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 , ∞ ) in a well-understood way. We study the corresponding effect of a power − β x p ( β ≠ 0 , p > 0 ) of the Bessel drift. The most interesting case occurs when β > 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 q th moments ( q > 0 ) as the exit time of Brownian motion. When p < 1 , the influence is much greater: the exit time has exponential moments.