  In this paper, we prove convergence in distribution of Langevin processes in
the overdamped asymptotics. The proof relies on the classical perturbed test
function (or corrector) method, which is used both to show tightness in path
space, and to identify the extracted limit with a martingale problem. The
result holds assuming the continuity of the gradient of the potential energy,
and a mild control of the initial kinetic energy.
