We consider an ordinary differential equation with a unique hyperbolic
attractor at the origin, to which we add a small random perturbation. It is
known that under general conditions, the solution of this stochastic
differential equation converges exponentially fast to an equilibrium
distribution. We show that the convergence occurs abruptly: in a time window of
small size compared to the natural time scale of the process, the distance to
equilibrium drops from its maximal possible value to near zero, and only after
this time window the convergence is exponentially fast. This is what is known
as the cut-off phenomenon in the context of Markov chains of increasing
complexity. In addition, we are able to give general conditions to decide
whether the distance to equilibrium converges in this time window to a
universal function, a fact known as profile cut-off.

通过给原有的超限吸引子的普通微分方程加入微小随机摄动，研究发现，解方程的收敛过程在一个时间窗口内突然发生，这一现象是 Markov 链逐渐复杂化中所称的截尾现象；另外，该论文还能够给出一般性的条件来确定平衡距离是否会在这个时间窗口内收敛到一个通用函数，这一现象被称为截尾剖面。