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

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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.