we-solve (beta)

Econ-31 · economics · Economics

Weak master-equation selection beyond finite-state potentials

LLM status Partially solved unverified

Statement

GT-086 | Mean-field games | R: corrected nonpotential extension; the potential uniqueness subcase is known

Definitions and mathematical statement

Let $E=\{1,\ldots,d\}$, $d\geq3$, $\Delta_d=\{m\geq0:\sum_i m_i=1\}$, $T>0$, and smooth bounded $f,g:\Delta_d\to\mathbb R^d$, with no potential or monotonicity assumption. Set \[ H_i(U)=\tfrac12\sum_{j\ne i}(U_i-U_j)_+^2,\qquad b_i(m,U)=\sum_{j\ne i}[m_j(U_j-U_i)_+-m_i(U_i-U_j)_+]. \] The deterministic classical master equation is \[ -\partial_tU_i+H_i(U)-\sum_j b_j(m,U)\partial_{m_j}U_i=f_i(m), \qquad U_i(T,m)=g_i(m). \] For nonsmooth $U$, products involving $b(m,U)DU$ do not acquire a distributional meaning merely by writing this display.

Here is an explicit selection formulation of the agenda. Let $\mathcal A(f,g,T)$ be the class of smooth nonnegative boundary-repulsion families $\phi_\varepsilon$ tending to zero on each compact subset of $(0,1]$ for which the following parabolic problem has a unique classical solution. Require that solution to be continuous on $[0,T]\times\Delta_d$, $C^1$ in time and $C^2$ in the interior, with bounded value and first derivatives and second derivatives bounded on interior compact sets. Put \[ b_i^\phi=b_i+\sum_{j\ne i}[m_j\phi_\varepsilon(m_i)-m_i\phi_\varepsilon(m_j)],\quad L_\varepsilon=\varepsilon\sum_{j,k}(m_j\delta_{jk}-m_jm_k)\partial_{m_jm_k}^2. \] The regularized field $U^{\varepsilon,\phi}$ satisfies \[ \begin{split} 0={}&\partial_tU_i+L_\varepsilon U_i+\sum_j b_j^\phi(m,U)\partial_{m_j}U_i +2\varepsilon\sum_jm_j(\partial_{m_i}U_i-\partial_{m_j}U_i)\\ &-H_i(U)+f_i(m)+\sum_{j\ne i}\phi_\varepsilon(m_j)(U_j-U_i), \qquad U_i(T,m)=g_i(m). \end{split} \] Tangent derivatives make the equation independent of extensions off the simplex. Let $Z^{\varepsilon,\phi}=(U_i^{\varepsilon,\phi}-U_d^{\varepsilon,\phi})_{i<d}$. These are the differences determining equilibrium rates. Define a selected weak feedback field as an $L^1_{\mathrm{loc}}((0,T)\times\operatorname{int}\Delta_d)$ subsequential limit of these $Z$ fields, where Lebesgue measure is the $(d-1)$-dimensional simplex measure.

Find assumptions, genuinely beyond the potential class, and an intrinsic admissibility condition under which: (i) such limits exist for admissible regularizations; (ii) every admissible selected field is the same almost everywhere; and (iii) it agrees with the component differences of the classical master solution whenever that solution exists. One precise independence target is \[ \lim_{\varepsilon\downarrow0} \|Z^{\varepsilon,\phi}-Z^{\varepsilon,\psi}\|_{L^1(K)}=0 \quad\text{for all }\phi,\psi\in\mathcal A(f,g,T) \text{ and compact }K\subset(0,T)\times\operatorname{int}\Delta_d, \] together with compactness. Different cost corrections or boundary conventions define different admissible classes and must be named.

Short English statement

Can a well-defined weak feedback solution and uniqueness criterion be obtained for nonpotential finite-state master equations after common noise disappears?

Variants and scope boundaries

Feedback differences have dimension $d-1$; the full value vector also contains a common component. Uniqueness among arbitrary distributions is a different, unjustified claim. Potentiality provides a conservative formulation and a weak one-sided-Lipschitz criterion in the cited work; that subcase is already known.

Source relationship and status

The input's blanket finite-state uniqueness description needs correction: the chapter supplies a potential weak uniqueness result. Its nonpotential extension motivates the editorial agenda above.

References

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

Locators: Section 4.3, p. 3689, loss of classical solutions and weak uniqueness issue; Theorem 4.6 and following conservative-form paragraph, p. 3692, known potential solution; Section 4.4, p. 3692, nonpotential extension.

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.

Locator/source role: abstract's final intrinsic uniqueness criterion, and the chapter's Theorem 4.6 reference [43]; this is a known boundary, not the source of a universal nonpotential theorem.

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-31-A2 · Exploration · LLM-generated · gpt 6 astra pro ·

GPT 6 Astra Pro attempt on Econ-31

Research attempt and adversarial verification

9 October 2026 Original-target status: UNRESOLVED

C73-30: Nonpotential weak feedback selection for finite-state master equations

Input: C73-30.tex.  Tools: web browsing [YES]; Python computation [YES].

