In the present paper, the noise-induced escape from stable invariant tori with the frame of general Langevin dynamics is investigated under weak random perturbations. Based on the large deviation theory, the quasi-potential, a quantity exponentially dominating the mean first escape time and the stationary or quasi-stationary probability density, is explored both analytically and numerically. The results show that whether the stochastic trajectories are ergodic on a torus has a crucial influence on the behavior of the quasi-potential. Specifically, there are two sources of ergodicity. One is the ergodicity of the deterministic flow, and the other requires the nondegeneracy of the noise in the tangential directions of a torus. It is found that if the ergodicity holds, the quasi-potential will be independent of the initial position on a torus, but not when the ergodicity is broken. In particular, it indicates that, for nonlinear systems driven by combined Gaussian white noise and multiple harmonic excitations, the mean first exit time varies discontinuously with respect to the frequency vector of these harmonic excitations as the noise intensity approaches zero. Adding noise to the phase of these harmonic excitations will eliminate the dependence on the initial position and thus, make the systems more robust. It reminds us that great care must be taken when dealing with noise-induced problems involving systems that possess unstable geometric structures.

The dynamics of a stochastic system that exhibits large fluctuations to a given state are almost deterministic due to weak random perturbations. Such large fluctuations occur with overwhelming probability in the vicinity of the so-called optimal path, which is a vital concept in physics, chemistry, and biology, as it uncovers the way in which a physical process escapes from the attractive domain of a metastable state and transitions between different metastable states. In this paper, we examine the prehistorical description of the optimal fluctuation with the unifying framework of Langevin dynamics by means of a quantity called prehistory probability density. We show that the optimal fluctuation has a strong connection with the time reversal of the associated diffusion process. Specifically, in the stationary or quasi-stationary cases, it is found that the prehistory probability density actually acts as the transition probability density of the reversed process. As noise intensity approaches zero, it focuses on the average dynamics of the reversed process due to the law of large numbers, which is then shown to coincide with the time reversal of the optimal path. The local dispersion of the prehistory probability density can thus be reformulated as a Gaussian distribution corresponding to the linearized part of the reversed process. In addition, as an analogue of the original definition in the stationary states, it is proven that the concept of prehistory probability density can also be extended to nonstationary cases where similar properties are still valid. Based on these theoretical results, an algorithm is designed and then successfully applied to a one-dimensional example at the end, showing its accuracy for pinpointing the location of the optimal path and its efficacy in cases where multiple optimal paths coexist.

This paper aims to derive accurate asymptotic estimates for the exit time probabilities of scalar Ornstein–Uhlenbeck (OU) bridges. The exit time probabilities are expressed as an asymptotic series in powers of a small parameter that characterizes the intensity of the noise inputs. It is shown that the series is valid in certain regions where all its terms are smooth functions. The results enable an accurate evaluation of the probability for a corresponding OU process to escape from a domain before a specified time, provided its initial and terminal states are known.