Profile cut-off phenomenon for the ergodic Feller root process

The present manuscript is devoted to the study of the convergence to equilibrium as the noise intensity ε>0 tends to zero for ergodic random systems out of equilibrium of the type {align*} d Xε_t(x) = (b-a Xε_t(x))d t+ε √{Xε_t(x)}d B_t, Xε_0(x) = x, t 0, {align*} where x 0, a>0 and b>0 are constants, and (Bₜ)t 0 is a one dimensional standard Brownian motion. More precisely, we show the strongest notion of asymptotic profile cut-off phenomenon in the total variation distance and in the renormalized Wasserstein distance when ε tends to zero with explicit cut-off time, explicit time window, and explicit profile function. In addition, asymptotics of the so-called mixing times are given explicitly.