The numerical evaluation of statistics plays a crucial role in statistical physics and its applied fields. It is possible to evaluate the statistics for a stochastic differential equation with Gaussian white noise via the corresponding backward Kolmogorov equation. The important notice is that there is no need to obtain the solution of the backward Kolmogorov equation on the whole domain; it is enough to evaluate a value of the solution at a certain point that corresponds to the initial coordinate for the stochastic differential equation. For this aim, an algorithm based on combinatorics has recently been developed. In this paper, we discuss a higher-order approximation of resolvent, and an algorithm based on a second-order approximation is proposed. The proposed algorithm shows a second-order convergence. Furthermore, the convergence property of the naive algorithms naturally leads to extrapolation methods; they work well to calculate a more accurate value with fewer computational costs. The proposed method is demonstrated with the Ornstein-Uhlenbeck process and the noisy van der Pol system.

翻译：统计量的数值计算在统计物理及其应用领域中起着至关重要的作用。通过高斯白噪声驱动的随机微分方程对应的向后科尔莫戈罗夫方程，可以评估其统计量。重要的一点是，无需获得整个定义域上向后科尔莫戈罗夫方程的解，只需评估解在对应于随机微分方程初始坐标的某一点处的值即可。为此，最近发展出基于组合数学的算法。本文讨论了预解算子的高阶逼近，并提出一种基于二阶逼近的算法。该算法具有二阶收敛性。此外，朴素算法的收敛特性自然引出了外推方法；这些方法能以更少的计算成本获得更精确的值。通过奥恩斯坦-乌伦贝克过程和噪声范德波尔系统验证了所提方法的有效性。