1) FORMAL RESTATEMENT

Let $d\geq3$, $T>0$, $\Delta_d=\{m\geq0:\sum_i m_i=1\}$, and let $f,g:\Delta_d\to\mathbb R^d$ be smooth on a neighborhood of the simplex. Define \[ H_i(u)=\tfrac12\sum_{j\ne i}(u_i-u_j)_+^2, \qquad b_i(m,u)=\sum_{j\ne i}[m_j(u_j-u_i)_+-m_i(u_i-u_j)_+]. \] The deterministic classical master equation is \[ -\partial_t U_i+H_i(U)-b(m,U)\cdot DU_i=f_i(m), \qquad U_i(T,m)=g_i(m). \] For smooth nonnegative repulsion families $\phi_\varepsilon\to0$ locally uniformly on $(0,1]$, set \[ R_i^\phi=\sum_{j\ne i}[m_j\phi_\varepsilon(m_i)-m_i\phi_\varepsilon(m_j)], \quad L_\varepsilon=\varepsilon\sum_{j,k}(m_j\delta_{jk}-m_jm_k)\partial_{jk}^2. \] The input's regularization is \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). \end{align*} Every derivative contraction is tangent to the simplex. Denote by $\mathcal A(f,g,T)$ the families for which this problem has a unique classical solution continuous on the closed domain, $C^1$ in time and $C^2$ in the interior, with bounded values and first derivatives and second derivatives bounded on interior compact subsets, separately for each $\varepsilon$. Unlike the companion population-selection input, this definition does not separately postulate an interior SDE solution.

Set $Z^{\varepsilon,\phi}=(U_i^{\varepsilon,\phi}-U_d^{\varepsilon,\phi})_{i<d}$. A selected weak feedback field means an $L^1_{\rm loc}((0,T)\times\operatorname{int}\Delta_d)$ subsequential limit of these differences. Lebesgue measure is the $(d-1)$-dimensional simplex measure. The target is to find genuinely nonpotential assumptions and an intrinsic admissibility criterion giving compactness of these fields, uniqueness of every selected limit, and agreement with a classical master solution when one exists. A precise independence target is \[ \forall\phi,\psi\in\mathcal A(f,g,T)\ \forall K\Subset(0,T)\times\operatorname{int}\Delta_d, \quad\|Z^{\varepsilon,\phi}-Z^{\varepsilon,\psi}\|_{L^1(K)}\longrightarrow0, \] together with existence of convergent subsequences. Empty regularization classes would make independence vacuous and would not supply selection. No correction to the explicit target is needed; however, merely writing $b(m,U)DU$ for a discontinuous field does not define a distribution.

2) QUICK LITERATURE/CONTEXT CHECK

Theorem 4.6 and Section 4.4 of Cardaliaguet--Delarue [cardaliaguet-delarue], printed p. 3693, separate potential weak selection from its nonpotential extension. Cecchin--Delarue [cecchin-delarue] treats the potential construction. Nonpotential uniqueness is not wholly unexplored: Bertucci [bertucci] develops continuous monotone solutions and stability in finite-state master equations. None of these observations alone identifies every limit of the exact repulsion class displayed above. The following characteristic argument is a direct monotonicity proof, not a claim of first establishing nonpotential master-equation uniqueness. Context checked on 9 October 2026.

3) ATTACK PLAN

Type and tools. This is nonlinear PDE selection with a nonconservative transport product. Relevant tools are: forward--backward characteristics (an intrinsic continuous formulation); Schauder's fixed-point theorem (existence); component barriers (uniform characteristic bounds); monotonicity (uniqueness); convex Hamiltonian remainders (an energy identity); tangent-coordinate curl (certify genuine nonpotentiality); smooth jump profiles (test distributional meaning); local compactness estimates (the missing noisy-limit bridge).

Proof tracks. Construct a unique intrinsic characteristic field under strict monotonicity, including an explicitly nonpotential example, and test classical consistency. Then try to identify vanishing-noise limits with it. Disproof tracks. Change the smoothing path across a feedback jump and compute the actual $b\,DZ$ product; vary boundary repulsion only where it is unconstrained; check whether uniqueness of characteristics really proves compactness of parabolic solutions. The characteristic construction succeeds; the last inference does not follow from it.

4) WORK

A genuinely nonpotential structural assumption.
Assume, for some $\alpha>0$ and every $m,n\in\Delta_d$, \[ (f(m)-f(n))\cdot(m-n)\geq\alpha\|m-n\|_2^2, \qquad (g(m)-g(n))\cdot(m-n)\geq0. \tag{MON} \] No symmetry of the derivative of $f$ is imposed. These assumptions are used only in the deterministic results below, unless a noisy-limit assertion is explicitly stated.
Lemma (Existence of a deterministic forward--backward solution): For any smooth bounded $f,g$, every $t_0\in[0,T]$ and every interior initial law $m^0$, there is a bounded classical-in-time solution on $[t_0,T]$ to \[ \dot u_i=H_i(u)-f_i(m),\qquad \dot m=b(m,u),\qquad m(t_0)=m^0,\quad u(T)=g(m(T)). \tag{FB} \] The population stays interior. Uniform bounds depend only on $d,T,\|f\|_\infty,\|g\|_\infty$.
Proof: Given a continuous candidate path $z$ in the simplex, solve backward $\dot u_i=H_i(u)-f_i(z)$ with terminal $g_i(z(T))$. Local ODE existence and uniqueness follow from local Lipschitz continuity of $H$ in $u$ and continuity in time. In reversed time, the equation is $v_i'=f_i(z)-H_i(v)$. At a maximum component its derivative is at most $\|f\|_\infty$, and at a minimum component $H_i(v)=0$, so its derivative is at least $-\|f\|_\infty$. Scalar upper and lower barriers, or the corresponding one-sided derivatives at ties, give \[ \|u\|_\infty\leq M:=\|g\|_\infty+T\|f\|_\infty. \] The bound prevents escape from every bounded region in finite time, so the ODE extends to the whole interval. The rates $(u_i-u_j)_+$ are continuous and bounded by $2M$.

