Potential theory of subordinate killed Brownian motion

Let W ^D be a killed Brownian motion in a domain D ⊂ R ^d and S an independent subordinator with Laplace exponent φ. The process Y ^D defined by Y _t ^D = W _S ^D _t is called a subordinate killed Brownian motion. It is a Hunt process with infinitesimal generator −φ(−Δ| _D ), where Δ| _D is the Dirichlet Laplacian. In this paper we study the potential theory of Y ^D under a weak scaling condition on the derivative of φ. We first show that non-negative harmonic functions of Y ^D satisfy the scale invariant Harnack inequality. Subsequently we prove two types of scale invariant boundary Harnack principles with explicit decay rates for non-negative harmonic functions of Y ^D . The first boundary Harnack principle deals with a C ^1,1 domain D and non-negative functions which are harmonic near the boundary of D, whilethesecondoneis for a more general domain D and non-negative functions which are harmonic near the boundary of an interior open subset of D. The obtained decay rates are not the same, reflecting different boundary and interior behaviors of Y ^D .