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.

近年来，瞬子微积分已成功地用于估计各种随机动力系统中罕见事件的尾部概率。然而，如果没有进一步的修正，这些估计只能捕捉指数缩放。在本文中，我们推导出了一个通用的封闭形式表达式，用于计算一般可观测量的概率密度函数的瞬时子轨迹周围波动的主要前因子贡献。关键技术是将 Gel'fand-Yaglom 递归评估方法应用于适当离散化的波动高斯路径积分，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与来自湍流理论的示例的直接采样之间的一致性。