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The local Turnpike Property in Mean Field Control and Games with quadratic Hamiltonian

Marco Cirant, Nicolò De Bernardi

TL;DR

This work addresses the local stability and turnpike behavior of mean field games with a quadratic Hamiltonian in a non-monotone setting. It identifies a stationary state $(\bar{u},\bar{m})$ satisfying a symmetry and a second-order positivity condition (S) that together yield a local exponential turnpike property for dynamic equilibria. By combining a perturbative linearization, a Lyapunov-type energy functional, and a nonlinear fixed-point argument, the authors prove the existence of stable finite-horizon solutions and, for small discount $\delta$, infinite-horizon equilibria that stay exponentially close to $(\bar{u},\bar{m})$. The approach extends turnpike phenomena beyond global uniqueness/monotonicity, leveraging probabilistic flow techniques, spectral-like positivity, and detailed $L^2$ and $L^{\infty}$ estimates on a torus, with potential extension to more general Hamiltonians and dissipative settings.

Abstract

We study the local stability properties of solutions to ergodic and discounted mean field games systems in the long time horizon, around stationary equilibria, when the Hamiltonian is quadratic. We replace the usual monotonicity of the coupling term with a weaker, local assumption on the stationary equilibrium (that need not be unique), stemming from a second-order strict positivity condition. This new stability assumption, together with a symmetry property of the system, allows us to derive an exponential turnpike property for those solutions that are close to the stationary one. Finally, through a fixed-point argument, we establish the actual existence of stable solutions, both on the finite horizon $[0,T]$ and on the infinite horizon, provided that the initial (and terminal) data are close enough to the stationary equilibrium.

The local Turnpike Property in Mean Field Control and Games with quadratic Hamiltonian

TL;DR

This work addresses the local stability and turnpike behavior of mean field games with a quadratic Hamiltonian in a non-monotone setting. It identifies a stationary state satisfying a symmetry and a second-order positivity condition (S) that together yield a local exponential turnpike property for dynamic equilibria. By combining a perturbative linearization, a Lyapunov-type energy functional, and a nonlinear fixed-point argument, the authors prove the existence of stable finite-horizon solutions and, for small discount , infinite-horizon equilibria that stay exponentially close to . The approach extends turnpike phenomena beyond global uniqueness/monotonicity, leveraging probabilistic flow techniques, spectral-like positivity, and detailed and estimates on a torus, with potential extension to more general Hamiltonians and dissipative settings.

Abstract

We study the local stability properties of solutions to ergodic and discounted mean field games systems in the long time horizon, around stationary equilibria, when the Hamiltonian is quadratic. We replace the usual monotonicity of the coupling term with a weaker, local assumption on the stationary equilibrium (that need not be unique), stemming from a second-order strict positivity condition. This new stability assumption, together with a symmetry property of the system, allows us to derive an exponential turnpike property for those solutions that are close to the stationary one. Finally, through a fixed-point argument, we establish the actual existence of stable solutions, both on the finite horizon and on the infinite horizon, provided that the initial (and terminal) data are close enough to the stationary equilibrium.

Paper Structure

This paper contains 8 sections, 19 theorems, 259 equations.

Table of Contents

  1. Introduction
  2. Discussion on the method
  3. The assumption (S)
  4. Final remarks
  5. $L^2$ turnpike a priori estimates
  6. $L^\infty$ turnpike a priori estimates
  7. Fixed point theorem and existence of solutions
  8. Appendix

Key Result

Theorem 1.1

Assume that (S) holds. Let $(u,m)$ and $(\bar{u},\bar{m})$ be (classical) solutions to dynsys and statsys respectively, such that satisfy the integrability conditions integrab. Then, there exist $\bar{\varepsilon} > 0, \bar{\lambda} > 0, \sigma > 0$ (which depend only on $\bar{m}, \bar{u}, \eta, f$, but not on the time horizon $T$), such that the following is true for all $\lambda \le \bar{\lambd

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 1.3: Equivalent formulations of (S)
  • proof
  • Remark 1.4: The principal eigenvalue
  • Remark 1.5: The Lasry-Lions monotone case
  • Lemma 1.6
  • proof
  • Remark 1.7
  • Remark 1.8
  • ...and 37 more