Propagation of Chaos for Nonlinear Markov Chains

TL;DR

This paper develops quantitative methods to analyze the propagation of chaos in nonlinear Markov chains and their particle systems, with applications to Euler schemes and particle filtering, under various regularity conditions.

Contribution

It provides a systematic, nonasymptotic framework for propagation of chaos using Lipschitz regularity, extending to uniform-in-time results and specific applications.

Findings

Quantitative propagation of chaos estimates established.

Enhanced convergence rates under stronger assumptions.

Applications to Euler-Maruyama schemes and particle filtering.

Abstract

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.