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，通常涉及确定一个随机过程(如布朗运动或L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从有界域内的初始点开始首先到达有界域边界所需时间的概率分布。我们在这个问题中考虑确定性，是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化，而量化随机系统的动态行为有助于人们了解随机性如何影响这些系统。在退出时间问题中，人们主要考虑的是平均退出时间和逃逸概率。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过渡到另一个区域的可能性。退出时间问题在许多领域有着广泛的应用，例如基金破产的时间，布朗尘埃粒子逃离一个区域的时间，蛋白质分子获得足够的能量改变形状并激活生物过程的时间，以及生态中物种的种群下降到临界阈值以下的时间等。退出时间问题的一个重要和有趣的应用是随机共振，最简单的例子是在双稳系统中除了随机力之外还增加了一个周期性的力。在这种情况下，存在一个最佳噪声强度，与周期性作用力相结合，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。当系统输出的信噪比最大化时，该噪声强度也是如此。这种现象，即噪声在周期性强迫的影响下增强信号，称为随机共振。随机共振在工程中有着广泛的应用，尤其是在微弱信号的检测方面，即从强噪声背景中提取有用信号或检测极微弱的信号。通过考虑出场时间来寻找最优噪声强度，从而提高输出噪声，对研究出场时间问题具有重要的意义。

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.