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

Abstract: We study the geometric ergodicity and the long time behavior of the Random Batch Method 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 or the batch size $p$. Moreover, the Wasserstein distance between the invariant distributions of the IPS and the RB-IPS is bounded by $O(\sqrt{\tau})$, showing that the RB-IPS can be used to sample the invariant distribution of the IPS accurately with greatly reduced computational cost.