  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 gradual, that is, the total variation
distance of the accelerated random dynamics and its equilibrium distribution
converges to a decreasing function, 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. This
completes the classification of all possible behaviors of the total variation
distance between the time marginal of the aforementioned stochastic dynamics
and its invariant measure for one dimensional smooth convex potentials. In
addition, there is no cut-off phenomenon for this one-parameter family of
random processes and asymptotics of the mixing times are derived.
