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Econ-26 · economics · Economics

Minimizer selection by the uncorrected vanishing-common-noise game

LLM status Solved unverified

Statement

GT-085 | Mean-field games | P: published question with an essential regularization boundary

Definitions and mathematical statement

Fix $E=\{1,\ldots,d\}$, $d\geq3$, $T>0$, $m_0\in\operatorname{int}\Delta_d$, where $\Delta_d=\{m\geq0:\sum_i m_i=1\}$. Let $\mathcal F,\mathcal G$ be $C^\infty$ potentials on a neighborhood of the simplex, with $f_i=\partial_i\mathcal F$, $g_i=\partial_i\mathcal G$. Write $r_+=\max(r,0)$ and \[ H_i(U)=\tfrac12\sum_{j\ne i}(U_i-U_j)_+^2. \] For $\varepsilon>0$, an admissible boundary repulsion is a smooth nonnegative function $\phi_\varepsilon$ with $\phi_\varepsilon\to0$ locally uniformly on $(0,1]$, for which the following penalized master equation has a unique classical solution $U^\varepsilon$. Require $U^\varepsilon$ to be continuous on $[0,T]\times\Delta_d$, $C^1$ in time and $C^2$ in the simplex interior, with bounded value and first derivatives and second derivatives bounded on interior compact sets; require the population SDE below to have a unique interior-valued solution. These are explicit admissibility conditions, ensured in the source by sufficiently strong repulsion at zero. Define \[ b_i^\varepsilon(m,U)=\sum_{j\ne i} \left[m_j\{\phi_\varepsilon(m_i)+(U_j-U_i)_+\} -m_i\{\phi_\varepsilon(m_j)+(U_i-U_j)_+\}\right]. \] The master equation, with the original costs $f,g$ kept unchanged, is \[ \begin{split} 0={}&\partial_tU_i^\varepsilon +\varepsilon\sum_{j,k}(m_j\delta_{jk}-m_jm_k)\partial_{m_jm_k}^2U_i^\varepsilon +\sum_j b_j^\varepsilon(m,U^\varepsilon)\partial_{m_j}U_i^\varepsilon\\ &+2\varepsilon\sum_jm_j(\partial_{m_i}U_i^\varepsilon-\partial_{m_j}U_i^\varepsilon) -H_i(U^\varepsilon)+f_i(m) +\sum_{j\ne i}\phi_\varepsilon(m_j)(U_j^\varepsilon-U_i^\varepsilon),\\ U_i^\varepsilon(T,m)={}&g_i(m). \end{split} \] $\delta_{jk}$ is the Kronecker delta. All derivatives are intrinsic tangent derivatives, equivalently derivatives of an extension; the tangent contractions displayed here are independent of the extension. In particular, this is a Kimura diffusion on the simplex, not an independent additive Brownian noise for each coordinate.

With independent Brownian motions $(W^{ij})_{i,j\in E}$, the selected population law solves \[ dm_i^\varepsilon=b_i^\varepsilon(m^\varepsilon,U^\varepsilon(t,m^\varepsilon))dt +\sqrt\varepsilon\sum_j\sqrt{m_i^\varepsilon m_j^\varepsilon}(dW^{ij}-dW^{ji}), \qquad m^\varepsilon(0)=m_0. \] Define $\mathcal M_{T,m_0}$ as the law paths in global minimizers of \[ \int_0^T\left[\tfrac12\sum_i\sum_{j\ne i}m_i a_{ij}^2+\mathcal F(m)\right]dt +\mathcal G(m(T)),\qquad \dot m_i=\sum_{j\ne i}(m_ja_{ji}-m_ia_{ij}),\quad a_{ij}\geq0, \] with $m(0)=m_0$, absolutely continuous $m\in\Delta_d$, and measurable finite-energy rates. For admissible repulsion families and convergent subsequences \[ \mathcal L(m^{\varepsilon_n})\Longrightarrow Q \quad\text{in }C([0,T],\Delta_d),\qquad \varepsilon_n\downarrow0, \] determine whether $Q(\mathcal M_{T,m_0})=1$. A full statement of any proposed theorem must specify its boundary-layer scale and the estimates ensuring tightness and removal of the repulsion in the limit.

Short English statement

Does the exact noisy game, with only the boundary repulsion and no extra cost correction, select deterministic potential minimizers as its common noise vanishes?

Variants and scope boundaries

The restriction to unchanged $f$ is essential. The source's Theorem 4.5 obtains minimizing selection after introducing additional costs $f^\varepsilon$ that restore a stochastic potential structure; that is a known positive result. Its approximate-equilibrium conclusion does not settle the exact uncorrected question. The finite-state Wright--Fisher formulation also uses a weighted representative-player interpretation, not an ordinary unweighted conditional jump law.

Source relationship and status

The uncorrected question appears explicitly before Theorem 4.5. The theorem's cost-corrected variant is already resolved in the cited source; its different quantifiers must be retained.

References

[S1] Cardaliaguet and Delarue. Selected topics in mean field games, ICM2022, vol. 5; DOI: 10.4171/ICM2022/17.

Locators: (4.14), p. 3686, penalized master equation; (4.15), Theorem 4.3, p. 3687; Definition 4.1, pp. 3687--3688; Section 4.3, "Selection of equilibria," p. 3690, exact uncorrected question; Theorem 4.5, pp. 3691--3692, cost-corrected positive result.

https://ems.press/content/book-chapter-files/33265

[S2] Alekos Cecchin and François Delarue. Selection by vanishing common noise for potential finite state mean field games. Communications in Partial Differential Equations 47(1) (2022), 89--168; DOI: 10.1080/03605302.2021.1955256.

Source role: primary paper for the corrected selection and intrinsic weak uniqueness results cited as [43] by the chapter; publication date differs from the chapter's early citation.

https://arxiv.org/abs/2005.12153

Common conventions: Source mathematical conventions

Finite games. For a finite set $A$, $\Delta(A)$ is its probability simplex. Players choose independent mixed strategies in Nash equilibrium; correlation is allowed only when explicitly part of the solution concept. In a game with payoff functions $u_i$ and strategy profile $x$, an additive $\varepsilon$ Nash equilibrium satisfies \[ u_i(x)\geq u_i(x_i',x_{-i})-\varepsilon \quad\text{for every player $i$ and every admissible deviation $x_i'$}. \] For numerical approximation, payoffs are normalized as locally specified. An $\varepsilon$ payoff residual is distinct from distance at most $\varepsilon$ to the set of exact equilibria. Point convergence, convergence of empirical play, and convergence in distance to an equilibrium set are also distinct.

Computational representations. Unless stated otherwise, a complexity question takes rational data with binary encoding; polynomial time is measured in total bit length. A fixed-accuracy polynomial algorithm is not necessarily polynomial in $1/\varepsilon$, and neither is necessarily polynomial in $\log(1/\varepsilon)$. Succinct games and value, demand, or sampling oracles must declare their interfaces. Exact real solutions require an explicit representation; an unspecified list of arbitrary real numbers is not a finite computational output. A claimed algorithmic barrier is meaningful only for its specified model and reductions.

Dynamic games. Histories, information, observation, and allowable randomization are part of the model. A stationary strategy depends only on the current state; a positional strategy in a deterministic graph game chooses one action at each controlled vertex. Nash equilibrium at an initial state and equilibrium after every history are different requirements. An expectation of a pathwise $\liminf$ payoff, a $\liminf$ of expected averages, a limiting expected average, and a uniform finite-horizon equilibrium are different criteria. None is silently substituted for another.

Allocation and mechanisms. A complete allocation assigns every item exactly once unless otherwise stated. Zero-valued goods, negative values, capacity constraints, feasibility, lotteries, and copies of goods affect fairness definitions and are specified locally. Dominant-strategy incentive compatibility, Bayesian incentive compatibility, universal truthfulness, and truthfulness in expectation impose different inequalities. Whenever an objective ratio has zero denominator, its convention is stated or those instances are excluded.

Infinite-dimensional models. Expectations, integrals, stochastic equations, and optimization objectives require their local regularity, integrability, filtrations, and admissible domains. A backward--forward mean-field system is a boundary-value problem; it is not an initial-value semiflow without an additional construction. Long-time, large-population, and vanishing-noise limits require an order of limits and a convergence topology. If a minimum need not be attained, the objective is its infimum or supremum and the approximation requirement is stated separately.

Statement licence: CC-BY-4.0 · Source file

Contributions 3

Econ-26-A2 · Exploration · LLM-generated · gpt 6 astra pro ·

GPT 6 Astra Pro attempt on Econ-26

Research attempt and adversarial verification

9 October 2026 Original-target status: UNRESOLVED

C73-25: Uncorrected vanishing-common-noise potential selection

Input: C73-25.tex.  Tools: web browsing [YES]; Python computation [YES].

1) FORMAL RESTATEMENT

Let $d\geq3$, $T>0$, $\Delta_d=\{m\in[0,1]^d:\sum_i m_i=1\}$, and $m_0\in\operatorname{int}\Delta_d$. Smooth potentials $\mathcal F,\mathcal G$ are defined on a neighborhood of the simplex; set $f_i=\partial_i\mathcal F$, $g_i=\partial_i\mathcal G$. Write \[ H_i(U)=\tfrac12\sum_{j\ne i}(U_i-U_j)_+^2, \quad b_i(m,U)=\sum_{j\ne i}[m_j(U_j-U_i)_+-m_i(U_i-U_j)_+], \] \[ R_i^\phi(m)=\sum_{j\ne i}[m_j\phi_\varepsilon(m_i)-m_i\phi_\varepsilon(m_j)], \quad L_\varepsilon=\varepsilon\sum_{j,k}(m_j\delta_{jk}-m_jm_k)\partial_{jk}^2. \] A family $\phi_\varepsilon$ is smooth, nonnegative and locally uniformly vanishing on $(0,1]$. It is admissible in this note precisely when the input's following PDE has a unique solution in its specified classical class and the associated SDE has a unique interior-valued solution: \begin{align*} 0={}&\partial_tU_i+L_\varepsilon U_i+(b+R^\phi)\cdot DU_i +2\varepsilon(e_i-m)\cdot DU_i-H_i(U)+f_i(m)\\ &+\sum_{j\ne i}\phi_\varepsilon(m_j)(U_j-U_i), \qquad U_i(T,m)=g_i(m),\\ dm_i^\varepsilon={}&[b_i(m^\varepsilon,U^\varepsilon)+R_i^\phi(m^\varepsilon)]dt +\sqrt\varepsilon\sum_j\sqrt{m_i^\varepsilon m_j^\varepsilon} (dW^{ij}-dW^{ji}),\qquad m^\varepsilon(0)=m_0. \end{align*} The $W^{ij}$ are independent Brownian motions. All displayed differential contractions are tangent to the simplex. The classical class means continuity on the closed time--simplex domain, $C^1$ time and $C^2$ interior space regularity, bounded values and first derivatives, and locally bounded interior second derivatives, separately for each $\varepsilon$. No bound uniform in $\varepsilon$ is presumed.

