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

Marco Cirant; Nicol\`o De Bernardi

arxiv: 2511.18923 · v2 · submitted 2025-11-24 · 🧮 math.AP · math.OC

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

Marco Cirant , Nicol\`o De Bernardi This is my paper

Pith reviewed 2026-05-17 05:48 UTC · model grok-4.3

classification 🧮 math.AP math.OC

keywords mean field gamesturnpike propertyquadratic Hamiltonianlocal stabilitystationary equilibriumexponential convergenceperiodic domainmean field control

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The pith

A local second-order positivity condition yields exponential turnpike for mean field games with quadratic Hamiltonians near stationary equilibria.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies long-time stability of solutions to mean field games and control systems with quadratic Hamiltonians around stationary equilibria. It replaces the usual global monotonicity requirement on the coupling with a weaker local assumption coming from a second-order strict positivity condition at the equilibrium. Together with a symmetry property, this local assumption produces an exponential turnpike property for nearby solutions on the flat torus or on Euclidean space. A fixed-point argument then shows that stable solutions exist on both finite and infinite horizons when the initial and terminal data lie close to the stationary equilibrium in the periodic case.

Core claim

We derive an exponential turnpike property for solutions to ergodic and discounted mean field games with quadratic Hamiltonians that are close to a stationary equilibrium satisfying a second-order strict positivity condition. This holds on the flat torus T^n and R^n. Moreover, we prove the existence of stable solutions on finite and infinite horizons in the periodic setting when initial and terminal data are sufficiently close to the stationary equilibrium.

What carries the argument

The local stability assumption derived from a second-order strict positivity condition on the stationary equilibrium, which replaces global monotonicity of the coupling term and enables the turnpike analysis.

If this is right

Solutions sufficiently close to the stationary equilibrium converge exponentially fast to it as the time horizon tends to infinity.
Existence of stable solutions on both finite and infinite horizons follows from a fixed-point argument when data are close to equilibrium.
The exponential turnpike holds on the torus and on R^n under the local assumption and symmetry.
The result applies to both ergodic and discounted mean field game systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The local viewpoint may allow treatment of problems with multiple stationary equilibria, each attracting its own basin of nearby solutions.
Numerical schemes for mean field games could exploit the local exponential decay to accelerate computation when data start near an equilibrium.
Similar local positivity conditions might be testable in related control problems beyond quadratic Hamiltonians.

Load-bearing premise

The stationary equilibrium satisfies a second-order strict positivity condition that provides local stability.

What would settle it

A numerical simulation or explicit example in which the second-order positivity condition fails at the equilibrium yet nearby solutions still fail to converge exponentially to it.

read the original abstract

We study the local stability properties of solutions to ergodic and discounted mean field games systems, as the time horizon $T \to +\infty$, 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, whenever the spatial domain $\Omega$ is either the flat torus $\mathbb{T}^n$ or $\mathbb{R}^n$. 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, in the periodic setting $\Omega=\mathbb{T}^n$, provided that the initial (and terminal) data are close enough to the stationary equilibrium.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper relaxes global monotonicity to a local second-order positivity condition at equilibrium and gets exponential turnpike plus existence for nearby solutions in quadratic-Hamiltonian mean field games on the torus or R^n.

