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Uniform-in-Time Convergence Rates to a Nonlinear Markov Chain for Mean-Field Interacting Jump Processes

Asaf Cohen, Ethan Huffman

TL;DR

This work establishes a uniform-in-time $1/N$ convergence rate for the empirical distribution of a finite-state, mean-field interacting particle system to the law of a nonlinear McKean–Vlasov Markov chain. It develops a master equation for the measure flow, proves regularity of its solutions, and shows that exponential stability of the linearized Kolmogorov equation implies stability of the full nonlinear system. The authors provide verifiable conditions for unique exponentially stable stationary distributions and demonstrate applicability to nonlinear Markov chains and ergodic mean-field games, including uniform-in-time propagation of chaos for $N$-player games with master-equation-based strategies. These results offer practical criteria for ergodicity and deliver robust convergence guarantees with implications for large-population stochastic games and their equilibria.

Abstract

We consider a system of $N$ particles interacting through their empirical distribution on a finite state space in continuous time. In the formal limit as $N\to\infty$, the system takes the form of a nonlinear (McKean--Vlasov) Markov chain. This paper rigorously establishes this limit. Specifically, under the assumption that the mean field system has a unique, exponentially stable stationary distribution, we show that the weak error between the empirical measures of the $N$-particle system and the law of the mean field system is of order $1/N$ uniformly in time. Our analysis makes use of a master equation for test functions evaluated along the measure flow of the mean field system, and we demonstrate that the solutions of this master equation are sufficiently regular. We then show that exponential stability of the mean field system is implied by exponential stability for solutions of the linearized Kolmogorov equation with a source term. Finally, we show that our results can be applied to the study of mean field games and give a new condition for the existence of a unique stationary distribution for a nonlinear Markov chain.

Uniform-in-Time Convergence Rates to a Nonlinear Markov Chain for Mean-Field Interacting Jump Processes

TL;DR

This work establishes a uniform-in-time convergence rate for the empirical distribution of a finite-state, mean-field interacting particle system to the law of a nonlinear McKean–Vlasov Markov chain. It develops a master equation for the measure flow, proves regularity of its solutions, and shows that exponential stability of the linearized Kolmogorov equation implies stability of the full nonlinear system. The authors provide verifiable conditions for unique exponentially stable stationary distributions and demonstrate applicability to nonlinear Markov chains and ergodic mean-field games, including uniform-in-time propagation of chaos for -player games with master-equation-based strategies. These results offer practical criteria for ergodicity and deliver robust convergence guarantees with implications for large-population stochastic games and their equilibria.

Abstract

We consider a system of particles interacting through their empirical distribution on a finite state space in continuous time. In the formal limit as , the system takes the form of a nonlinear (McKean--Vlasov) Markov chain. This paper rigorously establishes this limit. Specifically, under the assumption that the mean field system has a unique, exponentially stable stationary distribution, we show that the weak error between the empirical measures of the -particle system and the law of the mean field system is of order uniformly in time. Our analysis makes use of a master equation for test functions evaluated along the measure flow of the mean field system, and we demonstrate that the solutions of this master equation are sufficiently regular. We then show that exponential stability of the mean field system is implied by exponential stability for solutions of the linearized Kolmogorov equation with a source term. Finally, we show that our results can be applied to the study of mean field games and give a new condition for the existence of a unique stationary distribution for a nonlinear Markov chain.

Paper Structure

This paper contains 20 sections, 13 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Contributions
  4. Notation
  5. The Model and the Main Result
  6. Finite State System Setup
  7. Assumptions
  8. Main Result and the Master Equation
  9. Proof of the Main Result
  10. Proof of the properties of the master equation
  11. Proof of the bound on the first term
  12. Proof of the bound on the second term
  13. Exponential Stability from the Linearized System
  14. Applications to Nonlinear Markov Chains and Mean Field Games
  15. Nonlinear Markov Chains
  16. ...and 5 more sections

Key Result

Theorem 2.7

Assume that $\alpha$ is in (D-Lip) and satisfies (Erg), and that for all $N \geq 1$, we have that the $\xi^i$ in eq:markov are independent with distribution $\mu_0: = \text{Law}\,(Y_0)$. Then, there exists a constant $C$, independent of $\mu_0$, such that for any $N \geq 1$ and $\Phi$ in (D-Lip), we

Theorems & Definitions (30)

  • Remark 2.2
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Theorem 2.7
  • Proposition 2.8
  • Proposition 2.9
  • Proposition 2.10
  • proof : Proof of \ref{['prop:master']}
  • Lemma 3.1
  • ...and 20 more