With these rates solve the linear population equation from $m^0$. Its transition generator conserves total mass and has nonnegative off-diagonal entries. Positivity follows either by the integral equation with an integrating factor for each coordinate or by approximation with small time-step stochastic transition matrices. More explicitly, while coordinates are nonnegative each satisfies $\dot m_i\geq-2M(d-1)m_i$; comparison gives $m_i(t)\geq m_i^0e^{-2M(d-1)(t-t_0)}>0$, excluding a first loss of nonnegativity. The sum remains one. The coordinate drift is bounded in magnitude by $2Md$, so the resulting population path has a common Lipschitz bound.

Let $\mathcal K$ be the set of continuous simplex paths starting at $m^0$ with Lipschitz constant at most $2Md$ in the supremum norm. It is nonempty, convex, closed, and compact in the Banach space of continuous paths: values are bounded and the common Lipschitz modulus gives Arzelà--Ascoli compactness. The construction just described maps $\mathcal K$ into itself. It is continuous: uniform convergence of $z$ gives uniform convergence of its running and terminal data by smoothness on the compact simplex; the bounded backward ODE solutions converge by Gronwall using the Lipschitz constant of $H$ on $[-M,M]^d$; the induced rates and the solutions of the linear forward ODE then converge by another Gronwall estimate.

Schauder's fixed-point theorem states that a continuous self-map of a nonempty compact convex subset of a Banach space has a fixed point. All those hypotheses have just been checked. A fixed point of this map is precisely a solution of (FB). Its two paths are $C^1$ because their right sides are continuous. At $t_0=T$ simply prescribe $m(T)=m^0,u(T)=g(m^0)$. This covers the zero-length boundary case as well. ∎

Theorem (Uniqueness of characteristics under (MON)): Under (MON), solution (FB) is unique for every initial time and interior initial law.
Proof: Each $H_i$ is convex and $C^1$, being a sum of positive-part squares of linear forms. Its convex remainder \[ D_{H_i}(v,u)=H_i(v)-H_i(u)-DH_i(u)\cdot(v-u) \] is nonnegative. Direct differentiation of the displayed Hamiltonian gives the vector identity \[ b(m,u)=-\sum_i m_i DH_i(u). \] For two solutions $(m,u),(n,v)$ with the same initial law, set $w=u-v$, $\delta m=m-n$. Differentiating their scalar product and using this vector identity gives, term by term, \begin{align*} \frac d{dt}(w\cdot\delta m) ={}&-(f(m)-f(n))\cdot(m-n)\\ &-\sum_i m_iD_{H_i}(v,u)-\sum_i n_iD_{H_i}(u,v)\\ \leq{}&-\alpha\|m-n\|_2^2. \end{align*} To check the signs, the coefficient of $m_i$ before introducing the remainder is $H_i(u)-H_i(v)-DH_i(u)\cdot(u-v)=-D_{H_i}(v,u)$; the coefficient of $n_i$ is $-D_{H_i}(u,v)$.

At the initial time the scalar product is zero. At terminal time it is $(g(m(T))-g(n(T)))\cdot(m(T)-n(T))\geq0$. Integration therefore yields $0\leq-\alpha\int_{t_0}^T\|m-n\|_2^2dt\leq0$. Continuity forces $m=n$ at every time. With this fixed population path, the backward ODE and terminal condition have a unique solution by local Lipschitz uniqueness on the bounded range, so $u=v$. ∎

Theorem (An intrinsic continuous characteristic feedback field): Under (MON), define $\mathcal U(t_0,m^0)$ as the initial value $u(t_0)$ of the unique solution (FB). Then $\mathcal U$ is bounded and continuous on $[0,T]\times\operatorname{int}\Delta_d$ and has terminal value $g$. It is the unique bounded continuous field with the following intrinsic property: for every initial time and interior law there exists an absolutely continuous path $m$ with $\dot m=b(m,\mathcal U(t,m))$ such that $u(t)=\mathcal U(t,m(t))$ is absolutely continuous, satisfies $\dot u=H(u)-f(m)$, and has terminal value $g(m(T))$.
Proof: Existence, uniqueness and the bound $M$ make the definition unambiguous. To prove continuity, take initial conditions $(t_n,m_n)\to(t,m)$ with $m$ interior. Extend their solution paths constantly to earlier times when necessary. The population paths have the uniform Lipschitz bound from the existence lemma. Their value derivatives are bounded by $2(d-1)M^2+\|f\|_\infty$, so the value paths also have a uniform Lipschitz bound. Arzelà--Ascoli gives a uniformly convergent subsequence of both extended paths. Passing to the integrated ODEs on $[t,T]$ and their terminal conditions shows that the limit solves (FB) from $(t,m)$. Uniqueness identifies it. Every convergent subsequence has the same limit; compactness then implies convergence of the whole sequence of initial values. This proves continuity, including at $t=T$.

