The influence of a power law drift on the exit time of Brownian motion from a half-line

The addition of a Bessel drift 1/x to a Brownian motion affects the lifetime of the process in the interval (0, infinity) in a well-understood way. We study the corresponding effect of a power -beta/x(p) (beta not equal 0, p > 0) of the Bessel drift. The most interesting case occurs when beta > 0. If p > 1 then the effect of the drift is not too great in the sense that the exit time has the same critical value q(0) for the existence of qth moments (q > 0) as the exit time of Brownian motion. When p < 1, the influence is much greater: the exit time has exponential moments. (c) 2006 Elsevier B.V. All rights reserved.