read the letter

The main point is a local turnpike result under a weaker stability assumption than the usual global monotonicity on the coupling. With symmetry of the system, the authors obtain exponential convergence to the stationary equilibrium for solutions that remain close to it, both on the flat torus and on R^n. They then run a fixed-point argument to produce actual solutions that stay close when the initial and terminal data are sufficiently near the equilibrium, covering both finite-horizon and infinite-horizon cases in the periodic setting. The quadratic structure of the Hamiltonian is used throughout to simplify the estimates. This is a useful technical step because global monotonicity often fails in applications, and the local positivity condition at a (possibly non-unique) stationary point widens the range of couplings that can be treated. The existence result is obtained by standard contraction or fixed-point methods once the turnpike decay is in hand. The work is clearly aimed at specialists in mean-field games and long-time behavior of PDE systems. Readers who care about stability and turnpike properties in interacting-agent models will find the relaxed assumption and the symmetry-based argument worth seeing. The paper deserves a serious referee because the claim is new within the literature and rests on verifiable PDE estimates rather than hand-waving. One soft spot is that everything is local: the turnpike holds only for solutions already close to equilibrium, and the fixed-point map must keep the ball invariant. The nonlinear terms in the deviation system could in principle produce transient growth before decay takes over, especially on large but finite intervals; the paper must show that the local Hessian positivity plus the quadratic Hamiltonian closes the estimates without extra global Lipschitz control. If the proofs handle this carefully, the result stands; otherwise the scope narrows further. Overall I would send it to peer review with the expectation that referees focus on the details of the a-priori estimates and the fixed-point closure.

Referee Report

2 major / 2 minor

Summary. The paper establishes a local exponential turnpike property for solutions of ergodic and discounted mean-field games and control problems with quadratic Hamiltonians, around (possibly non-unique) stationary equilibria. It replaces global monotonicity of the coupling with a weaker local second-order strict positivity condition at equilibrium, combined with a symmetry property of the system, to obtain exponential decay to the stationary state for solutions that remain close to it (on the torus or R^n). Existence of such stable solutions on finite and infinite horizons is then proved via a fixed-point argument in the periodic setting, provided initial/terminal data are sufficiently close to equilibrium.

Significance. If the local stability and existence results hold, the work is significant for mean-field game theory: it weakens the standard global monotonicity assumption to a local condition at equilibrium, thereby extending turnpike analysis to settings where global monotonicity fails but local positivity holds. The combination of symmetry, local Hessian positivity, and fixed-point construction for both finite- and infinite-horizon problems adds a useful technical contribution to the PDE analysis of MFG systems.

major comments (2)

[turnpike derivation / stability section] The central turnpike derivation (likely §4 or the stability section following the local assumption): the second-order strict positivity controls only the linearization at equilibrium. For the nonlinear deviation system, the manuscript must supply explicit a-priori estimates showing that closeness to equilibrium is preserved uniformly in T and that nonlinear terms do not produce transient amplification before exponential decay dominates. Standard energy or Gronwall arguments on [0,T] with large T may fail to close without additional uniform Lipschitz control on the coupling or a quantitative closeness radius that is independent of T.
[fixed-point existence argument] Fixed-point construction for existence (§5 or the existence section): the map is required to send a small ball into itself on [0,T] and on the infinite horizon. This closure relies on the turnpike property already established for nearby solutions. If the turnpike estimates contain a gap for the nonlinear system, the contraction or invariance of the ball cannot be justified, particularly when T is large but finite.

minor comments (2)

[Introduction] Clarify at the outset how the symmetry property is used in conjunction with the local positivity condition; a short dedicated paragraph in the introduction would help.
[Introduction / literature review] Add a brief comparison paragraph with existing local monotonicity or local stability results in the MFG literature to situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We address each major comment below and will incorporate clarifications and additional details into the revised manuscript.

read point-by-point responses

Referee: The central turnpike derivation (likely §4 or the stability section following the local assumption): the second-order strict positivity controls only the linearization at equilibrium. For the nonlinear deviation system, the manuscript must supply explicit a-priori estimates showing that closeness to equilibrium is preserved uniformly in T and that nonlinear terms do not produce transient amplification before exponential decay dominates. Standard energy or Gronwall arguments on [0,T] with large T may fail to close without additional uniform Lipschitz control on the coupling or a quantitative closeness radius that is independent of T.

