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

随机动力系统视角下的退出时间及其在随机共振中的应用

• 批准号: 2752048
• 金额: --
• 依托单位: Imperial College London
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2752048
• 关键词: Exit; Time; perspective; random; dynamical

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.

退出时间问题或第一次通过时间问题通常涉及确定离开给定域的随机过程（例如布朗运动或莱维运动）所需的时间。更具体地说，它涉及计算从该域内的初始点开始，从而计算扩散过程所需时间的概率分布。我们考虑确定此问题中的数量，因为它们可以为我们提供随机系统的动态信息，即系统如何随着时间的流逝而演变，并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。人们主要考虑在退出时间问题中的平均退出时间和逃生概率。平均退出时间量化了一个随机系统将停留在一个区域中的时间，而逃生概率描述了系统从一个区域过渡到另一个区域的可能性。 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除了随机力外，还向双态系统增加了周期性的力。在这种情况下，存在一个最佳的噪声强度，与周期性的力量结合使用，该系统可以在一个时期内从一个亚稳态逃到另一个亚稳态到另一个稳定状态，这意味着平均出口时间为一半。当系统输出的信噪比最大化时，这种噪声强度也是如此。这种现象在周期性强迫的影响下增强了信号，称为随机共振。随机共振在工程中具有宽度应用，最著名的是检测弱信号，即从强噪声背景中提取有用的信号或非常准确地检测到有用的信号。通过考虑出口时间以找到最佳的噪声强度并从而增强输出噪声，我们可以掌握研究退出时间问题的重要价值。