Error Analysis of Time-Discrete Random Batch Method for Interacting Particle Systems and Associated Mean-Field Limits

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 $O(\sqrtτ + e^{-λt})$, where $τ$ is the time step and $λ$ is the convergence rate which does not depend on the time step $τ$ 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 $N\rightarrow\infty$.

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