Mixing times and cutoff phenomenon for Langevin SDEs.

In this presentation, we study mixing times and the so-called cutoff phenomenon for an ergodic overdamped nonlinear non-gradient Langevin dynamics with a strongly coercive potential and driven by an additive noise with small amplitude.

The cutoff phenomenon was introduced by Diaconis and Aldous in the study of a quantitative convergence to equilibrium for card-shuffling Markov models. When the driven noise is the Brownian motion, the total variation distance between the current state and its equilibrium distribution decays around the mixing time from one to zero abruptly. When the noise is the alpha-stable with index alpha>3/2, cutoff phenomenon still holds while for alpha\leq 3/2 our coupling techniques do not apply, and therefore we cannot conclude if the cutoff phenomenon still holds.

In the case of degenerate potential, we show that cutoff phenomenon does not hold.

The talk is based on series of papers with Milton Jara (IMPA, Brazil), Michael Högele (Universidad de los Andes, Colombia), Juan Carlos Pardo (CIMAT, Mexico) and Conrado da Costa (Durham University, UK).