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Description

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 Brownian motion. It is a Hunt process with infinitesimal generator $\phi(-\Delta|_D)$, where $\Delta|_D$ is the Dirichlet Laplacian. In this talk I will present several potential-theoretic results for $Y^D$ under a weak scaling condition on the derivative of $\phi$. These results include the scale invariant Harnack inequality for non-negative harmonic functions of $Y^D$, and two types of scale invariant boundary Harnack principles with explicit decay rates. The first boundary Harnack principle deals with a $C^{1,1}$ domain $D$ and non-negative functions which are harmonic near the boundary of $D$, while the second one is 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 behavior of $Y^D$. The results are new even in the case of the stable subordinator. Joint work with P.Kim and R.Song