We study how the resolvent-family of a diffusion behaves, as thedrift grows to infinity. The limit turns out to be a selfadjoint pseudo-resolvent.After reduction of the underlying Hilbert-space, this pseudo-resolvent becomesa resolvent to a strongly continuous semi-group of contractions. We prove thatthis semi-group is associated to some Hunt-process on some suitable state-space which is constructed from equivalence classes of the drifts trajectories.Finally, we show a distributional limit theorem for the accelerated diffusiontoward the associated Hunt process.