  We study Brownian motion in a drifted Brownian potential in the
subexponential regime. We prove that the annealed probability of deviating
below the almost sure speed has a polynomial rate of decay and compute the
exponent in this power law. This provides a continuous-time analogue of what
Dembo, Peres and Zeitouni proved for the transient random walk in random
environment. Our method takes a completely different route, making use of
Lamperti's representation together with an iteration scheme.
