Propagation of Chaos for Nonlinear Markov Chains

We study 1-Wasserstein propagation of chaos for “McKean-type” nonlinear Markov chains and their associated interacting particle systems. This paper is organized into two parts: the first part combines arguments from various areas of nonlinear Markov theory into a systematic treatment of quantitative, nonasymptotic empirical measure estimates and propagation of chaos, with Lipschitz regularity as the primary tool. We also study extensions to uniform-in-time propagation of chaos and improved convergence rates under stronger assumptions such as transportation inequalities, modified metrics, or geometric ergodicity. The second part of this work consists of two detailed applications of our results to specific systems of interest: an Euler-Maruyama scheme for the standard McKean-Vlasov diffusion, and particle filtering via Feynman-Kac distribution flows.

💡 Research Summary

This paper develops a comprehensive quantitative theory of propagation of chaos for discrete‑time “McKean‑type” nonlinear Markov chains and their associated interacting particle systems, measured in the 1‑Wasserstein metric. The authors begin by formalizing a class of nonlinear Markov kernels K_η(x,·) that depend on both the current state x and the law η of the process. Under the central assumption that the mapping η ↦ K_η is L‑Lipschitz on the space of probability measures equipped with the 1‑Wasserstein distance, the induced nonlinear transition operator Φ