For a one-dimensional smooth vector field in a neighborhood of an unstable equilibrium, we consider the associated dynamics perturbed by small noise. We give a revealing elementary proof of a result proved earlier using heavy machinery from Malliavin calculus. In particular, we obtain precise vanishing noise asymptotics for the tail of the exit time and for the exit distribution conditioned on atypically long exits. We also discuss our program on rare transitions in noisy heteroclinic networks.

中文翻译：

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线上不稳定平衡的退出时间的尾巴

对于不稳定平衡邻域中的一维平滑向量场，我们考虑由小噪声扰动的相关动力学。我们给出了一个揭示性的基本证明，证明了早先使用 Malliavin 微积分的重型机械证明的结果。特别是，我们获得了出口时间尾部和以非典型长出口为条件的出口分布的精确消失噪声渐近线。我们还讨论了我们关于嘈杂异宿网络中罕见转换的计划。