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 $\epsilon >0$ tends to zero for ergodic random systems out of equilibrium of the type \begin{align*} $\mathrm{d}$ X^{\varepsilon}_t(x) = (\mathfrak{b}-\mathfrak{a} X^{\varepsilon}_t(x))$\mathrm{d}$ t+\varepsilon \sqrt{X^{\varepsilon}_t(x)}$\mathrm{d}$ B_t, \quad X^{\varepsilon}_0(x) = x, \quad t\geqslant 0, \end{align*} where $x⩾0$, $\mathfrak{a}>0$ and $\mathfrak{b}>0$ are constants, and $(B_t{)}_{t⩾0}$ 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 $\epsilon$ 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.