For measurable nonnegative rates $a_{ij}$ with $\int_0^T\sum_{i,j\ne i}m_i a_{ij}^2dt<\infty$ and absolutely continuous simplex paths satisfying \[ \dot m_i=\sum_{j\ne i}(m_ja_{ji}-m_ia_{ij}),\qquad m(0)=m_0, \] define \[ J(m,a)=\int_0^T\left[\tfrac12\sum_{i,j\ne i}m_i a_{ij}^2+ \mathcal F(m)\right]dt+\mathcal G(m(T)). \] Let $\mathcal M_{T,m_0}$ be the population paths occurring in global minimizers of $J$. The literal question quantifies over all the above data, every admissible repulsion family, and every sequence $\varepsilon_n\downarrow0$ with $\mathcal L(m^{\varepsilon_n})\Rightarrow Q$ in $C([0,T],\Delta_d)$: must $Q(\mathcal M_{T,m_0})=1$? The cost functions $f,g$ are not replaced by corrected noisy costs. No correction to that target is needed. Boundary layers, the factor two in the Wright--Fisher covariance, and global versus stationary optimality are stress points.

2) QUICK LITERATURE/CONTEXT CHECK

Cardaliaguet and Delarue [cardaliaguet-delarue], printed p. 3691, explicitly separate the uncorrected question from the cost-corrected selection theorem on p. 3692 (Theorem 4.5). Cecchin and Delarue [cecchin-delarue] is the primary paper for that tailored potential-preserving construction. The extra costs in that theorem cannot be deleted from its hypotheses. These page numbers were checked against rendered source pages; some locators in the attachment are displaced by one page. A targeted check on 9 October 2026 did not supply a theorem covering the exact universal uncorrected quantifiers here.

3) ATTACK PLAN

Type and tools. This is a singular-noise limit and variational selection problem. Useful tools are: direct substitution in the master equation (an exact solvable subclass); stopped SDE estimates (separate interior behavior from a boundary layer); quadratic variation and Doob's inequality (control common noise); logarithmic barriers (verify interior admissibility); vector maximum arguments (prove the zero solution is unique); kinetic-energy nonnegativity (identify global minimizers); explicit nonconvex potentials (test whether the deterministic MFG equations suffice).

Proof tracks. Prove a quantitative limit for zero costs with arbitrary admissible repulsion, then seek a potential-energy comparison for nonconstant costs. Disproof tracks. Find a nonminimizing deterministic equilibrium, test whether it is genuinely selected by the uncorrected noise, and push repulsion into a thin boundary layer. The first track succeeds for zero costs. The constructed nonminimizing equilibrium is a verified obstruction to one inference, but its noisy selection is not proved.

4) WORK

Theorem (Zero-cost selection for every admissible repulsion family): If $\mathcal F=\mathcal G=0$, then for every admissible repulsion family, \[ \sup_{0\leq t\leq T}\|m^\varepsilon(t)-m_0\|_\infty \longrightarrow0\quad\text{in probability}. \] Moreover $\mathcal M_{T,m_0}=\{t\mapsto m_0\}$. Consequently every subsequential path-law limit has mass one on global minimizing paths in this subclass.
Proof: The vector $U^\varepsilon\equiv0$ solves every term of the master equation and its terminal condition. Its derivatives vanish and it is in the required classical class. Admissible uniqueness therefore forces this solution, so the controlled drift $b$ is zero.

Set $\delta=\frac12\min_i m_{0,i}>0$ and $\tau_\varepsilon=\inf\{t:\min_i m_i^\varepsilon(t)\leq\delta\}$. Let $a_\varepsilon=\sup_{r\in[\delta,1]}\phi_\varepsilon(r)$, which tends to zero by the exact local-uniform assumption. Before $\tau_\varepsilon$, $|R_i^\phi|\leq d a_\varepsilon$; this follows by bounding separately its nonnegative incoming and outgoing sums. Write the stopped equation as \[ m_i^\varepsilon(t\wedge\tau_\varepsilon)-m_{0,i} =\int_0^{t\wedge\tau_\varepsilon}R_i^\phi(m^\varepsilon(s))ds+M_i(t). \] The antisymmetric Brownian representation gives \[ d\langle M_i\rangle_t =2\varepsilon m_i^\varepsilon(1-m_i^\varepsilon) \mathbf1_{\{t\leq\tau_\varepsilon\}}dt\leq\frac\varepsilon2dt. \] Doob's $L^2$ inequality ($\mathbb E\sup_{t\leq T}|M_t|^2\leq4\mathbb E|M_T|^2$ for a square-integrable martingale starting at zero) implies $\mathbb E\sup_{t\leq T}|M_i(t)|^2\leq2\varepsilon T$.

On $\{\tau_\varepsilon\leq T\}$ some coordinate changes by at least $\delta$. For small $\varepsilon$ with $dTa_\varepsilon\leq\delta/2$, the union bound and Markov's inequality therefore give \[ \Pr(\tau_\varepsilon\leq T)\leq \Pr\left(\max_i\sup_{t\leq T}|M_i(t)|\geq\delta/2\right) \leq\frac{8d\varepsilon T}{\delta^2}. \] For any fixed $\eta>0$, also impose $dTa_\varepsilon\leq\eta/2$. On the event of no exit the full equation equals the stopped equation, and hence \[ \Pr\left(\sup_{t\leq T}\|m^\varepsilon(t)-m_0\|_\infty>\eta\right) \leq\frac{8d\varepsilon T}{\delta^2} +\frac{8d\varepsilon T}{\eta^2}\longrightarrow0. \] This proves convergence on the entire continuous-path space, not just at one time. In particular it supplies tightness and uniquely identifies every limit without assuming it in this subclass.

The deterministic objective is nonnegative kinetic energy. The zero rate matrix and constant path attain value zero. Any minimizer must satisfy $m_i a_{ij}^2=0$ almost everywhere for every edge. This implies $m_i a_{ij}=0$ almost everywhere, including where $m_i=0$. The forward equation then gives $\dot m=0$ almost everywhere. Absolute continuity and the initial condition force $m(t)=m_0$. Thus the minimizing-path set is the stated singleton. ∎

Boundary-layer scale in the theorem.
No prescribed shrinking width is needed for this particular zero-cost result. The proof localizes at the fixed interior distance $\delta=\frac12\min_i m_{0,i}$ and uses only $a_\varepsilon\to0$ there. It neither assumes nor proves that $\phi_\varepsilon$ is globally small. The probability of ever seeing the uncontrolled boundary region is explicitly $O(\varepsilon)$ after the local drift becomes small. This localization does not extend automatically when $b(m,U^\varepsilon)$ can drive the population toward that region.
Proposition (A nonvacuous admissible family for the zero-cost theorem): For $\mathcal F=\mathcal G=0$, the family $\phi_\varepsilon(r)=2\varepsilon$ is admissible for every $T>0$ and every interior initial point. Its unique classical solution is zero. For its population law, \[ \mathbb E m_i^\varepsilon(t)=\frac1d+ \left(m_{0,i}-\frac1d\right)e^{-2d\varepsilon t}. \]
Proof: The repulsion drift is $2\varepsilon(1-dm_i)$. The SDE coefficients are locally Lipschitz in the simplex interior, preserve the sum of coordinates, and give a unique strong solution up to an exit from that interior. To exclude such an exit let $B(m)=-\sum_i\log m_i$. Using an ambient extension of this function in the tangent contractions, the generator of this population diffusion satisfies \[ (L_\varepsilon+R^\phi\cdot D)B =\varepsilon\left(\sum_i\frac1{m_i}-d\right) -2\varepsilon\sum_i\frac1{m_i}+2\varepsilon d^2 \leq2\varepsilon d^2. \] Stop at the first coordinate below $\zeta>0$. Itô's formula gives $\mathbb E B(m_{T\wedge\tau_\zeta})\leq B(m_0)+2\varepsilon d^2T$. Since $B\geq0$ on the simplex and $B\geq\log(1/\zeta)$ at an exit, \[ \Pr(\tau_\zeta\leq T)\leq \frac{B(m_0)+2\varepsilon d^2T}{\log(1/\zeta)}\longrightarrow0. \] Localization therefore extends the unique strong solution to all finite times and keeps it interior. Taking expectations in its bounded-coefficient SDE gives the scalar linear equation $\dot{\bar m}_i=2\varepsilon(1-d\bar m_i)$, proving the formula.

It remains to verify PDE uniqueness rather than assume it for this example. Let $V(\tau,m)=U(T-\tau,m)$ be any bounded classical solution with initial value zero, and let $M=\sup_{\tau,m,i}|V_i|$. Its forward equation is \[ \partial_\tau V_i=(L_\varepsilon+B_i\cdot D)V_i-H_i(V) +2\varepsilon\sum_{j\ne i}(V_j-V_i), \quad B_i=b(m,V)+2\varepsilon(\mathbf1-dm)+2\varepsilon(e_i-m). \] Here $\mathbf1$ is the vector with every coordinate equal to one. The vector $B_i$ is tangent. Direct calculation shows \[ (L_\varepsilon+B_i\cdot D)B(m) \leq 2M d(d-1)+\varepsilon(2d^2+d)=:C_0. \] Indeed, writing $a_{jk}=(V_j-V_k)_+$ gives $b\cdot DB=\sum_{j,k\ne j}a_{jk}(1-m_j/m_k)\leq\sum_{j,k\ne j}a_{jk}\leq2Md(d-1)$. The mutation and diffusion terms were just calculated, and the extra drift contributes $-2\varepsilon/m_i+2\varepsilon d$, which has the required upper bound.

Fix $\alpha>0$ and set $C=C_0+1$. If some component of $V_i-\alpha[B(m)+C\tau]$ had a positive maximum on $[0,T]\times\operatorname{int}\Delta_d$, it would occur at positive time and an interior spatial point: the barrier diverges at the boundary and the initial expression is negative. At that point $V_i$ is maximal among the components. Thus $-H_i(V)\leq0$ and the switching sum is nonpositive. The spatial gradient of the subtracted expression vanishes, its diffusion contraction is nonpositive, and its time derivative at a maximum over past times is nonnegative. The PDE instead bounds that derivative by $\alpha(C_0-C)<0$, a contradiction. Therefore $V_i\leq\alpha[B+C\tau]$ everywhere. Apply the same argument to $-V_i-\alpha[B+C\tau]$: now $V_i$ is a minimum component, so $H_i(V)=0$ and the negative switching sum is nonpositive. It follows that $-V_i\leq\alpha[B+C\tau]$. Let $\alpha\downarrow0$ at each interior point, and use continuity at the boundary to obtain $V=0$ everywhere.

