计算随机系统中瞬子附近涨落的Gel'fand-Yaglom型方程
Gel'fand-Yaglom type equations for calculating fluctuations around Instantons in stochastic systems
摘要：
近年来，瞬子演算已成功地应用于各种随机动力系统中稀有事件的尾概率估计。然而，在没有进一步校正的情况下，这些估计只能捕获指数缩放。在本文中，我们推导出一个一般的，封闭形式的表达式的领先的前因子的贡献的波动周围的瞬子轨迹的概率密度函数的计算一般可观。其关键技术是将Gel'fand-Yaglom递推计算方法应用于波动的适当离散的高斯路径积分，以获得产生波动行列式的矩阵演化方程。我们证明了这些预测和直接采样的例子激励湍流理论之间的协议。
英文摘要：
In recent years, instanton calculus has successfully been employed to estimate tail probabilities of rare events in various stochastic dynamical systems. Without further corrections, however, these estimates can only capture the exponential scaling. In this paper, we derive a general, closed form expression for the leading prefactor contribution of the fluctuations around the instanton trajectory for the computation of probability density functions of general observables. The key technique is applying the Gel'fand-Yaglom recursive evaluation method to the suitably discretized Gaussian path integral of the fluctuations, in order to obtain matrix evolution equations that yield the fluctuation determinant. We demonstrate agreement between these predictions and direct sampling for examples motivated from turbulence theory.
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