噪声诱导同步的样本路径方法：双势阱中的随机共振
A sample-paths approach to noise-induced synchronization: Stochastic resonance in a double-well potential
摘要：
加性白噪声可以显著地增加MEMS系统对周期性驱动信号的响应。我们考虑两类双阱势，对称和非对称，周期调制的时间周期$1/\eps$，其中$\eps$是一个适度（而不是指数）的小参数。我们表明，当噪声强度$σ$跨越一个门限值时，系统的响应急剧变化。在阈值以下，路径集中在一个势阱附近，并且具有指数小的概率跳到另一个势阱。在阈值以上，阱之间的转变在对称情况下以指数接近1/2的概率发生，并且在非对称情况下以指数接近1的概率发生。过渡区在时间上定位在最小势垒高度的点附近。我们给出了一个数学上严格的描述个别路径的行为，这使我们，特别是要确定的幂律依赖的临界噪声强度上$\eps$和最小势垒高度，以及渐近的过渡和非转移概率。
英文摘要：
Additive white noise may significantly increase the response of bistable systems to a periodic driving signal. We consider two classes of double-well potentials, symmetric and asymmetric, modulated periodically in time with period $1/\eps$, where $\eps$ is a moderately (not exponentially) small parameter. We show that the response of the system changes drastically when the noise intensity $σ$ crosses a threshold value. Below the threshold, paths are concentrated near one potential well, and have an exponentially small probability to jump to the other well. Above the threshold, transitions between the wells occur with probability exponentially close to 1/2 in the symmetric case, and exponentially close to 1 in the asymmetric case. The transition zones are localised in time near the points of minimal barrier height. We give a mathematically rigorous description of the behaviour of individual paths, which allows us, in particular, to determine the power-law dependence of the critical noise intensity on $\eps$ and on the minimal barrier height, as well as the asymptotics of the transition and non-transition probabilities.
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