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Propagation of Chaos for Nonlinear Markov Chains

James Vuckovic

TL;DR

This work develops a general, quantitative theory of propagation of chaos for discrete-time McKean-type nonlinear Markov chains and their interacting-particle approximations under Lipschitz regularity of the nonlinear kernels. The authors derive nonasymptotic $1$-Wasserstein bounds that bound the distance between the empirical, interacting system and the mean-field limit, with explicit dependence on the number of particles $N$, the order $q$ of the marginal, the dimension $d$, and moment controls; they further extend to uniform-in-time results via modified metrics and geometric ergodicity. The framework is then applied to two key applications: (i) an Euler–Maruyama discretization of McKean–Vlasov SDEs in a uniformly convex setting, and (ii) nonlinear filtering via Feynman–Kac distribution flows, including SIS-R type filters, where kernel regularity and moment bounds yield propagation of chaos with explicit rates. The results illuminate how Lipschitz regularity and moment control suffice to obtain robust, nonasymptotic chaos estimates across a broad class of nonlinear Markov systems, including particle filters and mean-field MCMC schemes, and they discuss how transportation inequalities can improve the rates. Overall, the work provides a versatile, algebraic route to chaos bounds that connect nonlinear kernel regularity to practical particle-approximation guarantees with potential uniform-in-time applicability.

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.

Propagation of Chaos for Nonlinear Markov Chains

TL;DR

This work develops a general, quantitative theory of propagation of chaos for discrete-time McKean-type nonlinear Markov chains and their interacting-particle approximations under Lipschitz regularity of the nonlinear kernels. The authors derive nonasymptotic -Wasserstein bounds that bound the distance between the empirical, interacting system and the mean-field limit, with explicit dependence on the number of particles , the order of the marginal, the dimension , and moment controls; they further extend to uniform-in-time results via modified metrics and geometric ergodicity. The framework is then applied to two key applications: (i) an Euler–Maruyama discretization of McKean–Vlasov SDEs in a uniformly convex setting, and (ii) nonlinear filtering via Feynman–Kac distribution flows, including SIS-R type filters, where kernel regularity and moment bounds yield propagation of chaos with explicit rates. The results illuminate how Lipschitz regularity and moment control suffice to obtain robust, nonasymptotic chaos estimates across a broad class of nonlinear Markov systems, including particle filters and mean-field MCMC schemes, and they discuss how transportation inequalities can improve the rates. Overall, the work provides a versatile, algebraic route to chaos bounds that connect nonlinear kernel regularity to practical particle-approximation guarantees with potential uniform-in-time applicability.

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.
Paper Structure (35 sections, 33 theorems, 173 equations)

This paper contains 35 sections, 33 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. Applications
  4. Related Work
  5. Continuous Time
  6. Discrete Time
  7. Plan of the Paper
  8. Propagation of Chaos
  9. Preliminaries
  10. Notation
  11. Basic Constructions
  12. Main Result
  13. A Quantitative Empirical Measure Estimate
  14. Propagation of Chaos
  15. Sharpness and Rates
  16. ...and 20 more sections

Key Result

Lemma 1

\newlabellem:emp_mom0 Suppose that $(X^1_n, \dots, X^N_n)$ follows the interacting particle dynamics eq:ips, and $\eta^N_n$ is the empirical measure over the $N$ particles. Let $p\in [1,\infty)$; we have where the equalities are understood in $[0,\infty]$.

Theorems & Definitions (65)

  • Lemma 1
  • Proof 1
  • Proposition 2
  • Proof 2
  • Proposition 3
  • Proof 3
  • Remark 2.1
  • Lemma 3
  • Theorem 4
  • Proof 4
  • ...and 55 more