The zero field exists and has all the required bounded derivatives. This proves uniqueness in the stipulated class, completes admissibility, and confirms that the preceding selection theorem is not vacuous. ∎

Proposition (A smooth potential MFG equilibrium need not minimize the potential): There is a three-state smooth potential game, with an interior initial law, having a classical deterministic equilibrium of action zero whose population path is not a global potential minimizer.
Proof: Take $T=1$, $m_0=(1/3,1/3,1/3)$, $\mathcal F=0$ and $\mathcal G(m)=-3(m_1-1/3)^2$. Then $g_1=-6(m_1-1/3)$ and $g_2=g_3=0$. The constant path $m=m_0$, values $u=0$ and rates $a=0$ satisfy $\dot u_i=H_i(u)-f_i(m)$, the forward equation, and $u(1)=g(m(1))$. Its potential objective is zero.

For a competing admissible control set $a_{21}=a_{31}=c=\log2$ and all other off-diagonal rates to zero. Then \[ m_2(t)=m_3(t)=\tfrac13e^{-ct},\qquad m_1(t)=1-\tfrac23e^{-ct}. \] At time one the law is $(2/3,1/6,1/6)$. Its energy is $c/6$ and its terminal potential is $-1/3$, giving $J=(\log2)/6-1/3<0$. For the last strict inequality, $\log2<1<2$, for example by integrating $1/x<1$ over $(1,2)$. Thus the stationary equilibrium path cannot minimize $J$. ∎

Computation.
The competitor's objective evaluates to approximately $-0.2178088032$ in the script; the exact proof uses the analytic formula. For the admissible constant-repulsion example with $d=3$, $m_0=(0.5,0.3,0.2)$ and $T=1$, it evaluates the exact expectation formula at $\varepsilon=0.1,0.01,0.001$, obtaining first-coordinate means approximately $0.424802,0.490294,0.499003$. This is a deterministic formula check, not a simulated proof of tightness or of general selection. 
\begin{lstlisting} for eps in [0.1, 0.01, 0.001]:     mean = [1/d + (x-1/d)*exp(-2*d*eps*T) for x in m0]     assert abs(sum(mean)-1) < 1e-12 competitor_J = log(2)/6 - 1/3 assert competitor_J < 0 \end{lstlisting}

5) VERIFICATION

The noise bracket contains the factor two generated by $dW^{ij}-dW^{ji}$; replacing it by independent coordinate Brownian noise would invalidate the proof. Localization uses only local repulsion convergence and a fixed interior initial law. The explicit family's PDE uniqueness and interior SDE existence were both checked, rather than treating the adjective "admissible" as a proof. The negative-potential competitor is finite energy and solves its forward equation, but it is not claimed to be a noisy limit. Accordingly it refutes only "every deterministic equilibrium is a global minimizer." It does not refute the input's selection claim. No corrected running costs have been introduced anywhere in the positive theorem.

\Needspace{8\baselineskip}

6) FINAL

LABEL: UNRESOLVED for arbitrary smooth potentials and all admissible repulsion families.

Strongest checked partial result. For zero potentials, the full selected population path converges in probability to the unique minimizing path for every admissible family, with an explicit stopped-path error estimate. The constant family $2\varepsilon$ is independently proved admissible. An explicit smooth three-state example proves that identifying a limit as a deterministic equilibrium would not, by itself, finish the general problem.

Exact first gap. No inequality has been proved forcing the potential objective of a general uncorrected noisy limit to equal the global infimum of $J$; deterministic equilibrium equations alone do not imply it.

Top three next moves. (1) Derive an exact uncorrected Itô energy identity and isolate every term lacking a potential sign. (2) Obtain uniform interior/tightness estimates for nonzero controlled drifts, stating the repulsion width and height explicitly. (3) Test the displayed concave terminal-potential example by solving its actual parabolic system and checking whether any admissible limit selects its nonminimizing constant equilibrium.

Minimal counterexample perspective. A three-state nonconvex potential with multiple deterministic equilibria is a natural test bed. A genuine counterexample must additionally supply an admissible repulsion family, prove convergence of the actual noisy law, and verify positive limiting mass on nonminimizers. Those additional claims have not been established here.

Reproducibility and scope.
This is one of eleven companion reports. The bundle includes verify.py and its executed results. Finite experiments supplement the proofs; they are not proofs of asymptotic claims. Literature coverage is targeted, and no novelty or formal proof-assistant certification is claimed.

References

  • [cardaliaguet-delarue] Pierre Cardaliaguet and François Delarue. Selected topics in mean field games. Proceedings of the International Congress of Mathematicians 2022, volume 5. DOI: 10.4171/ICM2022/17. Sections 4.3--4.4; especially printed pp. 3691--3693. https://ems.press/content/book-chapter-files/33265.
  • [cecchin-delarue] Alekos Cecchin and François Delarue. Selection by vanishing common noise for potential finite state mean field games. Communications in Partial Differential Equations 47(1) (2022), 89--168. DOI: 10.1080/03605302.2021.1955256. Preprint: arXiv:2005.12153. https://arxiv.org/abs/2005.12153.

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Econ-26-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-26

A stochastic action certificate for the uncorrected selection problem

Problem C73-25. Status: unresolved. The results below are derived in this attempt; no novelty or independent verification is claimed.

Target and restriction

The target concerns the exact finite-state common-noise master equation with the original potential costs unchanged. The corrected-cost minimizing-selection theorem does not answer it [CD, Section 4.3]. The primary paper explicitly introduces a correction to restore potential structure [CeD, Sections 2 and 4]. We derive that structural defect directly and prove a conditional minimizing-selection theorem with a specified boundary layer and an explicit missing feedback estimate.

Fix $d\ge3$, $T>0$, $m_0\in\operatorname{int}\Delta_d$, and smooth potentials $\mathcal F,\mathcal G$. Write $f_i=\partial_i\mathcal F$, $g_i=\partial_i\mathcal G$, $H_i(p)=\frac12\sum_{j\ne i}(p_i-p_j)_+^2$, and \[ b_i(m,p)=\sum_{j\ne i}[m_j(p_j-p_i)_+-m_i(p_i-p_j)_+],\quad A_{jk}(m)=m_j\delta_{jk}-m_jm_k. \] For a rate matrix $a$, $b_i(m,a)=\sum_{j\ne i}(m_ja_{ji}-m_i a_{ij})$ instead denotes its linear population drift. The deterministic action is \[ J(m,a)=\int_0^T\left[\frac12\sum_{i,j\ne i}m_i a_{ij}^2+\mathcal F(m)\right]{\mathrm d} t+\mathcal G(m(T)). \] Its admissible paths are absolutely continuous, start from $m_0$, obey $\dot m=b(m,a)$, and have finite weighted quadratic rate energy.

For clarity, the exact uncorrected regularized equation used below is \begin{align*} 0={}&\partial_tU_i+\varepsilon A:D^2U_i +(b(m,U)+r^\varepsilon(m))\cdot DU_i-H_i(U)+f_i(m)\\ &+2\varepsilon\sum_jm_j(\partial_iU_i-\partial_jU_i) +\sum_{j\ne i}\phi_\varepsilon(m_j)(U_j-U_i), \qquad U_i(T,m)=g_i(m), \end{align*} where $r_i^\varepsilon=\phi_\varepsilon(m_i)-m_i\sum_j\phi_\varepsilon(m_j)$. An admissible solution is continuous on the closed time--simplex cylinder, $C^1$ in time and $C^2$ in the interior, with bounded values and first derivatives for each fixed $\varepsilon$, and locally bounded second derivatives. Admissibility also requires the selected population SDE, with drift $b(m,U^\varepsilon)+r^\varepsilon$ and covariance $2\varepsilon A(m){\mathrm d} t$, to have a unique interior-valued solution. These are assumptions on the regularization, not existence claims proved here.

The precise diffusion mismatch

Proposition: Work in an interior region where the boundary repulsion vanishes. Suppose a smooth scalar function $V$ solves the stochastic planner equation \[ 0=V_t+\varepsilon\sum_{j,k}A_{jk}V_{jk}-\sum_jm_jH_j(DV)+\mathcal F(m). \] Let $U_i=\partial_iV$ for a fixed smooth extension. The component differences of the residual in the uncorrected game master equation are \[ R_i-R_d=\varepsilon(V_{ii}-V_{dd}),\quad i<d. \] This residual one-form need not be a gradient. For the polynomial $V(m)=m_1^2m_2$ in $d=3$, the diffusion-mismatch one-form on the chart $(m_1,m_2)$ is $2\varepsilon m_2{\mathrm d} m_1$, whose exterior derivative is $-2\varepsilon{\mathrm d} m_1\wedge{\mathrm d} m_2$.
Proof: Direct differentiation gives \[ \partial_i\left(\varepsilon A:D^2V\right) =\varepsilon A:D^2U_i+ \varepsilon\left(V_{ii}-2\sum_jm_jV_{ij}\right). \] Also $D_p\sum_jm_jH_j(p)=-b(m,p)$, as follows by differentiating the positive-part squares. Thus differentiating the scalar planner equation produces all the deterministic terms of the game equation, and the extra diffusion term just displayed. The game instead contains \[ 2\varepsilon\sum_jm_j(\partial_iU_i-\partial_jU_i) =2\varepsilon\left(V_{ii}-\sum_jm_jV_{ij}\right). \] Their difference is $\varepsilon V_{ii}$. The scalar equation is prescribed on the simplex, so take the tangent derivative $\partial_i-\partial_d$ of that equation; this yields exactly the asserted residual difference. All component representatives here use the fixed extension: an arbitrary population-dependent common shift of $U$ is not being declared a symmetry of the noisy equation. For the polynomial, $V_{11}=2m_2$, $V_{22}=V_{33}=0$; differentiation of the indicated one-form proves its nonzero curl. This polynomial illustrates the operator defect alone; it is not asserted to solve the scalar planner equation for arbitrary prescribed data. ∎

A factor $\varepsilon$ does not make this harmless without uniform Hessian control. Conversely, a nonzero finite-$\varepsilon$ curl does not prove failure of zero-noise minimizing selection. The proposition diagnoses a blocked argument, not a counterexample to the target.

