Potential theory of subordinate killed Brownian motion
Panki Kim, Renming Song, Zoran Vondraček · Transactions of the American Mathematical Society · 2017
Let W D W^D be a killed Brownian motion in a domain D ⊂ R d D\subset \mathbb {R}^d and S S an independent subordinator with Laplace exponent ϕ \phi . The process Y D Y^D defined by Y t D = W S t D Y^D_t=W^D_{S_t} is called a subordinate killed Brownian motion. It is a Hunt process with infinitesimal generator − ϕ ( − Δ | D ) -\phi (-\Delta |_D) , where Δ | D \Delta |_D is the Dirichlet Laplacian. In this paper we study the potential theory of Y D Y^D under a weak scaling condition on the derivative of ϕ \phi . We first show that non-negative harmonic functions of Y D 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