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Let $W^D$ be a killed Brownian motion in a domain $D\subset {\mathbb R}^d$and $S$ an independent subordinator with Laplace exponent $\phi$. The process$Y^D$ defined by $Y^D_t=W^D_{S_t}$ is called a subordinate killed Brownianmotion. It is a Hunt process with infinitesimal generator $-\phi(-\Delta|_D)$,where $\Delta|_D$ is the Dirichlet Laplacian. In this paper we study thepotential theory of $Y^D$ under a weak scaling condition on the derivative of$\phi$. We first show that non-negative harmonic functions of $Y^D$ satisfy thescale invariant Harnack inequality. Subsequently we prove two types of scaleinvariant boundary Harnack principles with explicit decay rates fornon-negative harmonic functions of $Y^D$. The first boundary Harnack principledeals with a $C^{1,1}$ domain $D$ and non-negative functions which are harmonicnear the boundary of $D$, while the second one is for a more general domain $D$and non-negative functions which are harmonic near the boundary of an interioropen subset of $D$. The obtained decay rates are not the same, reflectingdifferent boundary and interior behaviors of $Y^D$.

Potential theory of subordinate killed Brownian motion