Stable processes have thorns

Abstract

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 [subscript] 2, . . . , x [subscript] d)| < f(x [subscript] 1)}. We give an integral criterion in terms of f for the existence of a random time s such that X(t) remains in X(s) + [line] D(f) for all t [is an element of the set] [s, s + 1).