Limiting distributions associated with moments of exponential

During the last decade, a number of explicit results about the distributions of exponential functionals of Brownian motion with drift: A _t^(μ) = ∫₀^t ds exp {2(B _s + μs)}, have been obtained, often originating with the works of D. Dufresne. In the present paper, we rely extensively on these results to show the existence of limiting measures as T → ∞, when the law of {B _t + μt, 0 ≦ t ≦ T} is perturbed by the Radon-Nikodym density consisting of either of the normalized functionals exp (-αA _T^(μ)) or 1/(A_T^(μ)) ^m. The results exhibit different regimes according to whether μ ≧ 0, or μ < 0 in the first case, and to a partition of the (μ, m)-plane in the second case. Although a large number of similar studies have been made for, say, one-dimensional diffusions, the present study, which focuses upon Brownian exponential functionals, appears to be new.