Profile cut-off phenomenon for the ergodic Feller root process

Abstract

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 driven by multiplicative non-linear noise of the type $d{X}_t^{\varepsilon }\left(x\right)=(b-a{X}_t^{\varepsilon }\left(x\right))dt+\varepsilon \sqrt{X_t^{\varepsilon }\left(x\right)}d{B}_t,X_0^{\varepsilon }\left(x\right)=x,t⩾0,$where $x⩾0$, $a>0$ and $b>0$ are constants, and ${\left(B_t\right)}_{t⩾0}$ 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 $\varepsilon$ 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.