Denote by αt(μ) the probability law of At(μ) =
∫0texp(2(Bs+μ s))ds for a Brownian motion
{Bs, s ≥ 0}. It is well known that αt(μ) is of interest in a number of domains, e.g. mathematical finance, diffusion processes in random environments, stochastic analysis on hyperbolic spaces and so on, but that it has complicated expressions. Recently, Dufresne obtained some remarkably simple expressions for αt(0) and
αt(1), as well as an equally remarkable relationship between
αt(μ) and
αt(ν) for two different drifts μ and ν. In this paper, hinging on previous results about αt(μ), we give different proofs of Dufresne's results and present extensions of them for the processes
{At(μ), t ≥ 0}.