Verification by a smooth deterministic value

Suppose, as an additional hypothesis, that a scalar $V\in C^1_tC^2_m([0,T]\times\Delta_d)$ satisfies \begin{equation}\label{wsref-0-48-4a-42} V_t-\sum_i m_iH_i(DV)+\mathcal F(m)=0,\qquad V(T,m)=\mathcal G(m), \end{equation} with bounded indicated derivatives. Tangent derivatives, or any fixed smooth extension, give the same formulas below. Set $p=DV$ and \[ \mathcal B(m,a,p)=\sum_{i,j\ne i}m_i\left[ \tfrac12a_{ij}^2+a_{ij}(p_j-p_i)+\tfrac12(p_i-p_j)_+^2\right]. \]
Lemma: For deterministic admissible paths, \[ J(m,a)-V(0,m_0)=\int_0^T\mathcal B(m,a,DV){\mathrm d} t\ge0. \] The feedback $a_{ij}=(p_i-p_j)_+$ attains equality, so $V(0,m_0)$ is the global minimum. For a continuous semimartingale population satisfying \[ {\mathrm d} m=b(m,a){\mathrm d} t+r{\mathrm d} t+{\mathrm d} M,\qquad {\mathrm d}\langle M_i,M_j\rangle_t=2\varepsilon A_{ij}(m_t){\mathrm d} t, \] with bounded rates and $m(0)=m_0$, its expected original action obeys the exact identity \begin{equation}\label{wsref-0-69-74-6f} {\mathbb{E}} J(m,a)-V(0,m_0) ={\mathbb{E}}\int_0^T\left[\mathcal B(m,a,DV)+DV\cdot r+\varepsilon A:D^2V\right]{\mathrm d} t. \end{equation} For $a_{ij}=(U_i-U_j)_+$ one further has \begin{equation}\label{wsref-0-73-71-75-61-72-65} 0\le\mathcal B(m,a,p)\le\frac12\sum_{i,j\ne i}m_i\big[(U_i-U_j)-(p_i-p_j)\big]^2. \end{equation}
Proof: The chain rule and (HJB) give the deterministic identity. For $q=p_i-p_j$, $\frac12a^2-aq+\frac12q_+^2=\frac12(a-q_+)^2+a(-q)_+\ge0$, with equality at $a=q_+$. The feedback is bounded and Lipschitz in $m$, since $DV$ is Lipschitz and positive part is Lipschitz. Its forward ODE therefore has a unique simplex-valued solution, yielding an admissible equality case.

For the stochastic identity, apply Itô's formula to $V(t,m_t)$, whose martingale part has zero expectation by boundedness of $DV$ and the quadratic variation. Substitute (HJB) and add the running costs to obtain (ito). Finally put $z=U_i-U_j$, $a=z_+$. If $q,z\ge0$ the scalar gap is $(z-q)^2/2$. If $q,z\le0$ it is zero. If $q\ge0\ge z$, it is $q^2/2\le(z-q)^2/2$. If $z\ge0\ge q$, it is $z^2/2-zq\le(z-q)^2/2$. Multiplying by $m_i$ and summing proves (square). ∎

An explicit boundary-layer removal estimate

Let $\phi_\varepsilon$ be any admissible repulsion family for the exact game satisfying additionally \begin{equation}\label{wsref-0-6c-61-79-65-72} 0\le\phi_\varepsilon\le\kappa,\qquad \phi_\varepsilon(r)=0\text{ for }r\ge\delta_\varepsilon,\qquad \delta_\varepsilon\downarrow0, \end{equation} where $\kappa$ is fixed. No rate relation between $\delta_\varepsilon$ and $\varepsilon$ is imposed beyond the source's admissibility; that admissibility includes existence and interior uniqueness for each $\varepsilon$. The estimate below does not assert admissibility of every cutoff with (layer).
Lemma: Suppose $0\le a_{ij}^\varepsilon\le M_*$ uniformly. The repelled population has \[ r_i^\varepsilon(m)=\phi_\varepsilon(m_i)-m_i\sum_j\phi_\varepsilon(m_j), \] and the common-noise martingale covariance of Lemma Lemma. Let $\mu=\min_i m_{0i}$, $C=(d-1)(M_*+\kappa)$ and $\delta_*=(\mu/2)e^{-CT}$. Then \[ {\mathbb{P}}\left(\min_{t\le T,i}m_i^\varepsilon(t)<\delta_*\right) \le\frac{32d\varepsilon T e^{2CT}}{\mu^2}. \] For all sufficiently small $\varepsilon$ with $\delta_\varepsilon<\delta_*$, ${\mathbb{E}}\int_0^T\|r^\varepsilon(m_t^\varepsilon)\|_1{\mathrm d} t=O(\varepsilon)$.
Proof: The total drift of coordinate $i$ is at least $-Cm_i$, because all incoming flows are nonnegative and its outgoing rate is at most $(d-1)(M_*+\kappa)$. Thus \[ m_i(t)\ge e^{-Ct}\left[m_{0i}+\int_0^te^{Cs}{\mathrm d} M_i(s)\right]. \] The bracket of $M_i$ is $2\varepsilon m_i(1-m_i){\mathrm d} t\le2\varepsilon{\mathrm d} t$. Doob's inequality gives ${\mathbb{E}}\sup_{t\le T}|\int_0^te^{Cs}{\mathrm d} M_i|^2\le8\varepsilon T e^{2CT}$. If these martingale suprema are at most $\mu/2$ for all $i$, every coordinate stays above $\delta_*$. Markov's inequality and a union bound give the claimed probability. On this good event all repulsions vanish for sufficiently small $\varepsilon$. On its complement, $\|r\|_1\le2\sum_i\phi_\varepsilon(m_i)\le2d\kappa$. Multiplication by $T$ and the probability bound proves the last assertion. ∎

A conditional selection theorem for unchanged costs

Theorem: Assume (HJB), (layer), and admissibility of the exact uncorrected master solutions $U^\varepsilon$. Suppose their rates $a^\varepsilon_{ij}=(U_i^\varepsilon-U_j^\varepsilon)_+$ are uniformly bounded by $M_*$, and along their population processes \begin{equation}\label{wsref-0-6d-69-73-73-69-6e-67} {\mathbb{E}}\int_0^T\sum_{i,j\ne i}m_i^\varepsilon \left[(U_i^\varepsilon-U_j^\varepsilon)(t,m_t^\varepsilon) -(V_i-V_j)(t,m_t^\varepsilon)\right]^2{\mathrm d} t\longrightarrow0. \end{equation} Then the population laws are tight in $C([0,T],\Delta_d)$, and every weak subsequential limit is supported on globally minimizing law paths of the original deterministic action $J$.
Proof: Lemma Lemma removes repulsion in expected $L^1$ time norm. The bounded Hessian in (HJB) gives $|\varepsilon A:D^2V|\le C_V\varepsilon$. Combining (ito), (square), and (missing) therefore gives ${\mathbb{E}} J(m^\varepsilon,a^\varepsilon)\to V(0,m_0)$.

Both original and repulsion drifts are uniformly bounded. The martingale has expected squared supremum $O(\varepsilon)$ in every coordinate. Thus $m^\varepsilon$ is the sum of a uniformly Lipschitz path starting at $m_0$ and a martingale tending to zero uniformly in probability. Arzelà--Ascoli and this approximation give tightness in the continuous-path space.

Here are details for passing the action to the limit without asserting that stochastic paths themselves are deterministic admissible paths. Adjoin the random occupation measures $\Lambda^\varepsilon({\mathrm d} t,{\mathrm d} a)={\mathrm d} t\,\delta_{a^\varepsilon(t)}({\mathrm d} a)$ on the compact set $[0,T]\times[0,M_*]^{d(d-1)}$. They have fixed first marginal ${\mathrm d} t$ and hence form a compact set. Take a joint convergent subsequence with $m^\varepsilon$, and realize it with almost sure uniform convergence of paths and weak convergence of occupation measures. Such a realization is available because both spaces are Polish. Extract a further subsequence so that the vanishing martingale and integrated repulsion also vanish almost surely. Write the limit measure as ${\mathrm d} t\,\pi_t({\mathrm d} a)$ and define $\bar a(t)=\int a\pi_t({\mathrm d} a)$. Passing to the integrated forward equations, using uniform path convergence and the continuity and linearity of $b(m,a)$ in $a$, gives $\dot m=b(m,\bar a)$ and $m(0)=m_0$.

The relaxed kinetic integral converges jointly because its integrand $\frac12\sum m_i(t)a_{ij}^2$ is continuous and bounded on the compact time-control domain after uniform path convergence. The potential terms converge as well. Jensen's inequality gives \[ J(m,\bar a)\le\lim J(m^\varepsilon,a^\varepsilon) \] on this realization. All actions are uniformly bounded, so expectations pass to the limit, giving ${\mathbb{E}} J(m,\bar a)\le V(0,m_0)$. The pair $(m,\bar a)$ is admissible, and Lemma Lemma gives $J(m,\bar a)\ge V(0,m_0)$ for every realization. Equality therefore holds almost surely. Its law paths belong to the original minimizing set, which proves the support statement. ∎

The theorem keeps $f,g$ unchanged and supplies both tightness and a boundary-layer estimate. Its unproved hypotheses are substantial: a globally smooth deterministic value excludes shocks, and (missing) is exactly the rate identification that the uncorrected master equation does not obtain automatically from a potential representation. A bounded-value estimate alone supplies neither. Extending the certificate through singularities or proving (missing) for the full admissible class remains the first gap. No unconditional selection theorem or counterexample for the catalogue target is claimed.

References

  • [CD] P. Cardaliaguet and F. Delarue, Selected topics in mean field games, Proc. ICM 2022, vol. 5, 3660--3703, EMS Press (2023), Section 4.3 and Theorem 4.5. https://doi.org/10.4171/ICM2022/17.
  • [CeD] A. Cecchin and F. Delarue, Selection by vanishing common noise for potential finite state mean field games, Comm. Partial Differential Equations 47 (2022), 89--168, doi:10.1080/03605302.2021.1955256. The construction and its cost correction are in Sections 2 and 4. https://arxiv.org/abs/2005.12153.

