In many real-world scenarios, the underlying random fluctuations are non-Gaussian, particularly in contexts where heavy-tailed data distributions arise. A typical example of such non-Gaussian phenomena calls for L\'evy noise, which accommodates jumps and extreme variations. We propose the Random Batch Method for interacting particle systems driven by L\'evy noises (RBM-L\'evy), which can be viewed as an extension of the original RBM algorithm in [Jin et al. J Compt Phys, 2020]. In our RBM-L\'evy algorithm, $N$ particles are randomly grouped into small batches of size $p$, and interactions occur only within each batch for a short time. Then one reshuffles the particles and continues to repeat this shuffle-and-interact process. In other words, by replacing the strong interacting force by the weak interacting force, RBM-L\'evy dramatically reduces the computational cost from $O(N^2)$ to $O(pN)$ per time step. Meanwhile, the resulting dynamics converges to the original interacting particle system, even at the appearance of the L\'evy jump. We rigorously prove this convergence in Wasserstein distance, assuming either a finite or infinite second moment of the L\'evy measure. Some numerical examples are given to verify our convergence rate.

翻译：在许多现实场景中，底层随机波动具有非高斯特性，特别是在出现重尾数据分布的背景下。这类非高斯现象的典型实例需要采用Lévy噪声，该噪声能够容纳跳跃和极端变异。我们提出了适用于Lévy噪声驱动的相互作用粒子系统的随机批处理方法（RBM-Lévy），可将其视为原始RBM算法[Jin et al. J Compt Phys, 2020]的扩展。在我们的RBM-Lévy算法中，$N$个粒子被随机分组为规模为$p$的小批次，相互作用仅在每个批次内短时发生。随后对粒子进行重新洗牌，并持续重复这种洗牌-相互作用过程。换言之，通过用弱相互作用力替代强相互作用力，RBM-Lévy将每个时间步的计算成本从$O(N^2)$显著降低至$O(pN)$。同时，即使在出现Lévy跳跃的情况下，所得动力学仍收敛于原始相互作用粒子系统。我们严格证明了在Lévy测度具有有限或无限二阶矩的假设下，该算法在Wasserstein距离意义下的收敛性。文中给出了若干数值算例以验证我们的收敛速率。