Slow diffusion for a Brownian motion with random reflecting barriers

Let β be a positive number: we consider a particle performing a one-dimensional Brownian motion with drift −β, 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 Γ(α + 1)1/αe2βX/2β2, provided that one of these tails is regularly varying with negative index −α. 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 α belongs to [12, 1].