生灭链的突然收敛与逃逸行为
Abrupt Convergence and Escape Behavior for Birth and Death Chains
摘要：
我们联系两个现象的渐近行为的随机过程：（i）突然收敛或截断现象，（ii）逃逸行为通常与退出亚稳态。前者的特点是收敛在渐近确定的时间，而后者的收敛时间是指数分布。我们比较和研究这两种现象的离散时间生灭链Z漂移向零。特别是，这包括能量驱动的演化与能量函数的形式，一个单一的井。在适当的漂移假设下，我们证明了在另一个方向上既有向零的突然收敛又有逃逸行为。此外，由于演化是可逆的，最终逃逸轨迹的规律与截止路径规律的时间反演相一致。因此，对于由具有足够陡峭的壁的一维能量阱定义的演化，截止和逃逸行为与时间反演相关。
英文摘要：
We link two phenomena concerning the asymptotical behavior of stochastic processes: (i) abrupt convergence or cut-off phenomenon, and (ii) the escape behavior usually associated to exit from metastability. The former is characterized by convergence at asymptotically deterministic times, while the convergence times for the latter are exponentially distributed. We compare and study both phenomena for discrete-time birth-and-death chains on Z with drift towards zero. In particular, this includes energy-driven evolutions with energy functions in the form of a single well. Under suitable drift hypotheses, we show that there is both an abrupt convergence towards zero and escape behavior in the other direction. Furthermore, as the evolutions are reversible, the law of the final escape trajectory coincides with the time reverse of the law of cut-off paths. Thus, for evolutions defined by one-dimensional energy wells with sufficiently steep walls, cut-off and escape behavior are related by time inversion.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
