  In this paper we return to the problem of Blumenthal-Getoor-Ray, published in
1961, which gave the law of the position of first entry of a symmetric
alpha-stable process into the unit ball. Specifically, we are interested in
establishing the same law, but now for a one dimensional alpha-stable process
which enjoys two-sided jumps, and which is not necessarily symmetric. Our
method is modern in the sense that we appeal to the relationship between
alpha-stable processes and certain positive self-similar Markov processes.
However there are two notable additional innovations. First, we make use of a
type of path censoring. Second, we are able to describe in explicit analytical
detail a non-trivial Wiener-Hopf factorisation of an auxiliary Levy process
from which the desired solution can be sourced. Moreover, as a consequence of
this approach, we are able to deliver a number of additional, related
identities in explicit form for alpha-stable processes.
