Freidlin-Wentzell theory of large deviations can be used to compute the likelihood of extreme or rare events in stochastic dynamical systems via the solution of an optimization problem. The approach gives exponential estimates that often need to be refined via calculation of a prefactor. Here it is shown how to perform these computations in practice. Specifically, sharp asymptotic estimates are derived for expectations, probabilities, and mean first passage times in a form that is geared towards numerical purposes: they require solving well-posed matrix Riccati equations involving the minimizer of the Freidlin-Wentzell action as input, either forward or backward in time with appropriate initial or final conditions tailored to the estimate at hand. The usefulness of our approach is illustrated on several examples. In particular, invariant measure probabilities and mean first passage times are calculated in models involving stochastic partial differential equations of reaction-advection-diffusion type.

中文翻译：

---

小噪声随机系统中期望、概率和平均首次通过时间的锐渐近估计

Freidlin-Wentzell 大偏差理论可用于通过优化问题的求解来计算随机动力系统中极端或罕见事件的可能性。该方法给出了指数估计，通常需要通过计算前因子来完善。这里展示了如何在实践中执行这些计算。具体来说，以一种面向数值目的的形式导出期望、概率和平均首次通过时间的尖锐渐近估计：它们需要求解适定矩阵 Riccati 方程，其中涉及 Freidlin-Wentzell 作用的最小值作为输入，或者向前或根据当前的估计调整适当的初始或最终条件。我们的方法的实用性通过几个例子来说明。特别地，在涉及反应-平流-扩散类型的随机偏微分方程的模型中计算不变测量概率和平均首次通过时间。