We present several constructions of a "censored stable process" in an open set D ⊂ R n , 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 α 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 1,1 open sets.

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We consider the Skorokhod problem in a time-varying interval. We prove existence and uniqueness for the solution. We also express the solution in terms of an explicit formula. Moving boundaries may generate singularities when they touch. We establish two sets of sufficient conditions on the moving boundaries that guarantee that the variation of the local time of the associated reflected Brownian motion is, respectively, finite and infinite. We also apply these results to study the semimartingale property of a class of two-dimensional reflected Brownian motions.