  The long-time/large-scale, small-friction asymptotic for the one dimensional
Langevin equation with a periodic potential is studied in this paper. It is
shown that the Freidlin-Wentzell and central limit theorem (homogenization)
limits commute. We prove that, in the combined small friction,
long-time/large-scale limit the particle position converges weakly to a
Brownian motion with a singular diffusion coefficient which we compute
explicitly. We show that the same result is valid for a whole one parameter
family of space/time rescalings. The proofs of our main results are based on
some novel estimates on the resolvent of a hypoelliptic operator.
