Ergodicity and long-time behavior of the Random Batch Method for interacting particle systems

Shi Jin, Lei Li, Xuda Ye, Zhennan Zhou

Published 2022-02-10Version 1

We study the geometric ergodicity and the long time behavior of the Random Batch Method [Jin et al., J. Comput. Phys., 400(1), 2020] for interacting particle systems, which exhibits superior numerical performance in recent large-scale scientific computing experiments. We show that for both the interacting particle system (IPS) and the random batch interacting particle system (RB-IPS), the distribution laws converge to their respective invariant distributions exponentially, and the convergence rate does not depend on the number of particles $N$, the time step $\tau$ for batch divisions and the batch size $p$. Moreover, the Wasserstein distance between $\tau$, showing that the RB-IPS can be used to sample the invariant distribution of the IPS accurately with greatly the invariant distributions of the IPS and the RB-IPS is bounded by $O(\sqrt{\tau})$ reduced computational cost.