Profile cut-off phenomenon for the ergodic Feller root process
Gerardo Barrera, Liliana Esquivel
TL;DR
This work establishes a sharp, parameter-explicit profile cut-off phenomenon for the ergodic CIR (Feller square-root) process in the weak-noise limit $\varepsilon\to 0^+$. By a Parseval–Plancherel Fourier framework, it derives a strong total variation cut-off with explicit cut-off time $t_\varepsilon=\frac{1}{\mathfrak{a}}\ln(1/\varepsilon)$ and window $\omega_\varepsilon=\frac{1}{\mathfrak{a}}$, along with a Gaussian-type profile $G^{\mathrm{TV}}_x(r)$ determined by $C_x=\frac{\sqrt{2\mathfrak{b}}}{\mathfrak{b}}(\mathfrak{a}x-\mathfrak{b})$. A parallel result holds for a renormalized Wasserstein distance $\varepsilon^{-(1\wedge p)}\mathcal{W}_p$, with profile $G^{\mathcal{W}_p}_x(r)=(\frac{\sqrt{2\mathfrak{b}}}{2\mathfrak{a}})^{1\wedge p}\mathcal{W}_p(|C_x|e^{-r}+\mathcal{G},\mathcal{G})$, where $\mathcal{G}$ is standard Gaussian and $|C_x|$ encodes the initial datum. The paper also provides explicit mixing-time asymptotics and identifies a non-cut-off case at $x=\mathfrak{b}/\mathfrak{a}$. The results illuminate how multiplicative noise interacts with mean reversion to yield precise, abrupt relaxation to equilibrium in both TV and Wasserstein metrics, with potential extension to broader affine diffusion models.
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 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.