On Random Batch Methods (RBM) for interacting particle systems driven by Lévy processes

The paper proves uniform-in-time RBM–Lévy convergence with rate O(kappa^{1/2}) in W2 under finite second moment and in W1 under finite first moment, based on a synchronous coupling, a dissipativity margin λV−2LK>0, and a decomposition of the batching cross term into I1+I2+I3 where I1=0 by unbiasedness, I2 and I3 are controlled via Hölder–type estimates and moment bounds, yielding a Grönwall inequality for E|Z1(t)|^2 (or a regularized |·|_ε argument for W1) and thus sup_t W_p ≤ C kappa^{1/2} (Theorems 1–2) . The candidate model’s solution also uses a synchronous coupling and the same dissipativity, but proceeds on each batching interval [tm,tm+1) via a “frozen versus motion” decomposition B_i(t)=ζ_i^m+R_i(t) and a variation-of-constants estimate, deriving a discrete recursion E H(tm+1) ≤ e^{−2λkappa} E H(tm) + C1 kappa^2/(p−1) + C2 kappa^2 and hence sup_t E|Z1(t)|^2 ≤ C kappa, giving the same O(kappa^{1/2}) Wasserstein rate. Thus, the core ideas agree (synchronous coupling, dissipativity, unbiased batching, and short-time increment control), but the technical scaffolding differs: the paper works with expectation-level decompositions I1–I3 and a regularized norm for the W1-case, while the model uses a pathwise Duhamel kernel and variance bounds for the frozen sampler. Both arguments are sound under the paper’s assumptions and deliver the same rate and uniform-in-time bounds .