Exit Time from the perspective of random dynamical systems and its application in stochastic resonance

退出时间问题，或首次通过时间问题，通常涉及确定随机过程（如布朗运动或lsamvy运动）离开给定域所需的时间。更具体地说，它涉及计算扩散过程首次到达有界域边界所需时间的概率分布，从该域内的初始点开始。我们在这个问题中考虑确定性量，因为它们可以为我们提供随机系统的动态信息，即系统如何随时间演变，而量化随机系统的动态行为有助于人们理解随机性如何影响这些系统。在撤离时间问题中，人们主要考虑的是平均撤离时间和撤离概率。平均退出时间量化了随机系统在一个区域停留的时间，而逃逸概率描述了系统从一个区域转移到另一个区域的可能性。退出时间问题在许多领域都有广泛的应用，例如基金破产的时间，布朗尘埃粒子逃离一个区域所需的时间，蛋白质分子获得足够能量改变形状并激活生物过程的时间，以及生态学中物种种群低于临界阈值的时间等。退出时间问题的一个重要而有趣的应用是随机共振，最简单的例子是在随机力之外向双稳态系统添加周期性力。在这种情况下，存在一个最佳噪声强度，与周期性力相结合，允许系统在一个周期内以近乎完美的精度从一个亚稳态逃逸到另一个亚稳态，这意味着平均退出时间是周期的一半。此噪声强度也是系统输出的信噪比最大的时候。这种在周期性强迫作用下噪声增强信号的现象被称为随机共振。随机共振在工程中有着广泛的应用，最显著的是微弱信号的检测，即从强噪声背景中提取有用的信号或检测极其微弱的信号。通过考虑退出时间来寻找最优噪声强度，从而提高输出噪声，具有研究退出时间问题的重要价值。

The exit time problem, or first passage time problem, typically involves determining the time it takes for a stochastic process (such as Brownian motion or Lévy motion) to leave a given domain. More specifically, it deals with calculating the probability distribution of the time it takes for a diffusion process to first reach the boundary of a bounded domain, starting from an initial point within that domain. We consider deterministic quantities in this problem because they can provide us with the dynamical information of random systems, that is, how the system evolves over time, and quantifying the dynamic behavior of random systems helps people understand how randomness affects these systems. People mainly consider the mean exit time and escape probability in the exit time problem. Mean exit time quantifies how long a random system will stay in a region, while escape probability describes the likelihood of the system transitioning from one region to another. The exit time problem has a wide range of applications in many fields, such as the time of a fund going bankrupt, the time it takes for a Brownian dust particle to escape from a region, the time for a protein molecule to gain sufficient energy to change shape and activate a biological process, and the time when the population of a species in ecology falls below a critical threshold, etc. One important and intriguing application of the exit time problem is stochastic resonance, the simplest example being the addition of a periodic force to a bistable system in addition to stochastic forces. In such a scenario, there exists an optimal noise intensity that, in conjunction with the periodic force, allows the system to escape from one metastable state to another with near-perfect precision within one period, meaning the mean exit time is half the period. This noise intensity is also when the signal-to-noise ratio of the system's output is maximized. This phenomenon, where noise enhances the signal under the influence of periodic forcing, is known as stochastic resonance. Stochastic resonance has widespread applications in engineering, most notably in the detection of weak signals, that is, extracting useful signals from a strong noise background or detecting extremely faint signals. By considering the exit time to find the optimal noise intensity and thus enhance the output noise, we grasp a significant value of studying the exit time problem.