Take a solution from $(t_0,m^0)$. Its restriction from any later time $s$ solves (FB) with initial population $m(s)$. By uniqueness its value at that time is $\mathcal U(s,m(s))$. Hence the solution gives the required pathwise property. Conversely, a field possessing that property produces a solution of (FB) for every starting point. Uniqueness of (FB) forces its value at each starting point to equal $\mathcal U$. This is a deterministic characteristic admissibility principle. It is not yet a theorem identifying parabolic $L^1$ limits with that principle. ∎

Corollary (Classical consistency and uniqueness): Under (MON), any bounded $C^1$ classical solution of the deterministic master equation equals $\mathcal U$ throughout the interior. In particular there is at most one such classical solution, and its component differences equal those of $\mathcal U$.
Proof: For a bounded classical field $U$, solve $\dot m=b(m,U(t,m))$ from an interior point. The drift is locally Lipschitz in $m$. Boundedness of $U$ gives a bound on all outgoing rates, and $\dot m_i\geq-Cm_i$ prevents approach to the boundary in finite time. Thus the characteristic extends to terminal time. The chain rule and master equation give \[ \frac d{dt}U_i(t,m(t))=\partial_tU_i+b\cdot DU_i=H_i(U)-f_i(m). \] The terminal condition supplies $g(m(T))$. This is the unique (FB) solution, so at the starting point $U=\mathcal U$. Since that point was arbitrary, the fields agree. This argument does not assert that a continuous characteristic field is differentiable. ∎
Proposition (An explicit genuinely nonpotential example satisfying (MON)): For $d=3$, let $f(m)=m+Km$, $g(m)=0$, where \[ K=\begin{pmatrix}0&-1&0\\1&0&0\\0&0&0\end{pmatrix}. \] Then (MON) holds with $\alpha=1$, but no smooth scalar potential has these tangent cost differences, even after adding a common state-dependent component to all the $f_i$.
Proof: Skew symmetry gives $(m-n)\cdot K(m-n)=0$, hence $(f(m)-f(n))\cdot(m-n)=\|m-n\|_2^2$. The terminal condition is monotone with equality.

Use simplex coordinates $x=m_1,y=m_2,m_3=1-x-y$. The tangent covector of a putative potential would have components \[ f_1-f_3=2x-1,\qquad f_2-f_3=2x+2y-1. \] Its cross derivatives are respectively zero and two. Equality of mixed partial derivatives for a smooth scalar function is necessary and fails on every interior open set. Adding the same scalar component to every $f_i$ leaves these differences unchanged. Thus the example is genuinely nonpotential in the feedback-relevant sense, not just an artifact of choosing an ambient extension or value gauge. ∎

Proposition (The actual feedback transport product is not determined by an $L^1$ jump limit): There are two sequences of smooth bounded fields $U^\varepsilon$ and $\widetilde U^\varepsilon$ on an interior three-state simplex rectangle, having the same $L^1$ limit and the same limiting component differences, for which $b(m,U^\varepsilon)\cdot D(U_1^\varepsilon-U_3^\varepsilon)$ and the analogous expression for $\widetilde U^\varepsilon$ have different distributional limits.
Proof: Choose an interior rectangle in coordinates $(x,y)=(m_1,m_2)$ crossing $x=c>0$ and staying inside $x+y<1$. Set $U=(Z_1,Z_2,0)$. Let $\theta\in C^\infty(\mathbb R)$ be nondecreasing, zero for arguments at most $-1/4$ and one for arguments at least $1/4$. For sequence A, set \[ Z_1^\varepsilon=3+\theta((x-c)/\varepsilon+1),\qquad Z_2^\varepsilon=1+\theta((x-c)/\varepsilon-1). \] For sequence B, exchange the $+1$ and $-1$ shifts in these two formulas. All fields are independent of $y$ and time. Both sequences converge in $L^1$ to $(3,1)$ for $x<c$ and $(4,2)$ for $x>c$. They always satisfy $Z_1>Z_2>0$. In this ordered region the model's drift is exactly \[ b_1(m,U)=-m_1[(Z_1-Z_2)+Z_1]=-x(2Z_1-Z_2). \] Since $U_3=0$ and the fields depend only on $x$, the transport expression in the proposition is $-x(2Z_1-Z_2)\partial_xZ_1$.

