In this paper, we study an ordinary differential equation with a degenerate
global attractor at the origin, to which we add a white noise with a small
parameter that regulates its intensity. Under general conditions, for any fixed
intensity, as time tends to infinity, the solution of this stochastic dynamics
converges exponentially fast in total variation distance to a unique
equilibrium distribution. We suitably accelerate the random dynamics and show
that the preceding convergence is sharp, that is, the total variation distance
of the accelerated random dynamics and its equilibrium distribution tends to a
decreasing profile, which corresponds to the total variation distance between
the marginal of a stochastic differential equation that comes down from
infinity and its corresponding equilibrium distribution. In particular, there
is no cutoff phenomenon for this one-parameter family of random processes.