Authors: We appreciate this observation. In the stability analysis, after linearizing around the equilibrium, we consider the nonlinear terms as perturbations. By assuming the solution starts sufficiently close to the equilibrium (with the distance quantified by the local positivity constant), we apply a Gronwall inequality that incorporates the exponential decay from the linear part. The key is that the radius of the neighborhood is chosen independently of T, based on the local Hessian positivity and the Lipschitz constant of the coupling, ensuring that the nonlinear contributions remain smaller than the decaying linear terms uniformly in time. We will revise the manuscript to include an explicit statement of this uniform radius and a detailed bootstrap argument to prevent any potential transient growth. revision: yes
Referee: Fixed-point construction for existence (§5 or the existence section): the map is required to send a small ball into itself on [0,T] and on the infinite horizon. This closure relies on the turnpike property already established for nearby solutions. If the turnpike estimates contain a gap for the nonlinear system, the contraction or invariance of the ball cannot be justified, particularly when T is large but finite.

Authors: The fixed-point map is defined on a ball whose radius is selected small enough to lie within the region where the turnpike estimates hold, as established in the preceding stability section. Since the turnpike decay rate is uniform with respect to T (depending only on the local positivity), the estimates for the deviation remain valid for all T, including large finite horizons. We will add a precise reference to the stability theorem when proving the invariance of the ball, and include an estimate showing that the map indeed maps the ball to itself without relying on circular reasoning. This should resolve the concern for both finite and infinite horizon cases. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via PDE estimates and external fixed-point theorems

full rationale

The paper introduces a local stability assumption from a second-order strict positivity condition at the (possibly non-unique) stationary equilibrium, combined with a symmetry property of the system. This is used to derive exponential turnpike estimates for solutions close to equilibrium on T^n or R^n, followed by a fixed-point argument to prove existence of stable solutions on finite and infinite horizons when initial/terminal data are sufficiently close. These steps rely on standard energy estimates, Gronwall-type arguments, and external mathematical theorems rather than self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citations. The central claims remain independent of the inputs by construction and are supported by rigorous PDE analysis without circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard PDE well-posedness for quadratic Hamiltonians plus the new local positivity condition; no free parameters or invented entities are introduced.

axioms (1)

domain assumption Second-order strict positivity condition on the stationary equilibrium
Invoked to replace global monotonicity and enable local stability analysis.

pith-pipeline@v0.9.0 · 5468 in / 1231 out tokens · 83283 ms · 2026-05-17T05:48:00.149999+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

the third one is way more delicate. This is a sort of second-order positivity condition... (S) can be also seen as the positivity of a certain 'principal eigenvalue' for the linearized stationary system
IndisputableMonolith/Foundation/BranchSelection.lean RCLCombiner_isCoupling_iff echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

the symmetry property(ii) is a symmetry property that is clear within the quadratic Hamiltonian framework by identity D¯u=-D¯m/¯m... (BB*)A*=A(BB*)
IndisputableMonolith/Cost.lean Jcost_pos_of_ne_one echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

−˙Φ(t)≥σ|Φ(t)|... exponential decay... turnpike property

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

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M. Cirant, A. Cosenza, and G. Verzini. Ergodic mean field games: existence of local minimizers up to the Sobolev critical case.Calc. Var. Partial Differential Equations, 63(5):Paper No. 134, 23, 2024

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Cirant and D

M. Cirant and D. Ghilli. Existence and non-existence for time-dependent mean field games with strong aggregation.Math. Ann., 383(3-4):1285–1318, 2022

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M. Cirant and A. Porretta. Long time behavior and turnpike solutions in mildly non-monotone mean field games.ESAIM Control Optim. Calc. Var., 27:Paper No. 86, 40, 2021

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N. Mimikos-Stamatopoulos and S. Munoz. Regularity and long time behavior of one- dimensional first-order mean field games and the planning problem.SIAM J. Math. Anal., 56(1):43–78, 2024

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E. Trélat and C. Zhang. Integral and measure-turnpike properties for infinite-dimensional opti- mal control systems.Math. Control Signals Systems, 30(1):Art. 3, 34, 2018. 38

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E. Trélat, C. Zhang, and E. Zuazua. Steady-state and periodic exponential turnpike property for optimal control problems in Hilbert spaces.SIAM J. Control Optim., 56(2):1222–1252, 2018