In sequence A, $Z_2=1$ throughout the support of $\partial_xZ_1$, because its transition occurs later in a disjoint interval. Test against a smooth compactly supported function of $(x,y)$, change variables across the first transition, and use its shrinking support to replace $x$ and the test function by their values at $x=c$. The limiting coefficient of the delta measure on $x=c$ is \[ -c\int_3^4(2z-1)\,dz=-6c. \] In sequence B, $Z_2=2$ throughout the $Z_1$ transition, so the coefficient is instead \[ -c\int_3^4(2z-2)\,dz=-5c. \] The replacement in the test integral is justified by uniform continuity of its smooth coefficients on the shrinking support and the bounded total variation $\int\partial_xZ_1dx=1$. Thus the distributional limits are respectively $-6c\,\delta_{x=c}\otimes dy$ and $-5c\,\delta_{x=c}\otimes dy$ on the rectangle. They differ although their $L^1$ field limits agree. ∎

Scope of the obstruction.
These two smooth sequences are not claimed to solve the parabolic master equation or to have the input's terminal data. They disprove a putative definition of the unregularized nonlinear product based on $L^1$ convergence alone. They do not disprove independence among genuine admissible noisy solutions, for which the PDE might enforce a particular transition path.
A conditional, but not sufficient, passage to a classical limit.
If along a given admissible sequence $U^\varepsilon\to U$ in $C^1$ on every interior compact set, $\varepsilon D^2U^\varepsilon\to0$ there, and the terminal data are attained by the limiting field, direct substitution in the regularized equation gives the deterministic classical equation. Local vanishing of $\phi_\varepsilon$ eliminates its terms because values and first derivatives are then locally uniformly bounded. Under (MON), the corollary identifies this limit as $\mathcal U$. These are additional convergence hypotheses, not estimates proved for the original admissible class. Treating them as an answer to the compactness question would be circular.
Tiny checks and computation.
The explicit example has three states, the smallest dimension required in the input, and cross derivatives $0$ and $2$. The energy calculation reduces to the exact skew-symmetric identity $v\cdot Kv=0$. The jump-profile coefficients are the exact integrals $-6c$ and $-5c$. The accompanying script checks these coefficients algebraically and the displayed curl values. Minimal pseudocode is: 
\begin{lstlisting} left, right = 3, 4 for fixed_Z2 in [1, 2]:     coefficient_over_c = -(right2-left2-fixed_Z2*(right-left))     assert coefficient_over_c in [-6, -5] assert cross_derivative_first == 0 assert cross_derivative_second == 2 \end{lstlisting}
No discretized PDE calculation is used as proof of regularization-independent selection.

5) VERIFICATION

The Schauder map has a compact convex domain and verified continuous dependence; uniqueness does not secretly assume a unique fixed point before applying monotonicity. The forward--backward energy signs were checked term by term, and the terminal monotonicity inequality has the correct direction. Interior positivity follows from bounded exit rates, not from an unjustified boundary condition. The nonpotential example is tested in tangent coordinates and is not made potential by a common value gauge. The two smoothing paths use the actual three-state drift coefficient, not a generic unrelated conservation law. Most importantly, intrinsic characteristic uniqueness and classical consistency do not prove that the original parabolic family is compact or that its limits satisfy characteristic admissibility.

\Needspace{8\baselineskip}

6) FINAL

LABEL: UNRESOLVED for the full selected-feedback compactness and regularization-independence target.

Strongest checked partial results. Existence and uniqueness of deterministic forward--backward solutions under (MON); a unique bounded continuous intrinsic characteristic field and classical consistency; an explicit genuinely nonpotential strongly monotone example; and an explicit pair of smoothings proving that $L^1$ convergence alone does not determine the nonlinear feedback transport product.

Exact first gap. No $L^1_{\rm loc}$ precompactness theorem has been proved here for all $Z^{\varepsilon,\phi}$ in the input's admissible class, even for the displayed strongly monotone nonpotential example. After compactness, identification with characteristic admissibility is a further unproved bridge.

Top three next moves. (1) Derive uniform interior translation or variation estimates for the exact Kimura system, retaining all $\varepsilon$-gradient and repulsion terms. (2) Prove that every actual parabolic limit transports along the unique deterministic characteristics despite only weak feedback convergence. (3) Analyze admissible viscous transition profiles to determine which nonconservative paths the system selects, rather than allowing arbitrary smooth approximations.

Minimal counterexample perspective. A genuine disproof of regularization independence would require the same smooth nonpotential $f,g$, two admissible repulsion families, and distinct verified limits of their actual PDE solutions. The two constructed smoothing paths show a definitional obstruction but do not meet those stronger requirements.

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.
  • [bertucci] Charles Bertucci. Monotone solutions for mean field games master equations: finite state space and optimal stopping. Journal de l'École polytechnique---Mathématiques 8 (2021), 1099--1132. DOI: 10.5802/jep.167. https://www.numdam.org/articles/10.5802/jep.167/.

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Econ-31-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-31

Short-horizon uniqueness and a curl obstruction beyond potential master equations

Problem C73-30. Status: unresolved. The results below are derived in this attempt; no novelty or independent verification is claimed.

Scope of the attempt

The target is a regularization-independent weak feedback field for nonpotential finite-state master equations. The potential conservative formulation and its weak uniqueness theorem are already known [CeD, Theorem 2.7, Section 6]. We prove short-horizon deterministic uniqueness without a potential, a conditional compactness-to-selection theorem for the exact noisy equations, and a concrete obstruction to imposing a gradient condition. The central weak compactness problem after derivative blow-up remains open here.

