由高斯过程平方驱动的随机演化的样本路径大偏差
Sample-path large deviations for stochastic evolutions driven by the square of a Gaussian process
AI总结：
本文研究快速高斯过程平方驱动的随机演化，借助 Fredholm 行列式与 Widom 定理得到显式样本路径大偏差泛函，并揭示噪声可诱发亚稳态及瞬子轨迹。
摘要：
近来出现了许多物理模型，它们由一个随机过程描述，其增量由快速高斯过程的二次型给出。我们发现，描述此类过程样本路径大偏差的速率函数，可以通过某个 Fredholm 行列式在大区域尺寸下的渐近行为计算。后者可利用 Widom 的一个定理进行解析求解，该定理将著名的 Szegő-Kac 公式推广到多维情形。这提供了一大类具有时间尺度分离的随机动力系统，对它们可以得到显式的样本路径大偏差泛函。受流体动力学和大气动力学问题启发，我们构造了一个简单例子：单个慢自由度由快速多元高斯过程的平方驱动，并利用我们的一般结果分析其大偏差泛函。尽管该例子的无噪声极限只有一个不动点，但相应的大偏差有效势具有多个不动点。换言之，正是噪声的加入导致了亚稳态。我们利用速率函数的显式结果构造连接这些亚稳态的瞬子轨迹。
英文摘要：
Recently, a number of physical models has emerged described by a random process with increments given by a quadratic form of a fast Gaussian process. We find that the rate function which describes sample-path large deviations for such a process can be computed from the large domain size asymptotic of a certain Fredholm determinant. The latter can be evaluated analytically using a theorem of Widom which generalizes the celebrated Szegő-Kac formula to the multi-dimensional case. This provides a large class of random dynamical systems with time scale separation for which an explicit sample-path large deviation functional can be found. Inspired by problems in hydrodynamics and atmosphere dynamics, we construct a simple example with a single slow degree of freedom driven by the square of a fast multi-variate Gaussian process and analyse its large deviations functional using our general results. Even though the noiseless limit of this example has a single fixed point, the corresponding large deviations effective potential has multiple fixed points. In other words, it is the addition of noise that leads to metastability. We use the explicit answers for the rate function to construct instanton trajectories connecting the metastable states.
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