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E. Trélat and E. Zuazua. The turnpike property in finite-dimensional nonlinear optimal control. J. Differential Equations, 258(1):81–114, 2015

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[1] [1]

Bakry, F

D. Bakry, F. Barthe, P . Cattiaux, and A. Guillin. A simple proof of the Poincaré inequality for a large class of probability measures including the log-concave case.Electron. Commun. Probab., 13:60–66, 2008

work page 2008

[2] [2]

Bardi and H

M. Bardi and H. Kouhkouh. Long-time behavior of deterministic mean field games with non- monotone interactions.SIAM J. Math. Anal., 56(4):5079–5098, 2024

work page 2024

[3] [3]

Bayraktar and X

E. Bayraktar and X. Zhang. Solvability of infinite horizon McKean-Vlasov FBSDEs in mean field control problems and games.Appl. Math. Optim., 87(1):Paper No. 13, 26, 2023

work page 2023

[4] [4]

Berry, O

J. Berry, O. Ley, and F. J. Silva. Approximation and perturbations of stable solutions to a station- ary mean field game system.Journal de Mathématiques Pures et Appliquées, 194:103666, 2025

work page 2025

[5] [5]

V . I. Bogachev, N. V . Krylov, M. Röckner, and S. V . Shaposhnikov.Fokker-Planck-Kolmogorov equations, volume 207 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2015

work page 2015

[6] [6]

Briani and P

A. Briani and P . Cardaliaguet. Stable solutions in potential mean field game systems.Nonlinear Differential Equations and Applications NoDEA, 25(1):1, 2017

work page 2017

[7] [7]

Cardaliaguet, J.-M

P . Cardaliaguet, J.-M. Lasry, P .-L. Lions, and A. Porretta. Long time average of mean field games. Netw. Heterog. Media, 7(2):279–301, 2012

work page 2012

[8] [8]

Cardaliaguet, J.-M

P . Cardaliaguet, J.-M. Lasry, P .-L. Lions, and A. Porretta. Long time average of mean field games with a nonlocal coupling.SIAM J. Control Optim., 51(5):3558–3591, 2013

work page 2013

[9] [9]

Cardaliaguet and M

P . Cardaliaguet and M. Masoero. Weak KAM theory for potential MFG.J. Differential Equations, 268(7):3255–3298, 2020

work page 2020

[10] [10]

Cardaliaguet and A

P . Cardaliaguet and A. Porretta. Long time behavior of the master equation in mean field game theory.Anal. PDE, 12(6):1397–1453, 2019

work page 2019

[11] [11]

Carmona, Q

R. Carmona, Q. Cormier, and H. M. Soner. Synchronization in a Kuramoto mean field game. Comm. Partial Differential Equations, 48(9):1214–1244, 2023

work page 2023

[12] [12]

Cecchin, G

A. Cecchin, G. Conforti, A. Durmus, and K. Eichinger. The exponential turnpike phenomenon for mean field game systems: weakly monotone drifts and small interactions.arXiv:2409.09193, 2024

work page arXiv 2024

[13] [13]

Cesaroni and M

A. Cesaroni and M. Cirant. Concentration of ground states in stationary mean-field games systems.Anal. PDE, 12(3):737–787, 2019

work page 2019

[14] [14]

Cesaroni and M

A. Cesaroni and M. Cirant. Brake orbits and heteroclinic connections for first order mean field games.Trans. Amer. Math. Soc., 374(7):5037–5070, 2021

work page 2021

[15] [15]

Cesaroni and M

A. Cesaroni and M. Cirant. Stationary equilibria and their stability in a Kuramoto MFG with strong interaction.Comm. Partial Differential Equations, 49(1-2):121–147, 2024. 37

work page 2024

[16] [16]

M. Cirant. On the existence of oscillating solutions in non-monotone mean-field games.J. Differential Equations, 266(12):8067–8093, 2019

work page 2019

[17] [17]