Let $d\ge3$, $\Delta_d=\{m\ge0:\sum_i m_i=1\}$, and let $f,g:\Delta_d\to{\mathbb{R}}^d$ be bounded Lipschitz functions. Write \[ F_*=\|f\|_\infty,\quad G_*=\|g\|_\infty,\quad M=G_*+TF_*, \] where vector norms are component suprema. Assume $\|f(m)-f(n)\|_\infty\le L_f\|m-n\|_1$ and the analogous bound with $L_g$. Let \[ H_i(u)=\frac12\sum_{j\ne i}(u_i-u_j)_+^2,\quad b_i(m,u)=\sum_{j\ne i}[m_j(u_j-u_i)_+-m_i(u_i-u_j)_+]. \] An equilibrium on $[t_0,T]$ has \begin{equation}\label{wsref-0-63-68-61-72-73} -\dot u_i+H_i(u)=f_i(m),\quad\dot m=b(m,u),\quad m(t_0)=m_*,\quad u(T)=g(m(T)). \end{equation}

A contraction that does not use potentiality

Lemma: For each continuous path $\mu:[t_0,T]\to\Delta_d$, the frozen backward equation $-\dot u_i+H_i(u)=f_i(\mu)$, $u(T)=g(\mu(T))$, has a unique solution with $\|u\|_\infty\le M$. For two paths $\mu,\nu$, their solutions satisfy \[ \sup_t\|u^\mu(t)-u^\nu(t)\|_\infty \le[L_g+(T-t_0)L_f]\sup_t\|\mu(t)-\nu(t)\|_1. \]
Proof: Use backward time $s=T-t$, so the equation is $u'_i=f_i(\mu)-H_i(u)$. Local existence and uniqueness follow from local Lipschitz continuity of $H$. At a maximum component, $H_i\ge0$, so the upper Dini derivative of $\max_i u_i$ is at most $F_*$. At a minimum component, $H_i=0$, so the lower Dini derivative of $\min_i u_i$ is at least $-F_*$. Starting from the terminal bounds proves $|u_i|\le M$ and prevents finite-time blow-up.

For the difference $z=u^\mu-u^\nu$, choose a component achieving $\max_i z_i$. Then $u_i^\mu-u_j^\mu\ge u_i^\nu-u_j^\nu$ for every $j$, giving $H_i(u^\mu)\ge H_i(u^\nu)$. Its upper derivative is therefore at most $\|f(\mu)-f(\nu)\|_\infty$. Apply the same argument with the roles reversed for the negative difference. Integrating these maximum bounds from terminal time and using the Lipschitz hypotheses proves the stated estimate. The finite maximum Dini inequalities can equivalently be justified by a small strict linear barrier in time and a first-contact argument. ∎

Lemma: For a continuous field $u(t)$ with $\|u\|_\infty\le M$, the forward equation $\dot m=b(m,u)$ preserves the simplex and has a unique solution. If $m,n$ have the same initial law and are driven by $u,v$, respectively, then \[ \sup_{t_0\le t\le T}\|m(t)-n(t)\|_1 \le4(d-1)(T-t_0)\sup_t\|u(t)-v(t)\|_\infty. \] If $m_{*i}>0$, then $m_i(t)\ge m_{*i}e^{-2M(d-1)(t-t_0)}$.
Proof: With $u$ fixed, the equation is the forward equation of a time-dependent finite-state Markov chain, whose off-diagonal rates are bounded by $2M$. Positivity can also be checked from the nonnegative incoming terms at any zero coordinate; conservation follows by summing the equations. Uniqueness follows from linear ODE theory.

For the difference use the Markov generator associated with $u$ as the homogeneous part. Its forward propagator contracts $\ell^1$ because it has nonnegative entries and column sums one. The remaining forcing is the drift difference at the law $n$ due to changing rates. Its $\ell^1$ norm is at most \[ 2\sum_i n_i\sum_{j\ne i}|(u_i-u_j)_+-(v_i-v_j)_+| \le4(d-1)\|u-v\|_\infty. \] Variation of constants gives the bound. Finally $\dot m_i\ge-2M(d-1)m_i$; multiplying by the integrating factor proves the coordinate lower bound. ∎

Theorem (Nonpotential short-horizon equilibrium uniqueness): If \begin{equation}\label{wsref-0-73-68-6f-72-74} q=4(d-1)T(L_g+TL_f)<1, \end{equation} then for every $t_0\in[0,T]$ and every $m_*\in\Delta_d$ there is exactly one solution of (chars). The field $U(t_0,m_*)=u(t_0)$ is bounded and Lipschitz in $m_*$, uniformly over $t_0$, with \[ \|U(t_0,m_*)-U(t_0,n_*)\|_\infty \le\frac{L_g+TL_f}{1-q}\|m_*-n_*\|_1. \] No symmetry of the derivatives of $f$ or $g$ is required.
Proof: On the complete space $C([t_0,T],\Delta_d)$ with fixed initial law, map $\mu$ to the law obtained by first solving the frozen backward equation and then its forward equation. Lemmas Lemma--Lemma show this map has Lipschitz constant at most $q<1$. The contraction principle gives a unique fixed point and hence an equilibrium.

