The classical Fokker-Planck equation (FPE) is a key tool in physics for describing systems influenced by drag forces and Gaussian noise, with applications spanning multiple fields. We consider the fractional Fokker-Planck equation (FFPE), which models the time evolution of probability densities for systems driven by L\'evy processes, relevant in scenarios where Gaussian assumptions fail. The paper presents an efficient and accurate numerical approach for the free-space FFPE with constant coefficients and Dirac-delta initial conditions. This method utilizes the integral representation of the solutions and enables the efficient handling of very high-dimensional problems using fast algorithms. Our work is the first to present a high-precision numerical solver for the free-space FFPE with Dirac-delta initial conditions. This opens the door for future research on more complex scenarios, including those with variable coefficients and other types of initial conditions.

翻译：经典福克-普朗克方程是物理学中描述受阻力与高斯噪声影响系统的关键工具，其应用遍及多个领域。本文研究分数阶福克-普朗克方程，该方程模拟了由Lévy过程驱动系统的概率密度时间演化，适用于高斯假设失效的场景。本文针对常系数自由空间分数阶福克-普朗克方程提出一种高效精确的数值方法，该方法采用狄拉克-δ初始条件。本方法利用解的积分表示形式，通过快速算法实现对超高维问题的高效处理。本研究首次提出了针对狄拉克-δ初始条件的自由空间分数阶福克-普朗克方程高精度数值求解器，为未来研究更复杂场景（包括变系数及其他类型初始条件）奠定了基础。