A semiclassical approach to the Kramers--Smoluchowski equation

We consider the Kramers–Smoluchowski equation at a low temperature regime and show how semiclassical techniques developed for the study of the Witten Laplacian and Fokker–Planck equation provide quantitative results. This equation comes from molecular dynamics and temperature plays the role of a semiclassical paramater. The presentation is self-contained in the one dimensional case, with pointers to the recent paper \cite{Mi16} for results needed in higher dimensions. One purpose of this note is to provide a simple introduction to semiclassical methods in this context.