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Econ-26-A3 · Full disproof claimed · LLM-generated · gpt-6 astra, claude opus 5.5, gpt-6 astra pro

A nonminimizing limit of the uncorrected noisy game

A nonminimizing limit of the uncorrected noisy game

Ji Ho Bae
LawAll AI

11 October 2026

\begin{abstract} The answer to C73-25, with its stated admissible class, is negative. We construct fixed smooth potential costs in three states and smooth, nonnegative repulsions tending locally uniformly to zero. Their exact uncorrected master equations have unique classical solutions. The actual selected population laws converge to a deterministic path which is not the population path of any global potential minimizer. The selection argument is an area identity along the actual noisy population. \end{abstract}

Scope. The canonical statement permits a vanishing interior perturbation of the boundary repulsion, with no monotonicity or support restriction. The example uses that permission. It does not settle the more restricted question in which the penalty must be a decreasing cutoff supported only near zero. The original costs remain unchanged. The cost-corrected selection theorem in [CD, Theorem 4.5, p. 3692] and [CeD] has different equations.

The equation and the conclusion

Write $\Delta_3=\{m\geq0:\sum_i m_i=1\}$ and $A_m=\operatorname{diag}(m)-mm^\top$. Set $T=1$ and $\mathcal F=0$. For $U\in\mathbb R^3$ put \[ a_{ij}=(U_i-U_j)_+, \quad H_i(U)=\tfrac12\sum_{j\ne i}a_{ij}^2, \quad b_i^a=\sum_{j\ne i}(m_ja_{ji}-m_ia_{ij}). \] Given a penalty $\phi$, set \[ r_i^\phi=\phi(m_i)-m_i\sum_j\phi(m_j),\qquad Q_i=\sum_j\phi(m_j)(U_j-U_i),\qquad b=b^a+r^\phi. \] The exact master equation used below is \begin{equation}\label{wsref-0-6d-61-73-74-65-72} \partial_tU_i+\varepsilon A_m:D^2U_i +[b+2\varepsilon(e_i-m)]\cdot DU_i-H_i(U)+Q_i=0, \qquad U_i(1,m)=\partial_i\mathcal G(m). \end{equation} All derivatives are tangent derivatives. The selected population has drift $b(m,U(t,m))$ and covariance $2\varepsilon A_m\,dt$, with the Brownian realization specified in C73-25. In particular, its noise is the original Kimura noise.
Theorem: There are a fixed smooth ambient potential $\mathcal G$, a fixed $m_0\in\operatorname{int}\Delta_3$, and an admissible penalty family $\phi_\varepsilon$ for which the actual solutions of (master) and their selected populations satisfy \[ m^\varepsilon\longrightarrow \bar m \quad\hbox{in probability in }C([0,1],\Delta_3), \qquad \bar m\notin\mathcal M_{1,m_0}. \] Thus their limiting law $Q=\delta_{\bar m}$ satisfies $Q(\mathcal M_{1,m_0})=0$.

More precisely, for one sufficiently small fixed $\delta>0$, the construction has \begin{equation}\label{wsref-0-73-63-61-6c-65-73} \eta_\varepsilon=\frac1{\log(e+1/\varepsilon)},\qquad \theta_\varepsilon=\min\{\varepsilon^2,1/100,1/(16\sqrt3)\}, \qquad \phi_\varepsilon(s)=\kappa_\varepsilon \chi(s/\theta_\varepsilon)^2+\eta_\varepsilon\psi_\delta(s). \end{equation} Here $\chi$ is smooth, nonincreasing, one on $[0,1]$ and zero on $[2,\infty)$; $\psi_\delta\geq0$ is fixed and smooth. A sufficient height $\kappa_\varepsilon$ is specified in Lemma Lemma. There is no restriction on the product $\kappa_\varepsilon\theta_\varepsilon$. In addition, \[ \mathbb E\int_0^1\|r^{\phi_\varepsilon}(m_t^\varepsilon)\|_1\,dt \longrightarrow0, \] and the population laws are tight in the required path space.

Constants denoted $C$ are independent of the small parameters until those parameters are fixed. A subscript $\delta$ permits dependence on the fixed construction, including its small radial scale. The order of choices is: fixed cutoffs, a small fixed $\rho_0$, the absorption constant $K_0$ determined by the uniform profile and forward Poincaré bounds, then a sufficiently small fixed $\delta$ and $R=\delta^2$, and finally $\eta=\eta_\varepsilon\to0$ and $\varepsilon\to0$.

Exact flat coordinates in one ordering chamber

On the chamber $U_1>U_2>U_3$ introduce \begin{equation}\label{wsref-0-66-6c-61-74} y_1=\sqrt{2m_1},\qquad y_2=\sqrt{2m_1+4m_2},\qquad P_1=y_1\left(U_1-\tfrac{U_2+U_3}{2}\right),\qquad P_2=\tfrac{y_2}{2}(U_2-U_3). \end{equation} The inverse map is \[ m_1=y_1^2/2,\quad m_2=(y_2^2-y_1^2)/4,\quad m_3=1-(y_1^2+y_2^2)/4. \] Consequently $U\cdot dm=P\cdot dy$ and \begin{equation}\label{wsref-0-66-6c-61-74-48} \overline H(m,U):=\sum_i m_iH_i(U)=\tfrac12|P|^2, \qquad U_1>U_2>U_3\quad\Longleftrightarrow\quad P_1/y_1>P_2/y_2>0. \end{equation} Indeed, the three positive differences are $P_1/y_1-P_2/y_2$, $P_1/y_1+P_2/y_2$, and $2P_2/y_2$. Inserting them into $\sum_i m_iH_i$ proves (flatH). Thus the original deterministic feedback dynamics with zero running cost, in this chamber and without repulsion, are $\dot y=-P$, $\dot P=0$.

For an interior penalty $\phi=\eta\psi$, put $\Psi=\sum_i\psi(m_i)$ and define \begin{equation}\label{wsref-0-52-44} \begin{split} R_1(y)&=\frac{\psi(m_1)}{y_1}-\frac\Psi2y_1,\\ R_2(y)&=\frac{2\psi(m_2)+\psi(m_1)}{y_2}-\frac\Psi2y_2,\\ D(y)&=\operatorname{diag}(\Psi+R_1/y_1,\Psi+R_2/y_2). \end{split} \end{equation} The exact deterministic equations of the uncorrected game are \begin{equation}\label{wsref-0-4f-44-45} \dot y=-P+\eta R(y),\qquad \dot P=\eta D(y)P. \end{equation} For example, with $J_1=y_1$, $J_2=y_2/2$ and $p_{v,k}=P_k/J_k$, the equations for component differences give $\dot p_{v,k}=c_kp_{v,k}^2/2+\eta\Psi p_{v,k}$, where $c_1=2,c_2=1$. Also $\dot J_k=-c_kP_k/2+\eta(R_k/y_k)J_k$. The quadratic terms cancel in $\dot P_k$ and give (ODE). There is no derivative of $\psi$ in this costate equation.

A fixed nonnegative profile with centered rotation

Let \[ m_*=(1/5,3/10,1/2),\quad u=\sqrt{2/5},\quad y_*=(u,2u), \quad p=(1,1),\quad P_*=\delta p,\quad y_0=y_*+\delta p. \] Set $m_0=m(y_0)$, using the inverse map in Section 2. Take disjoint fixed smooth cutoffs $\zeta_1,\zeta_2$, equal to one near $1/5,3/10$ respectively and supported in small interior intervals disjoint from $1/2$. For parameters $h_1,h_2,d_1,d_2$ set \[ h_a(s)=\sum_{j=1}^2[h_j+d_j(s-s_j)]\zeta_j(s), \qquad s_1=1/5,\quad s_2=3/10,\qquad \psi_{\delta,a}(s)=s+\delta h_a(s). \] The baseline $\psi(s)=s$ has $R=0$ and $D=I$ in (RD). Linearity therefore gives \begin{equation}\label{wsref-0-62-61-73-65-6c-69-6e-65} R_{\delta,a}=\delta R_h,\qquad D_{\delta,a}=I+\delta D_h, \qquad M_h=D_h-DR_h. \end{equation} Here $DR_h$ means the ordinary Jacobian with respect to $y$.

Fix the two slopes at \[ d_1=-\frac5{7u},\qquad d_2=-\frac1{7u}. \] Choose the two values by the following exact linear equations: \begin{equation}\label{wsref-0-5a} Z_\delta(a):=-\int_0^1 \left[R_{\delta,a}(y_*+s\delta p) +(1-s)D_{\delta,a}(y_*+s\delta p)\delta p\right]ds=0. \end{equation} The map $Z_\delta/\delta$ is smooth at $\delta=0$, where it equals $-R_h(y_*)-p/2$. Direct substitution in (RD) gives \[ \frac{R_{h1}(y_*)}{u}=2h_1-\tfrac12h_2, \qquad \frac{R_{h2}(y_*)}{2u}=\tfrac18h_1+\tfrac34h_2. \] The values-to-drift matrix has determinant $25u^2/8\ne0$. Thus, for all sufficiently small $\delta$, inversion of a smooth two-by-two matrix gives unique smooth values $h_1(\delta),h_2(\delta)$ satisfying (Z), with limits \begin{equation}\label{wsref-0-6a-65-74-73} h_1(0)=-\frac8{25u},\qquad h_2(0)=-\frac7{25u}. \end{equation} The functions $h_{a(\delta)}$ have uniform smooth bounds. Outside their fixed interior supports $\psi_\delta(s)=s$; on the supports $s$ is bounded below. Hence $\psi_\delta=s+\delta h_{a(\delta)}\geq0$ on $[0,1]$ after reducing $\delta$. This fixes one smooth scalar profile.

We record the matrix calculation responsible for the selection. At $y_*$ let $K_h=15h_1/4-5h_2/2$. Differentiating (RD) yields \[ \begin{split} (M_h)_{11}-(M_h)_{22}&=K_h-4d_1/5+d_2/2,\\ (M_h)_{12}+(M_h)_{21}&=-d_1/10+d_2/2,\\ (M_h)_{12}&=d_2/5. \end{split} \] At the limiting values (jets) these give \begin{equation}\label{wsref-0-4d-6d-61-74-72-69-78} M_h(y_*)=\frac1u \begin{pmatrix}-113/70&-1/35\\1/35&-113/70\end{pmatrix} =\mu_0I+\omega_0J, \qquad J=\begin{pmatrix}0&1\\-1&0\end{pmatrix},\quad \omega_0=-\frac1{35u}<0. \end{equation} For our values at positive $\delta$, the error in this matrix is $O(\delta)$. Freeze at $y_*$ the matrices \begin{equation}\label{wsref-0-41-42-43} A=DR_\delta(y_*),\quad D=D_\delta(y_*),\quad B=D-A, \quad C=J\left(B-\tfrac12\operatorname{tr}(B)I\right),\quad k=\operatorname{tr} D-\tfrac12\operatorname{tr} B. \end{equation} Every traceless two-by-two matrix becomes symmetric after left multiplication by $J$. By (baseline)--(Mmatrix), \begin{equation}\label{wsref-0-63-6f-65-72-63-69-76-65-6d-61-74-72-69-63-65-73} A=O(\delta),\quad D=I+O(\delta),\quad k=1+O(\delta),\quad c_0\delta I\leq C\leq c_1\delta I,\quad A^\top+D-kI=O(\delta) \end{equation} for positive fixed constants $c_0,c_1$. In fact $C=\delta|\omega_0|I+O(\delta^2)$.