Cirant, A

M. Cirant, A. Cosenza, and G. Verzini. Ergodic mean field games: existence of local minimizers up to the Sobolev critical case.Calc. Var. Partial Differential Equations, 63(5):Paper No. 134, 23, 2024

work page 2024

[18] [18]

Cirant and D

M. Cirant and D. Ghilli. Existence and non-existence for time-dependent mean field games with strong aggregation.Math. Ann., 383(3-4):1285–1318, 2022

work page 2022

[19] [19]

Cirant, F

M. Cirant, F. Kong, J. Wei, and X. Zeng. Critical mass phenomena and blow-up behaviors of ground states in stationary second order mean-field games systems with decreasing cost.J. Math. Pures Appl. (9), 198:Paper No. 103687, 60, 2025

work page 2025

[20] [21]

Cirant and A

M. Cirant and A. Porretta. Long time behavior and turnpike solutions in mildly non-monotone mean field games.ESAIM Control Optim. Calc. Var., 27:Paper No. 86, 40, 2021

work page 2021

[21] [22]

Gilbarg and N

D. Gilbarg and N. S. Trudinger.Elliptic partial differential equations of second order, volume 224. Springer, 1977

work page 1977

[22] [23]

F. Kong, Y. Tong, and X. Zeng. Local minimizers in second order mean-field games systems with choquard coupling, 2025. arXiv:2501.15362

work page arXiv 2025

[23] [24]

F. Kong, Y. Tong, X. Zeng, and H. Zhou. Blow-up behaviors of ground states in ergodic mean- field games systems with hartree-type coupling, 2024. arXiv:2412.02922

work page arXiv 2024

[24] [25]

O. A. Ladyˇ zenskaja, V . A. Solonnikov, and N. N. Uralceva.Linear and quasilinear equations of parabolic type, volume Vol. 23 ofTranslations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1968. Translated from the Russian by S. Smith

work page 1968

[25] [26]

Lasry and P .-L

J.-M. Lasry and P .-L. Lions. Jeux à champ moyen. II. Horizon fini et contrôle optimal.C. R. Math. Acad. Sci. Paris, 343(10):679–684, 2006

work page 2006

[26] [27]

G. M. Lieberman.Second order parabolic differential equations. World Scientific Publishing Co., Inc., River Edge, NJ, 1996

work page 1996

[27] [28]

A. R. Mészáros and F. J. Silva. On the variational formulation of some stationary second-order mean field games systems.SIAM J. Math. Anal., 50(1):1255–1277, 2018

work page 2018

[28] [29]

Mimikos-Stamatopoulos and S

N. Mimikos-Stamatopoulos and S. Munoz. Regularity and long time behavior of one- dimensional first-order mean field games and the planning problem.SIAM J. Math. Anal., 56(1):43–78, 2024

work page 2024

[29] [30]

Pellacci, A

B. Pellacci, A. Pistoia, G. Vaira, and G. Verzini. Normalized concentrating solutions to nonlinear elliptic problems.J. Differential Equations, 275:882–919, 2021

work page 2021

[30] [31]

Porretta and E

A. Porretta and E. Zuazua. Long time versus steady state optimal control.SIAM J. Control Optim., 51(6):4242–4273, 2013

work page 2013

[31] [32]

Trélat and C

E. Trélat and C. Zhang. Integral and measure-turnpike properties for infinite-dimensional opti- mal control systems.Math. Control Signals Systems, 30(1):Art. 3, 34, 2018. 38

work page 2018

[32] [33]

Trélat, C

E. Trélat, C. Zhang, and E. Zuazua. Steady-state and periodic exponential turnpike property for optimal control problems in Hilbert spaces.SIAM J. Control Optim., 56(2):1222–1252, 2018

work page 2018

[33] [34]

Trélat and E

E. Trélat and E. Zuazua. The turnpike property in finite-dimensional nonlinear optimal control. J. Differential Equations, 258(1):81–114, 2015

work page 2015

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Tr´ elat and E

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