In this work, we present the correction formulas of the $k$-step BDF convolution quadrature at the starting $k-1$ steps for the fractional Feynman-Kac equation with L\'{e}vy flight. The desired $k$th-order convergence rate can be achieved with nonsmooth data. Based on the idea of [{\sc Jin, Li, and Zhou}, SIAM J. Sci. Comput., 39 (2017), A3129--A3152], we provide a detailed convergence analysis for the correction BDF$k$ scheme. The numerical experiments with spectral method are given to illustrate the effectiveness of the presented method. To the best of our knowledge, this is the first proof of the convergence analysis and numerical verified the sapce fractional evolution equation with correction BDF$k$.

中文翻译：

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用 L\'{e}vy 飞行修正分数阶 Feynman-Kac 方程的 BDFk

在这项工作中，我们提出了具有 L\'{e}vy 飞行的分数 Feynman-Kac 方程在起始 $k-1$ 步的 $k$-step BDF 卷积正交校正公式。使用非平滑数据可以达到所需的 $k$th 阶收敛速度。基于 [{\sc Jin, Li, and Zhou} 的思想，SIAM J. Sci. Comput., 39 (2017), A3129--A3152]，我们为修正 BDF$k$ 方案提供了详细的收敛分析。给出了光谱法的数值实验来说明所提出方法的有效性。据我们所知，这是收敛性分析的第一个证明，数值验证了带修正 BDF$k$ 的空间分数演化方程。