Slow diffusion for a Brownian motion with random reflecting barriers

Let [beta] be a positive number: we consider a particle performing a one-dimensional Brownian motion with drift -[beta], diffusion coefficient 1, and a reflecting barrier at 0. We prove that the time R, needed by the particle to reach a random level X, has the same distribution tails as [Gamma]([alpha] + 1)1/[alpha]e2[beta]X/2[beta]2, provided that one of these tails is regularly varying with negative index -[alpha]. As a consequence, we discuss the asymptotic behaviour of a Brownian motion with random reflecting barriers, extending some results given by Solomon when X is exponential and [alpha] belongs to [, 1].