We consider a class of one-dimensional absorbed diffusion processes with small singular noises that exhibit multi-scale dynamics in the sense that typical trajectories of the solution process first quickly approach to transient states, captured by quasi-stationary distributions (QSDs), then stay with transient states for a very long period, and finally deviate from transient states and slowly relax to the absorbing state. The main purpose of the present paper is to give qualitative characterizations of such multi-scale dynamics with particular interest in the intriguing transient dynamics governed by transient states. This is achieved by the establishment of (i) noise-vanishing asymptotics of the first eigenvalue and the gap between the first and second eigenvalues of the generator that are respectively the rates for solutions to get absorbed by the absorbing state and attracted by QSDs; (ii) sophisticated estimates quantifying the distance between solutions and convex combinations of QSDs and the absorbing state. Applications to stochastic models arising in chemical reactions and population dynamics are discussed.

我们考虑一类具有小的奇异噪声的一维吸收扩散过程，该现象表现出多尺度的动力学，即解决过程的典型轨迹首先快速接近瞬态，然后由准稳态分布（QSD）捕获，然后保持不变处于很长一段时间的瞬态状态，最终偏离瞬态状态并逐渐松弛到吸收状态。本文的主要目的是对这种多尺度动力学进行定性表征，特别关注瞬态控制的瞬态动力学。这是通过建立（i）消除第一特征值的消噪渐近性和发生器的第一特征值与第二特征值之间的差距来实现的，它们分别是溶液被吸收态吸收并被QSD吸引的速率；（ii）复杂的估计，量化了QSD的溶液和凸组合与吸收状态之间的距离。讨论了在化学反应和种群动态中产生的随机模型的应用。