In this presentation, we study an ordinary differential equation with a unique degenerate attractor at the origin, perturbed by the addition of Brownian noise with a small parameter that regulates its magnitude. Under general conditions, for any fixed noise magnitude, the solution to this SDE converges exponentially fast in total variation distance to its unique equilibrium distribution as time goes by.

We suitably accelerate the random dynamics and establish that this convergence occurs in a sharp form. More precisely, the total variation distance between the accelerated dynamics and its equilibrium distribution converges to a non-degenerate limiting profile. This profile corresponds to the total variation distance between the marginal distribution of an appropriately defined SDE, which comes down from infinity, and its associated equilibrium distribution. We point out that the limiting profile that emerges from this convergence is not a step function, which is typically observed in the context of the cut-off phenomenon for random processes.

This talk is based in a paper in SPA 2025 with Conrado da Costa (Durham University, UK) and Milton Jara (IMPA, Brazil).