Sharp convergence for degenerate Langevin dynamics

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.Comment: 37 page