Sharp bounds on the density, Green function and jumping function of subordinate killed BM (Q1434100)

Let \(D\subset \mathbb{R}^d\) be a \(C^{1,1}\)- or an exterior \(C^{1,1}\)-domain. On \(D\) consider the Laplacian \(\Delta_D\) with Dirichlet boundary data. Use an \(\alpha/2\)-stable subordinator to define a Markov jump process \((Z_t)_{t>0}\) with generator \(-(-\Delta_D)^{\alpha/2}\). The author proves sharp upper and lower bounds to the densities of the Green's and the transition function of \((Z_t)_t\). Here ``sharp'' means that the upper and lower bounds only differ by a multiplicative constant. When the classical Brownian motion is first subordinated and then killed, the result is of course different from \((Z_t)_t\).