${\chi }^2$-cut-off phenomenon for Galerkin projections of Fokker-Planck equations with monomial potentials

In this manuscript, we establish the existence/non-existence of the cut-off phenomenon for the Langevin–Kolmogorov random dynamics with monomial convex potentials, possible singular, and driven by a Brownian motion with small strength. We consider a truncated ${\chi }^2$-distance, that is, a distance based on Galerkin projections of the eigensystem, and show that not only a refined knowledge of the eigenvalues is needed but also a refined asymptotics of the growth for the eigenfunctions of the Fokker–Planck equations associated to the Langevin–Kolmogorov dynamics. In addition, this explicit analysis yields asymptotics of the mixing times and, in some regimes, information on the limiting profile, going beyond the product condition and the cut-off window alone.