Fixed smooth costs and a deterministic reference

Let \[ c=(3\delta/(2u),\delta/u,0),\qquad \mathcal G_{\rm b}(m)=c\cdot m. \] Its flat gradient at $y_*$ is $P_*$. Choose once and for all a smooth bounded function $q$ on $[0,\infty)$, equal to $-2$ on $[0,1]$ and zero on $[4,\infty)$. With $R=\delta^2$ set \begin{equation}\label{wsref-0-72-61-64-69-61-6c} H_R(r)=\int_0^r t q(t^2/R^2)\,dt. \end{equation} Then $H_R(r)=-r^2$ for $r\leq R$, its gradient vanishes for $r\geq2R$, and \begin{equation}\label{wsref-0-72-61-64-69-61-6c-62-6f-75-6e-64-73} |\nabla H_R|\leq CR,\qquad \|D^2H_R\|+|H_R'(r)/r|\leq C,\qquad \|D^3H_R\|\leq C/R. \end{equation} The value at $r=0$ in the quotient is interpreted by continuity. The composition is a smooth radial function of $y$.

On a flat ball of fixed radius $\rho_0$ about $y_*$ define the desired local terminal potential \begin{equation}\label{wsref-0-6c-6f-63-61-6c-47} \widetilde\mathcal G(y)=\mathcal G_{\rm b}(m_*)+P_*\cdot(y-y_*)+H_R(|y-y_*|). \end{equation} Multiply $\widetilde\mathcal G-\mathcal G_{\rm b}(m(y))$ by a smooth cutoff equal to one on radius $\rho_0/2$ and zero outside radius $\rho_0$, and add $\mathcal G_{\rm b}$. Choose the ball compactly inside the simplex chart. This defines a smooth potential on the whole simplex. Its degree-one homogeneous extension $\mathcal G(m)=S\mathcal G(m/S)$, $S=\sum_i m_i>0$, defines a smooth ambient potential on a neighborhood of the simplex. Since the linear flat term and $\mathcal G_{\rm b}$ agree to first order at $y_*$, this extension has \begin{equation}\label{wsref-0-67-73-6d-61-6c-6c} \max_i\|g_i-c_i\|_\infty\leq C(\delta\rho_0+R),\qquad K:=\max_i\|g_i\|_\infty\leq C\delta. \end{equation} All these costs are fixed before $\varepsilon$ varies.

For small $\eta\geq0$, solve (ODE) backward with terminal position $z$ near $y_*$ and momentum $\nabla\widetilde\mathcal G(z)$. At $\eta=0,z=y_*$ this is $\bar y^0(t)=y_*+(1-t)P_*$, $\bar P^0(t)=P_*$. The terminal-to-initial position derivative is $I+D^2\widetilde\mathcal G(y_*)=-I$, hence is invertible. Differentiating the backward ODE with respect to $\eta$ at this reference gives exactly the initial-position derivative $Z_\delta$ in (Z): in backward time $s$ the momentum derivative is $-\int_0^sD_\delta(y_*+r\delta p)\delta p\,dr$, and its position derivative is its integral minus $\int_0^sR_\delta(y_*+r\delta p)\,dr$. Therefore (Z) and the implicit function theorem give an exact deterministic reference $(\bar y^\eta,\bar P^\eta)$ with \begin{equation}\label{wsref-0-72-65-66-65-72-65-6e-63-65} \bar y^\eta(0)=y_0,\qquad \bar y^\eta(1)=y_*+s_\eta,\quad |s_\eta|\leq C_\delta\eta^2, \qquad |\bar y^\eta-y_*|+|\bar P^\eta|\leq C\delta. \end{equation} It converges uniformly to $(\bar y^0,P_*)$. This reference is used for centering.

Admissibility and estimates independent of layer height

Lemma: Let $q_0\geq0$ be any fixed smooth interior addition with $\|q_0\|_\infty\leq Q$. For fixed $0<\varepsilon<1/2$ and $0<\theta<1/(4\sqrt3)$, the penalty $\phi(s)=\kappa\chi(s/\theta)^2+q_0(s)$ is admissible for (master) if \begin{equation}\label{wsref-0-68-65-69-67-68-74} \kappa\geq\max\{\kappa_H(\sqrt{2\varepsilon},2K+Q+2\varepsilon), 1+\varepsilon\}. \end{equation} Here $\kappa_H(\sigma,B_0)$ is the sufficient threshold of [BCCD, Theorem 2.10]. It depends on noise and the bound $B_0$ for the nonnegative inward coefficient, and not on $\theta$ or the large outflow bound. The solution is unique in the canonical classical class and satisfies $|U_i|\leq K$. The same statement holds on any fixed finite horizon, with $K=\|g\|_\infty$ since $\mathcal F=0$.
Proof: We give the reduction to the cited linear theory, since the printed master theorem [BCCD, Theorem 3.4] assumes a decreasing cutoff and cannot directly be applied to our total penalty. Put $a_M(r)=\min(M,r_+)$, $M=2K$, and use $H_M^{-,i}(v)=-\sum_{j\ne i}\int_0^{(v_i-v_j)_+}\min(M,r)\,dr$. With $\phi_0=\kappa\chi(s/\theta)^2$, the generic fixed-point system of [BCCD, Section 4.3.2, $4.16$--$4.18$] has component drift $\phi_0(m_j)+\beta_j^i+m_j\beta_j^\circ$, where we insert \[ \begin{split} \beta_j^i&=q_0(m_j)+\sum_\ell m_\ell a_M(v_\ell-v_j) +2\varepsilon\delta_{ij},\\ \beta_j^\circ&=-\sum_\ell[\phi(m_\ell)+a_M(v_j-v_\ell)] -2\varepsilon. \end{split} \] These coefficients sum to zero in the complete drift and give precisely the clipped population drift plus $2\varepsilon(e_i-m)$. Crucially, $0\leq\beta_j^i\leq Q+M+2\varepsilon$; the large $\kappa$ appears in the outflow coefficient only. If $P_\Gamma$ clips to $[-\Gamma,\Gamma]$, use the bounded source \[ P_\Gamma\left[H_M^{-,i}(v) +\sum_j\phi(m_j)P_\Gamma(v_j-v_i)\right]. \] For fixed parameters all the coefficient regularity assumptions of that construction hold. Its frozen linear solvability [BCCD, Lemma 4.4], the Hölder estimate [BCCD, Theorem 2.10], and the Schauder argument in its Section 4.3.2 yield a Wright--Fisher classical solution of this clipped system, with bounded first derivatives. Their constants may depend on the fixed outflow and source sizes; the repulsion threshold uses the displayed inward bound. This is why (height) is noncircular.

At a maximum over all components, $H_M^{-,i}\leq0$ and the switching source is nonpositive; at a minimum $H_M^{-,i}=0$ and the switching source is nonnegative. The maximum principle is justified at the degenerate boundary using $B_{\rm ent}=\log3+\sum_jm_j\log m_j$. For any positive $\alpha$, a maximum of $U_i-\alpha B_{\rm ent}$ cannot lie on the boundary: an inward displacement changes $U_i$ by $O(h)$ and $-\alpha B_{\rm ent}$ by a positive multiple of $-\alpha h\log h+O(h)$. The component generator applied to $B_{\rm ent}$ is bounded above for fixed parameters, since $\varepsilon A_m:D^2B_{\rm ent}=2\varepsilon$ and the other entropy drifts have finite upper bounds. Subtract also this bound plus one times $\alpha(1-t)$, and apply the interior maximum principle in reverse time. Letting $\alpha\downarrow0$ gives $|U_i|\leq K$. Thus $M=2K$ removes the control and Hamiltonian clips. After $\kappa$ is fixed choose $\Gamma>\max\{2K,4K^2+6K\|\phi\|_\infty\}$; both source clips are then inactive. This proves existence for the exact unchanged costs. If $K=0$, $U=0$ works, or one can use any positive control clip.

For bounded feedback rates $a_{ij}\leq M_1$, $b_j\geq\phi(m_j)-C_\varepsilon m_j$. On $m_j\leq\theta$, the generator of the population, or of any component diffusion with added drift $2\varepsilon(e_i-m)$, applied to $-\log m_j$ is at most $-(\kappa-\varepsilon)/m_j+C_\varepsilon+2\varepsilon$. Off that layer it is bounded above. Stopped logarithmic barriers exclude hitting any face. Interior local Lipschitz coefficients give a unique strong solution extending throughout $[0,1]$.

Finally, if $U,V$ are canonical classical solutions, their difference $w_i$ satisfies a linear equation with the component generator based on $U$ and source \[ \sum_j\phi(m_j)(w_j-w_i)-H_i(U)+H_i(V) +[b(m,U)-b(m,V)]\cdot DV_i. \] For fixed $\varepsilon$ this source is bounded by $C_\varepsilon\max_j\|w_j\|_\infty$, including the drift mismatch term because $DV$ is bounded. Itô's formula along the interior component diffusion, followed by removal of compact stopping times, gives backward Gronwall with zero terminal data. Hence $w=0$. ∎

Apply the lemma in (scales) with $q_0=\eta_\varepsilon\psi_\delta$ and $Q=\|\psi_\delta\|_\infty$, taking $\kappa_\varepsilon$ at least the right side of (height) and $1/\varepsilon$. The noise convention in [BCCD] is generator $\sigma^2 A_m/2$, so $\sigma=\sqrt{2\varepsilon}$ here. The cutoff is supported in $[0,2\theta_\varepsilon]$; hence the family is smooth, nonnegative and locally uniformly vanishing on $(0,1]$. Only small $\varepsilon$ matters. If an index for every $\varepsilon>0$ is wanted, put $a=\max(1,4\varepsilon)$ and apply the lemma to $V(s,m)=U(s/a,m)/a$ with horizon $a$, noise $\varepsilon/a$, terminal data $g/a$ and penalty $\phi/a$; rescaling back supplies the remaining indices without changing $g$.

