Let X(t) be the symmetric [alpha]-stable process in R [to the power of] d, [alpha is an element of the set] (0, 2), d [is greater than or equal to] 2. For f : (0, 1) [approaching] (0,[infinity]) let D(f) be the thorn {x [is an element of the set] R [to the power of] d : x [subscript]1 [is an element of the set] (0, 1), |(x ...

We present several constructions of a "censored stable process" in an open set D [is an element of the subset] R [to the power of] 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 ...