Censored symmetric L\'evy processes

We examine three equivalent constructions of a censored symmetric purely discontinuous L'evy process on an open set $D$; via the corresponding Dirichlet form, through the Feynman-Kac transform of the L'evy process killed outside of $D$ and from the same killed process by the Ikeda-Nagasawa-Watanabe piecing together procedure. By applying the trace theorem on $n$-sets for Besov-type spaces of generalized smoothness associated with complete Bernstein functions satisfying certain scaling conditions, we analyze the boundary behaviour of the corresponding censored L'evy process and determine conditions under which the process approaches the boundary $\mathrm{\partial }D$ in finite time. Furthermore, we prove a stronger version of the 3G inequality and its generalized version for Green functions of purely discontinuous L'evy processes on $\kappa$-fat open sets. Using this result, we obtain the scale invariant Harnack inequality for the corresponding censored process.