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

TL;DR

This paper investigates the cut-off phenomenon for Langevin--Kolmogorov dynamics with monomial potentials, using a Galerkin projection-based distance to analyze mixing times and eigenfunction asymptotics.

Contribution

It provides new criteria for the existence of the cut-off phenomenon and detailed asymptotics of eigenfunctions and mixing times for Fokker--Planck equations with monomial potentials.

Findings

Established conditions for cut-off and non-cut-off regimes.

Derived asymptotics for eigenvalues and eigenfunctions.

Provided detailed mixing time estimates and limiting profiles.

Abstract

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.