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

The addition of a Bessel drift 1x to a Brownian motion affects the lifetime of the process in the interval (0,∞)(0,∞) in a well-understood way. We study the corresponding effect of a power −βxp(β≠0,p>0) of the Bessel drift. The most interesting case occurs when β>0β>0. If p>1p>1 then the effect of the drift is not too great in the sense that the exit time has the same critical value q0q0 for the existence of qqth moments (q>0)(q>0) as the exit time of Brownian motion. When p<1p<1, the influence is much greater: the exit time has exponential moments.