Exit times of symmetric α-stable processes from unbounded convex domains

Let X-t be a d-dimensional symmetric stable process with parameter alpha is an element of (0, 2). Consider tau(D) the first exit time of X-t from the domain D = {(x, y) is an element of R x Rd-1 : 0 < x, vertical bar y vertical bar < phi(x)}, where phi is concave and lim(x ->infinity) phi(x) = infinity. We obtain upper and lower bounds for P-z {tau(D) > t} and for the harmonic measure of X-t killed upon leaving D boolean AND B(0, r). These estimates are, under some mild assumptions on phi, asymptotically sharp as t -> infinity. In particular, we determine the critical exponents of integrability of tau(D) for domains given by phi(x) = x(beta) [ln(x + 1)](gamma), where 0 <= beta i < 1, and gamma is an element of R. These results extend the work of R. Banuelos and R. Bogdan (2).