We study how the resolvent-family of a diffusion behaves, as the drift grows to infinity. The limit turns out to be a selfadjoint pseudo-resolvent. After reduction of the underlying Hilbert-space, this pseudo-resolvent becomes a resolvent to a strongly continuous semi-group of contractions. We prove that this 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 diffusion toward the associated Hunt process.