Two useful uniform estimates also explain why an arbitrarily high layer causes no hidden impulse. With $M=2K$, put $\mu=\min_i m_{0i}$, $C_*=2(M+1)$ and $\ell=(\mu/2)e^{-C_*}$. Let $\alpha_\varepsilon=\sup_{s\in[\ell,1]}\phi_\varepsilon(s)\to0$. Stop the actual population on its first coordinate reaching $\ell$. Before this time $b_i+C_*m_i\geq0$, for small $\varepsilon$. The integrating-factor martingale has expected squared supremum at most $2\varepsilon e^{2C_*}$, since $d\langle M_i^m\rangle\leq\varepsilon\,dt/2$. A coordinate exit needs a martingale drop of at least $\mu/2$. Thus \begin{equation}\label{wsref-0-67-6c-6f-62-61-6c-65-78-69-74} \mathbb P(\hbox{exit before }1)\leq24\varepsilon e^{2C_*}/\mu^2. \end{equation} On nonexit paths the integrated repulsion is bounded by $3\alpha_\varepsilon$. The controlled drift is uniformly bounded and the population martingale tends uniformly to zero in $L^2$. These facts prove tightness and repulsion removal in probability.

For expected absolute removal, set $F_{\rm ent}=\sum_i m_i\log m_i$ and $k_\ell=\log(1/(3\ell))>0$. Its repulsion drift is \[ r^\phi\cdot DF_{\rm ent} \leq-k_\ell\sum_{i:m_i<\ell}\phi(m_i)+3\alpha_\varepsilon\log3. \] Its controlled drift is at most $2M\log3$, its diffusion drift is $2\varepsilon$, and its martingale bracket is bounded by $24\varepsilon/e^2\,dt$. Stop first inside the simplex and then pass to the limit; the negative term is integrable by Fatou and the other terms have the displayed bounds. On the exit event, apply the resulting entropy inequality from the exit time to $1$ conditionally on that event. The entropy oscillation is at most $\log3$. Together with (globalexit), this gives \begin{equation}\label{wsref-0-69-6d-70-75-6c-73-65} \mathbb E\int_0^1\sum_i\phi_\varepsilon(m_i^\varepsilon)\,dt \leq3\alpha_\varepsilon+C_\ell\varepsilon\longrightarrow0. \end{equation} Since $\|r^\phi\|_1\leq2\sum_i\phi(m_i)$, this proves the claimed expected removal without a height-times-width assumption.

The actual feedback remains in the protected chamber

We now reduce $\rho_0$ and then $\delta$ as necessary. The value estimate and (gsmall) give $|U_i|\leq C\delta$ globally, so all actual rates are $O(\delta)$. For a component $i$, use the auxiliary diffusion with generator \[ \varepsilon A_m:D^2+[b(m,U)+2\varepsilon(e_i-m)]\cdot D. \] Start in the flat ball of radius $\rho_0/2$ and stop on radius $\rho_0$. On this protected set the boundary cutoff is absent, $\phi=\eta\psi_\delta$, and the drift is $O(\delta+\eta+\varepsilon)$. Choose $\delta$ small compared with $\rho_0$ and then $\eta,\varepsilon\leq\delta$. A deterministic drift displacement is less than $\rho_0/4$, so Doob's inequality bounds the exit probability by $C\varepsilon/\rho_0^2$. Along this auxiliary diffusion (master) gives $dU_i=(H_i-Q_i)\,dt+$ martingale. Before exit, $|H_i|\leq C\delta^2$ and $|Q_i|\leq C\eta\delta$. Using (gsmall) at the terminal time, and the global value bound at a stopped exit, proves uniformly on the inner ball and in time \begin{equation}\label{wsref-0-63-68-61-6d-62-65-72-65-73-74-69-6d-61-74-65} |U_i(t,m)-c_i| \leq C(\delta\rho_0+R+\delta^2+\eta\delta +\delta\varepsilon/\rho_0^2). \end{equation} The gaps of $c$ are positive fixed multiples of $\delta$. Choose $\rho_0$, then $\delta$ with $R=\delta^2$, and finally $\eta,\varepsilon$ so that the right side is less than one quarter of the smallest gap. This proves $U_1>U_2>U_3$ for the actual global master field throughout the inner ball.

Let $L$ be a fixed sufficiently large constant and stop the actual selected flat population at \[ \tau=1\wedge\inf\{t:|y_t-y_0|\geq L\delta\}. \] Reduce $\delta$ so that this ball and the reference (reference) are inside radius $\rho_0/4$ about $y_*$. The actual flat population drift is $O(\delta)$ and its bracket is $O(\varepsilon)$. Taking $L$ greater than twice the deterministic drift bound gives \begin{equation}\label{wsref-0-6c-6f-63-61-6c-65-78-69-74} \mathbb P(\tau<1)\leq C\varepsilon/\delta^2. \end{equation} Up to this stop the actual and reference positions are $O(\delta)$ from $y_*$, their momenta are $O(\delta)$, and the exact flat and radial formulas hold.

Original weighted energy and exact finite-noise errors

Along the actual selected population write $z_i=(e_i-m)\cdot DU_i$ and $\Gamma_i=DU_i^\top A_mDU_i$. For $V=\sum_i m_iU_i^2$, the generator including time gives \begin{equation}\label{wsref-0-65-6e-65-72-67-79-69-64-65-6e-74-69-74-79} (\partial_t+\varepsilon A_m:D^2+b\cdot D)V =2\varepsilon\sum_i m_i\Gamma_i -\sum_{i,j\ne i}m_iU_j a_{ij}^2 +\mathcal R, \quad \mathcal R=\sum_j\phi(m_j)\sum_i m_i(U_j-U_i)^2\geq0. \end{equation} Indeed, the cross term $4\varepsilon U_i\Gamma_{A_m}(m_i,U_i) =4\varepsilon m_iU_iz_i$ cancels the weighted-player contribution $-4\varepsilon m_iU_iz_i$. On an active edge, $a_{ij}(U_j^2-U_i^2)+U_ia_{ij}^2=-U_ja_{ij}^2$. The repulsion terms combine as $\sum_i r_iU_i^2-2\sum_i m_iU_iQ_i=\mathcal R$. These identities prove (energyidentity).

Stop on $\min_i m_i\geq1/n$, apply Itô's formula and take expectations. The right side after rearrangement is bounded by $K^2+2K(2K)^2$. The nonnegative left integrands pass by monotone convergence, and the bounded remaining terms pass by bounded convergence, because each continuous interior path stays in some compact interior set. Hence \begin{equation}\label{wsref-0-65-6e-65-72-67-79-62-6f-75-6e-64} 2\varepsilon\mathbb E\int_0^1\sum_i m_i\Gamma_i\,dt +\mathbb E\int_0^1\mathcal R\,dt\leq C_E, \qquad C_E=K^2+8K^3=O(\delta^2). \end{equation} In particular, on the actual stopped protected set, $\varepsilon\mathbb E\int_0^\tau|D_{\rm tan}U|^2\,dt\leq C_\delta$ and $\varepsilon\mathbb E\int_0^\tau|D_{\rm tan}U|\,dt\leq C_\delta\sqrt\varepsilon$.

Let $B_v$ have rows $(1,-1/2,-1/2)$ and $(0,1,-1)$, so that $p_v=B_vU$. Let $M^m$ be the population martingale, $K_y=Dy\,A_m(Dy)^\top$, and put \[ \ell_k=\sum_i(B_v)_{ki}z_i,\qquad t_k=\sum_i(B_v)_{ki}DU_i^\top A_mD_my_k. \] On $[0,\tau]$, the exact equations are \begin{equation}\label{wsref-0-73-74-6f-63-68-61-73-74-69-63-66-6c-61-74} \begin{split} dy_k&=[-P_k+\eta R_k-\varepsilon K_{y,kk}/y_k]\,dt+dM_k^y,\\ dP_k&=[\eta D_{kk}(y)P_k+\varepsilon e_k]\,dt+dN_k^P,\\ e_k&=-K_{y,kk}P_k/y_k^2-2J_k\ell_k+c_kt_k,\\ dM_k^y&=D_my_k\cdot dM^m,\\ dN_k^P&=J_k\sum_i(B_v)_{ki}DU_i\cdot dM^m +(P_k/y_k)D_my_k\cdot dM^m. \end{split} \end{equation} To check them, the original equation along the population is $dU_i=(H_i-Q_i-2\varepsilon z_i)\,dt+DU_i\cdot dM^m$. Thus $dp_{v,k}=(c_kp_{v,k}^2/2+\eta\Psi p_{v,k} -2\varepsilon\ell_k)\,dt+\sum_i(B_v)_{ki}DU_i\cdot dM^m$. In $d(J_kp_{v,k})$ the quadratic drift cancels as before, the coordinate Itô drift gives the first term of $e_k$, and the product bracket gives $\varepsilon c_kt_k$. This proves (stochasticflat). Bounded geometry on the protected set and (energybound) imply \begin{equation}\label{wsref-0-6e-6f-69-73-65-62-6f-75-6e-64-73} \begin{split} \operatorname{TV}(E_X)&\leq C_\delta\varepsilon,\qquad \mathbb E\operatorname{TV}(E_p)\leq C_\delta\sqrt\varepsilon,\\ \mathbb E\operatorname{tr}\langle M^y\rangle_\tau&\leq C_\delta\varepsilon,\qquad \mathbb E\operatorname{tr}\langle N^P\rangle_\tau\leq C_\delta, \end{split} \end{equation} where $dE_{X,k}=-\varepsilon K_{y,kk}/y_k\,dt$ and $dE_{p,k}=\varepsilon e_k\,dt$. In particular the costate martingale is bounded in energy. Cauchy--Schwarz for covariations gives \begin{equation}\label{wsref-0-63-72-6f-73-73-6e-6f-69-73-65} \mathbb E\operatorname{TV}\langle M_i^y,N_j^P\rangle_\tau \leq C_\delta\sqrt\varepsilon. \end{equation}

An area estimate for the actual centered processes

Center the actual variables at the reference with $\eta=\eta_\varepsilon$: $X=y-\bar y^\eta$, $p_c=P-\bar P^\eta$, stopping both at the same $\tau$. In this section abbreviate $p_c$ to $v$. Equations (reference), (baseline) and Taylor's theorem give the exact decomposition \begin{equation}\label{wsref-0-63-65-6e-74-65-72-65-64} \begin{split} dX&=[-v+\eta AX+\eta r_X]\,dt+dE_X+dM^y,\\ dv&=[\eta Dv+\eta r_v]\,dt+dE_p+dN^P,\\ |r_X|&\leq C\delta^2|X|,\qquad |r_v|\leq C\delta^2(|X|+|v|),\qquad X_0=0. \end{split} \end{equation} For the second remainder, explicitly write $[D(y)-D(y_*)]v+[D(y)-D(\bar y)]\bar P$. Both derivatives of the nonconstant part of $D$ are $O(\delta)$; the positions and $\bar P$ are $O(\delta)$. The first remainder follows similarly from $DR(y)-DR(y_*)=O(\delta|y-y_*|)$.

