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 >0 tends to zero for ergodic random systems out of equilibrium of the type align* d X^ₜ (x) = (b-a X^ₜ (x) ) d t+ X^ₜ (x) d Bₜ, X^₀ (x) = x, t 0, align* where x 0, a>0 and b>0 are constants, and (Bₜ) ₓ 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 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.