Mean field error estimate of the random batch method for large interacting particle system

The random batch method (RBM) proposed in [Jin et al., J. Comput. Phys., 400(2020), 108877] for large interacting particle systems is an efficient with linear complexity in particle numbers and highly scalable algorithm for N-particle interacting systems and their mean-field limits when N is large. We consider in this work the quantitative error estimate of RBM toward its mean-field limit, the Fokker-Planck equation. Under mild assumptions, we obtain a uniform-in-time O(τ² + 1/N) bound on the scaled relative entropy between the joint law of the random batch particles and the tensorized law at the mean-field limit, where τ is the time step size and N is the number of particles. Therefore, we improve the existing rate in discretization step size from O(√τ) to O(τ) in terms of the Wasserstein distance.