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Long Time Behavior and Stabilization for Displacement Monotone Mean Field Games

Marco Cirant, Alpár R. Mészáros

TL;DR

This work advances the understanding of long-time behavior in mean field games by proving exponential stabilization for displacement-monotone, non-separable MFGs on unbounded state spaces, valid for both deterministic and stochastic settings. It leverages Pontryagin's maximum principle and forward–backward SDEs to characterize Nash equilibria without relying on a stationary state, and shows convergence to an infinite-horizon MFG with an ergodic constant $\lambda$. The authors establish uniform second-moment bounds, pointwise gradient control, and exponential decay in Wasserstein distance of the population measures, culminating in a rigorous passage to the infinite-horizon problem and a unique ergodic limit. Their framework accommodates generalized confining Hamiltonians and provides concrete mechanical and non-separable examples, broadening applicability to realistic multi-agent dynamics. The results yield a robust turnpike-type behavior with explicit decay rates, offering quantitative insights into stabilization mechanisms in large-population control problems.

Abstract

This paper is devoted to the study of the long time behavior of Nash equilibria in Mean Field Games within the framework of displacement monotonicity. We first show that any two equilibria defined on the time horizon $[0,T]$ must be close as $T \to \infty$, in a suitable sense, independently of initial/terminal conditions. The way this stability property is made quantitative involves the $L^2$ distance between solutions of the associated Pontryagin system of FBSDEs that characterizes the equilibria. Therefore, this implies in particular the stability in the 2-Wasserstein distance for the two flows of probability measures describing the agent population density and the $L^2$ distance between the co-states of agents, that are related to the optimal feedback controls. We then prove that the value function of a typical agent converges as $T \to \infty$, and we describe this limit via an infinite horizon MFG system, involving an ergodic constant. All of our convergence results hold true in a unified way for deterministic and idiosyncratic noise driven Mean Field Games, in the case of strongly displacement monotone non-separable Hamiltonians. All these are quantitative at exponential rates.

Long Time Behavior and Stabilization for Displacement Monotone Mean Field Games

TL;DR

This work advances the understanding of long-time behavior in mean field games by proving exponential stabilization for displacement-monotone, non-separable MFGs on unbounded state spaces, valid for both deterministic and stochastic settings. It leverages Pontryagin's maximum principle and forward–backward SDEs to characterize Nash equilibria without relying on a stationary state, and shows convergence to an infinite-horizon MFG with an ergodic constant . The authors establish uniform second-moment bounds, pointwise gradient control, and exponential decay in Wasserstein distance of the population measures, culminating in a rigorous passage to the infinite-horizon problem and a unique ergodic limit. Their framework accommodates generalized confining Hamiltonians and provides concrete mechanical and non-separable examples, broadening applicability to realistic multi-agent dynamics. The results yield a robust turnpike-type behavior with explicit decay rates, offering quantitative insights into stabilization mechanisms in large-population control problems.

Abstract

This paper is devoted to the study of the long time behavior of Nash equilibria in Mean Field Games within the framework of displacement monotonicity. We first show that any two equilibria defined on the time horizon must be close as , in a suitable sense, independently of initial/terminal conditions. The way this stability property is made quantitative involves the distance between solutions of the associated Pontryagin system of FBSDEs that characterizes the equilibria. Therefore, this implies in particular the stability in the 2-Wasserstein distance for the two flows of probability measures describing the agent population density and the distance between the co-states of agents, that are related to the optimal feedback controls. We then prove that the value function of a typical agent converges as , and we describe this limit via an infinite horizon MFG system, involving an ergodic constant. All of our convergence results hold true in a unified way for deterministic and idiosyncratic noise driven Mean Field Games, in the case of strongly displacement monotone non-separable Hamiltonians. All these are quantitative at exponential rates.

Paper Structure

This paper contains 17 sections, 31 theorems, 239 equations.

Table of Contents

  1. Introduction
  2. Technical setting and assumptions
  3. First set of assumptions
  4. Global second moment estimates along MFG Nash equilibria
  5. Second moment estimates
  6. Second set of standing assumptions
  7. Back to the moment estimates
  8. Examples of $H,g$ satisfying the 'generalized confining' assumptions
  9. Mechanical Hamiltonians
  10. General classes of Hamiltonians
  11. Localization arguments and global in time estimates on $D_xu$
  12. Long-time behavior of $\rho$ and $D_x u$
  13. Some preparatory results
  14. Decay results in time for $\rho$ and for $D_xu$
  15. Localization arguments for $D_x u$
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $H:{\mathbb R}^{d}\times{\mathbb R}^{d}\times{\mathscr P}_{2}({\mathbb R}^{d})\to{\mathbb R}$ be displacement $c_{0}$-monotone with $c_{0}>0$ and suppose that it satisfies our standing assumptions. Let $\left(u^{1,T}_{s},\rho^{1,T}_{s}\right)_{s\in[0,T]}$ and $\left(u^{2,T}_{s},\rho^{2,T}_{s}\ri and, if we assume in addition that $\rho^1_0 = \rho^2_0$,

Theorems & Definitions (78)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Remark 3.6
  • ...and 68 more