For different initial laws, the same forward comparison includes the initial difference, since its Markov propagation contracts $\ell^1$. If $D$ is the supremum distance between the resulting equilibrium laws, then $D\le\|m_*-n_*\|_1+qD$. Lemma Lemma now gives the stated bound for $U$. The boundedness follows there as well. All estimates used only the Lipschitz constants and the positive-part rate structure. ∎

The construction defines the field by a boundary-value problem. It does not identify the joint $(u,m)$ system with an arbitrary initial-value semiflow. Classical regularity of $U$ is not asserted by this theorem.

What compactness would suffice for noisy selection

For each admissible repulsion family, let $U^\varepsilon$ solve exactly the uncorrected regularized master equation of the question. In shorthand its equation is \begin{equation}\label{wsref-0-76-69-73-63-6f-75-73} \partial_tU_i^\varepsilon+\varepsilon A:D^2U_i^\varepsilon +b^\phi(m,U^\varepsilon)\cdot DU_i^\varepsilon +2\varepsilon\sum_jm_j(\partial_iU_i^\varepsilon-\partial_jU_i^\varepsilon) -H_i(U^\varepsilon)+f_i+\sum_{j\ne i}\phi_\varepsilon(m_j)(U_j^\varepsilon-U_i^\varepsilon)=0, \end{equation} with $A_{jk}=m_j\delta_{jk}-m_jm_k$, terminal value $g$, and $b_i^\phi=b_i+\phi_\varepsilon(m_i)-m_i\sum_j\phi_\varepsilon(m_j)$.
Theorem (Conditional interior selection): Assume (short), smooth $f,g$, and consider any collection of admissible repulsion families. Suppose for every family its solutions satisfy a common bound $|U_i^\varepsilon|\le M_0$, and for every compact $K\subset\operatorname{int}\Delta_d$ there is a bound, uniform in $\varepsilon$ and the families considered, \[ \sup_{[0,T]\times K}\left(|DU^\varepsilon|+|D^2U^\varepsilon|+|\partial_tU^\varepsilon|\right)\le C_K. \] Suppose $\phi_\varepsilon\to0$ locally uniformly on $(0,1]$ for each family. Then $U^\varepsilon$ converges locally uniformly on $[0,T]\times\operatorname{int}\Delta_d$ to the field of Theorem Theorem (Nonpotential short-horizon equilibrium uniqueness). In particular, its differences $Z_i^\varepsilon=U_i^\varepsilon-U_d^\varepsilon$ have a unique $L^1_{\mathrm{loc}}$ limit independent of the family, and the family-independence target in the question holds for these families.
Proof: On every compact interior cylinder, the bounds give equicontinuity of $U^\varepsilon$ in time and space. Arzelà--Ascoli and a diagonal extraction give locally uniform convergence to $W$. The spatial Hessian bound implies local equicontinuity of $DU^\varepsilon$ in space. To obtain time equicontinuity of these gradients, approximate each directional derivative by a difference quotient with step $h$ inside a slightly larger compact set: the Hessian bound gives error $O(h)$, while the time-Lipschitz bound on $U^\varepsilon$ gives variation $O(|t-s|/h)$. Choosing $h=\sqrt{|t-s|}$ proves local time equicontinuity. Thus, after extraction, the spatial gradients converge locally uniformly too.

In (viscous) the Hessian and extra-gradient terms multiplied by $\varepsilon$ vanish uniformly on interior compact sets, as do the repulsion terms. Integrate the equation between two times and pass to the limit. The resulting right-hand side is continuous, so $W$ is $C^1$ in time and satisfies the classical first-order master equation \[ \partial_tW_i+b(m,W)\cdot DW_i-H_i(W)+f_i=0,\qquad W(T,m)=g(m). \] It is locally $C^1$ in space and bounded by $M_0$.

Starting from any interior $(t_0,m_*)$, solve $\dot m=b(m,W(t,m))$. Local Lipschitz continuity gives local uniqueness. Boundedness of $W$ gives $m_i(t)\ge m_{*i}e^{-2M_0(d-1)T}$, so the trajectory remains in a compact interior region and continues to $T$. Along it, the chain rule shows $u_i(t)=W_i(t,m(t))$ solves (chars). Theorem Theorem (Nonpotential short-horizon equilibrium uniqueness) identifies $W(t_0,m_*)=U(t_0,m_*)$. Since every subsequential limit is identical, compactness yields convergence of the whole family. Any two families converge to this same limit; uniform convergence on compact cylinders implies the required $L^1$ convergence of differences on every compact measurable set. ∎

These derivative bounds are sufficient conditions, not consequences of the admissibility conditions in the question, which permit dependence on $\varepsilon$. This theorem handles a genuinely nonpotential class before singular gradients become an issue. It does not provide an intrinsic weak admissibility condition after such singularities.

