In this thesis, we first consider a subordinate Brownian motion X with Gaussian components when the scaling order of purely discontinuous part is between 0 and 2 including 2. We establish sharp two-sided bounds for transition density of X in Rd and C1,1 open sets. As a corollary, we obtain a sharp Green function estimates. Second, we show that, when potentials are in appropriate Kato classes, Dirichlet heat kernel estimates for a large class of non-local operators are stable under (non-local) Feynman-Kac perturbations. Especially, our operators include infinitesimal generators for killed subordinate Brownian motions whose scaling order is 2.