On the good event $G=\{\tau=1\}$, the terminal radial formula gives \begin{equation}\label{wsref-0-74-65-72-6d-69-6e-61-6c} v_1=h_1X_1+e_1,\qquad |h_1|\leq H,\qquad |e_1|\leq C_\delta\eta^2|X_1|. \end{equation} Indeed, $v_1=\nabla H_R(X_1+s_\eta)-\nabla H_R(s_\eta)$. Take $h_1=H_R'(|X_1|)/|X_1|$. The mixed difference from $\nabla H_R(X_1)$ is bounded by $\|D^3H_R\|\,|s_\eta|\,|X_1|$. Use (radialbounds) and (reference); $H$ is uniform in small $\delta$, whereas $C_\delta$ is allowed to depend on $R$.

First, the forward equation in (centered), Gronwall, Doob and Cauchy--Schwarz give \begin{equation}\label{wsref-0-50-6f-69-6e-63-61-72-65} \mathbb E\sup_{t\leq1}|X_{t\wedge\tau}|^2 \leq C\mathbb E\int_0^\tau|v_t|^2\,dt+C_\delta\varepsilon. \end{equation} For example, its square root follows from the pathwise bound $\sup|X|\leq2(\int_0^\tau|v|\,dt+\operatorname{TV}(E_X)+\sup|M^y|)$, valid once $\eta\|A\|+C\eta\delta^2$ is small. It also bounds $\mathbb E\int_0^\tau w|X|^2\,dt$ by $C\mathbb E\int_0^\tau w|v|^2\,dt+C_\delta\varepsilon$ for $w(t)=e^{-\eta kt}$, since $1/2\leq w\leq1$ for small $\eta$.

Choose $K_0$ large, put $\lambda=K_0\delta^2$ and set \[ N=J+\eta\lambda I,\quad Q=X^\top Nv,\quad E=\tfrac12X^\top CX,\quad L_c=A^\top+D-kI. \] The two-by-two identity $A^\top J+JD=C+kJ$, together with Itô's formula, gives \[ \begin{split} dQ-\eta kQ\,dt={}&[\eta X^\top Cv-\eta\lambda|v|^2 +\eta^2\lambda X^\top L_cv]\,dt\\ &+\eta[r_X^\top Nv+X^\top Nr_v]\,dt+d\mathcal V_Q+d\mathcal M_Q,\\ dE={}&[-X^\top Cv+\eta X^\top CAX+\eta X^\top Cr_X]\,dt +d\mathcal V_E+d\mathcal M_E, \end{split} \] where the finite-variation noise terms are \[ \begin{split} d\mathcal V_Q&=(dE_X)^\top Nv+X^\top N\,dE_p +\sum_{ij}N_{ij}\,d\langle M_i^y,N_j^P\rangle,\\ d\mathcal V_E&=X^\top C\,dE_X +\tfrac12\sum_{ij}C_{ij}\,d\langle M_i^y,M_j^y\rangle. \end{split} \] All martingale terms have mean zero after stopping, by bounded $X,v$ and (noisebounds). Integrating with factor $w$, substituting the energy equation and using $X_0=0$ gives the exact identity \begin{equation}\label{wsref-0-61-72-65-61-69-64-65-6e-74-69-74-79} \begin{split} &\mathbb E[w(\tau)(Q_\tau+\eta E_\tau)] +\eta\lambda\mathbb E\int_0^\tau w|v|^2\,dt\\ &\quad+\eta^2\mathbb E\int_0^\tau wX^\top[\tfrac{k}{2}C-\operatorname{Sym}(CA)]X\,dt\\ ={}&\eta^2\lambda\mathbb E\int_0^\tau wX^\top L_cv\,dt +\eta\mathbb E\int_0^\tau w[r_X^\top Nv+X^\top Nr_v]\,dt\\ &\quad+\eta^2\mathbb E\int_0^\tau wX^\top Cr_X\,dt +\mathbb E\int_0^\tau w\,d\mathcal V_Q +\eta\mathbb E\int_0^\tau w\,d\mathcal V_E. \end{split} \end{equation} By (noisebounds)--(crossnoise), the last two terms in absolute value are at most $C_\delta\sqrt\varepsilon$. By (coercivematrices), after reducing $\delta$, \[ \tfrac{k}{2}C-\operatorname{Sym}(CA)\geq c_2\delta I. \] The first-order remainder on the right of (areaidentity) is bounded by $C\eta\delta^2\mathbb E\int w(|X|^2+|X||v|)\,dt$. Use (Poincare) and $2|X||v|\leq|X|^2+|v|^2$. Choosing $K_0$ sufficiently large absorbs it into one quarter of $\eta\lambda\mathbb E\int w|v|^2\,dt$, with error $C_\delta\varepsilon$. The $L_c=O(\delta)$ term is absorbed into another quarter by the same bound and small $\eta$. The remaining deterministic term is at most $C\eta^2\delta^3\mathbb E\int w|X|^2\,dt$ and is absorbed by the positive bulk matrix for small $\delta$.

On $G$, (terminal) yields \[ Q_1+\eta E_1 \geq\eta\big[c_0\delta/2-\lambda H -C_\delta\eta-C_\delta\lambda\eta^2\big]|X_1|^2 \geq c_3\eta\delta|X_1|^2. \] Here choose $\delta$ so that $\lambda H\leq c_0\delta/8$, then $\eta$ small for that fixed $\delta$. On exit, $|X|+|v|\leq C\delta$; the negative endpoint contribution has expectation at most $C\delta^2\mathbb P(\tau<1)\leq C\varepsilon$ by (localexit). Consequently (areaidentity) proves \begin{equation}\label{wsref-0-73-65-6c-65-63-74-69-6f-6e-65-73-74-69-6d-61-74-65} \eta\delta\mathbb E[\mathbf 1_G|X_1|^2] +\eta\delta^2\mathbb E\int_0^\tau|v|^2\,dt \leq C_\delta\sqrt\varepsilon. \end{equation} Since $\sqrt\varepsilon/\eta_\varepsilon\to0$, (Poincare), (localexit) and (selectionestimate) imply $\sup_t|y_t-\bar y^{\eta_\varepsilon}_t|\to0$ in probability. The reference convergence in (reference) proves the path convergence in Theorem Theorem.

The limiting population path is not minimizing

The limit is the original central straight path $\bar y(t)=y_*+(1-t)P_*$, with $\bar m=m(\bar y)$. Its feedback stays in the strict chamber: at $y_*$, $P_{*1}/y_{*1}=\delta/u>\delta/(2u)=P_{*2}/y_{*2}>0$, and the inequalities persist along this short path.

It is necessary to exclude every control producing this population path, not just its equilibrium feedback. For any nonnegative rates $a$ and cotangent $U$, completing the square edge by edge gives \begin{equation}\label{wsref-0-46-65-6e-63-68-65-6c} L(m,a)+U\cdot b^a(m,a)+\overline H(m,U)\geq0, \qquad L(m,a)=\tfrac12\sum_{i,j\ne i}m_i a_{ij}^2. \end{equation} Indeed each edge contributes $m_i[\tfrac12a_{ij}^2-a_{ij}(U_i-U_j) +\tfrac12(U_i-U_j)_+^2]\geq0$. Along the central path choose the cotangent corresponding to $P_*$ in (flat). If any rates generate that same path, then $U\cdot\dot m=P_*\cdot\dot{\bar y}=-|P_*|^2$ and $\overline H=|P_*|^2/2$. Thus (Fenchel) gives $L\geq|P_*|^2/2$ almost everywhere. The central feedback attains this lower bound.

Take any fixed $w\in\mathbb R^2$ with $|w|=R/2$ and use the straight competitor from $y_0$ to $y_*+w$ with constant momentum $P_*-w$. For sufficiently small fixed $\delta$, $R=\delta^2$ ensures that this entire path lies in the protected strict chamber, so its original rates are nonnegative and its kinetic action is $|P_*-w|^2/2$. Its terminal potential is, by (localG) and (radial), $\mathcal G_{\rm b}(m_*)+P_*\cdot w-|w|^2$. Its total original action minus the minimum action among controls generating the central population path is therefore \[ \tfrac12|P_*-w|^2+P_*\cdot w-|w|^2 -\tfrac12|P_*|^2=-\tfrac12|w|^2<0. \] This is a fixed positive action gap, independent of $\varepsilon$ and $\eta$. No control producing $\bar m$ can be a global minimizer. Together with the actual path convergence, this proves Theorem Theorem. $\square$

Use of AI tools

The author used research AI agents that the author built and configured (GPT-6 Astra max, Claude Opus 5.5 max and GPT-6 Astra Pro) during the proof work, for proof search and construction, proof checking, translation, Python computation and similar tasks. The final manuscript was written, and the proofs were checked, by the human author. All responsibility for this work rests with the author personally.

References

  • [BCCD] E. Bayraktar, A. Cecchin, A. Cohen and F. Delarue, Finite state mean field games with Wright--Fisher common noise, Journal de Mathématiques Pures et Appliquées 147 $2021$, 98--162. DOI: 10.1016/j.matpur.2021.01.003. arXiv:1912.06701v2 (10 January 2021). Theorem 2.10, journal p. 112, Lemma 4.4, p. 134, and Section 4.3.2, p. 136, especially $4.16$--$4.18$, supply the analytic construction. https://arxiv.org/abs/1912.06701v2.
  • [CeD] A. Cecchin and F. Delarue, Selection by vanishing common noise for potential finite state mean field games, Communications in Partial Differential Equations 47$1$ $2022$, 89--168. DOI: 10.1080/03605302.2021.1955256. https://arxiv.org/abs/2005.12153.
  • [CD] P. Cardaliaguet and F. Delarue, Selected topics in mean field games, Proc. ICM 2022, vol. 5, pp. 3660--3703, EMS Press $2023$. DOI: 10.4171/ICM2022/17. The uncorrected question is in Section 4.3, p. 3691; $4.14$, p. 3687, is the master equation, and Theorem 4.5 is the cost-corrected result. https://ems.press/content/book-chapter-files/33265.

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