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 $\varepsilon>0$ 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 $x\geqslant 0$, $\mathfrak{a}>0$ and $\mathfrak{b}>0$ are constants, and $(B_t)_{t \geqslant 0}$ 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 $\varepsilon$ 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.