant under the semigroup of the subordinate killed Brownian motion. By the two-sided estimates of the Martin kernel and the canonical representation of the positive harmonic functions, we also show that the Harnack inequality and boundary Harnack principle hold for all the positive harmonic functions of the subordinate killed Brownian motion in the unbounded domains mentioned above.