On the distribution of first hits for the symmetric stable processes.

Here t>0, x and t are points in RN, dS is N-dimensional Lebesgue measure, (x, t) is the usual inner product in RN, and j | 2=Q(, t). Throughout this paper integrals will be over all of RN unless explicitly stated otherwise. Of course, to determine our process we must also specify the distribution of X(O). We will always assume that our process starts from some fixed point x in RN; that is, X(O) = x with probability one. We will write P. and E. for probabilities and expectations under the condition X(O) = x. We will assume that the sample functions are normalized to be right continuous and to have left limits everywhere. See [2, ?2] for a complete description of this setup. Define