The random batch method provides an efficient algorithm for computing statistical properties of a canonical ensemble of interacting particles. In this work, we study the error estimates of the fully discrete random batch method, especially in terms of approximating the invariant distribution. Using a triangle inequality framework, we show that the long-time error of the method is $O(\sqrt{\tau} + e^{-\lambda t})$, where $\tau$ is the time step and $\lambda$ is the convergence rate which does not depend on the time step $\tau$ or the number of particles $N$. Our results also apply to the McKean-Vlasov process, which is the mean-field limit of the interacting particle system as the number of particles $N\rightarrow\infty$.
翻译：随机批量方法为计算一个交互式粒子共集的统计属性提供了一个高效的算法。 在这项工作中, 我们研究完全离散随机批量方法的错误估计, 特别是接近变量分布的错误估计。 我们使用三角不平等框架, 显示该方法的长期错误是$O (\\ sqrt\tau} + e ⁇ \\\\\ lumbda t}$, $tau$ 是时间步骤, $\lambda$ 是不取决于时间步骤$\tau$ 或粒子数量$ $的趋同率。 我们的结果也适用于麦肯- Vlasov 进程, 这是互动粒子系统的平均值限制 $N\rightrowleinfty$ 。
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