Censored stable processes

We present several constructions of a censored stable process in an open set D subset of R, i.e., a symmetric stable process which is not allowed to jump outside D. We address the question of whether the process will approach the boundary of D in a finite time - we give sharp conditions for such approach in terms of the stability index alpha and the thickness of the boundary. As a corollary, new results are obtained concerning Besov spaces on non-smooth domains, including the critical exponent case. We also study the decay rate of the corresponding harmonic functions which vanish on a part of the boundary. We derive a boundary Harnack principle in C-1,C-1 open sets.