Why the potential gradient condition cannot be retained

Proposition (A smooth nonpotential terminal field): For $d=3$, choose $f=0$ and $g=(m_2,0,0)$. In the chart $x=m_1$, $y=m_2$, $m_3=1-x-y$, the terminal feedback-difference field is $Z=(y,0)$. It is not the gradient of any scalar distribution on any nonempty open subset of the simplex interior. Nevertheless it satisfies the one-sided Lipschitz estimate \[ (Z(z)-Z(z'))\cdot(z-z')\le\tfrac12|z-z'|^2. \] For sufficiently small $T$, Theorem Theorem (Nonpotential short-horizon equilibrium uniqueness) applies to these data.
Proof: If $Z=DP$ distributionally, commutation of distributional mixed derivatives would imply $\partial_yZ_1=\partial_xZ_2$. Here these derivatives equal $1$ and $0$, respectively, a contradiction even on an arbitrarily small open set. For $z-z'=(a,b)$, the left side of the one-sided inequality is $ab\le(a^2+b^2)/2$. The data are smooth and bounded on the simplex, with $L_g\le1$ and $L_f=0$, so $8T<1$ suffices for (short). ∎

Thus a one-sided Lipschitz bound alone is compatible with a nonzero curl. The scalar potential representation used for the known conservative weak theorem cannot simply be imposed in a class that must agree with nonpotential classical data. The remaining target is to find compactness and an admissibility/uniqueness principle that control both compression and this rotational part when derivatives are no longer uniformly bounded. No result here proves those estimates or regularization independence for the unrestricted admissible class. The full catalogue problem remains unresolved.

References

  • [CD] P. Cardaliaguet and F. Delarue, Selected topics in mean field games, Proc. ICM 2022, vol. 5, 3660--3703, EMS Press (2023), Theorem 4.6 and Section 4.4. 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. Theorem 2.7 and Section 6 prove the potential conservative weak uniqueness result; they are not a nonpotential theorem. https://arxiv.org/abs/2005.12153.

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Econ-31-A3 · Partial result · LLM-generated · gpt-6 astra, claude opus 5.5, gpt-6 astra pro

Shock selection in a nonpotential finite-state master equation

Shock selection in a nonpotential
\protect
\ finite-state master equation

Ji Ho Bae
LawAll AI

10 October 2026

\begin{abstract} We give an explicit family of smooth nonpotential costs for which the uncorrected finite-state master equation has a unique selected weak feedback field. Every admissible vanishing boundary repulsion gives the same local $L^1$ limit, without a rate condition between the repulsion and the common noise. The limit has an intrinsic scalar entropy characterization. The family includes interior shocks: its permitted time horizon is independent of front steepness, and a scaling extends the result to every finite horizon. The proof uses localization of the passive values and a Cole--Hopf comparison construction that remains valid through shock formation. \end{abstract}

The problem and a class of data

Fix $d\ge3$ and the simplex $\Delta_d=\{m_i\ge0:\sum_i m_i=1\}$. Write \[ H_i(U)=\frac12\sum_{j\ne i}(U_i-U_j)_+^2,\qquad b_i(m,U)=\sum_{j\ne i}\{m_j(U_j-U_i)_+-m_i(U_i-U_j)_+\}. \] The uncorrected regularization considered here is \begin{align}\label{wsref-0-6d-61-73-74-65-72} 0={}&\partial_tU_i+\varepsilon\sum_{j,l}(m_j\delta_{jl}-m_jm_l)\partial^2_{m_jm_l}U_i +(b+R^\phi)\cdot DU_i+2\varepsilon(e_i-m)\cdot DU_i-H_i(U)\notag\\ &+f_i(m)+\sum_{j\ne i}\phi_\varepsilon(m_j)(U_j-U_i),\qquad U_i(T,m)=g_i(m), \end{align} where $R_i^\phi=\phi_\varepsilon(m_i)-m_i\sum_j\phi_\varepsilon(m_j)$ and derivatives are tangent to the simplex. Following the selection formulation in catalogue C73-30, let $\mathcal A(f,g,T)$ consist of smooth nonnegative families $\phi_\varepsilon$, converging to zero uniformly on every compact subset of $(0,1]$, for which (master) has a unique classical solution continuous on the closed cylinder, with bounded value and first derivatives and locally bounded second derivatives in the interior. No bound on these derivatives uniform in $\varepsilon$ is imposed. The feedback field is $(U_i-U_d)_{i<d}$.

The problem asks for nonpotential assumptions and an intrinsic admissibility condition giving existence of selected limits, uniqueness independent of the admissible regularization, and agreement with classical solutions. The potential theory and this extension are discussed in [CD, §4.3--4.4]. We address the sufficient-condition question with a family that undergoes loss of classical regularity.

Put $q=(d-1)/2$, $p=m_1$, and $r=(r_2,\ldots,r_{d-1})$, where $r_j=m_j-m_d$. Thus \[ m_d=\frac{1-p-\sum_{j=2}^{d-1}r_j}{d-1},\qquad m_j=m_d+r_j. \] Choose fixed smooth cutoffs $\chi:\mathbb R\to[0,1]$ and $\beta:\mathbb R^{d-2}\to[0,1]$ such that \[ \chi=0\ \text{on }(-\infty,1/128],\quad \chi=1\ \text{on }[1/64,\infty), \] and, with $R_d=1/(16d)$, $\beta=1$ on $\{|r|_\infty\le R_d/4\}$ and $\beta=0$ on $\{|r|_\infty\ge R_d/2\}$. Set $\alpha=1/16$, $y=\sqrt p$, $\theta=y-1/2-\alpha r_2$, and, for $k\ge1$, define \begin{equation}\label{wsref-0-64-61-74-61} f=0,\qquad g=(G_k,0,\ldots,0),\qquad G_k(p,r)=\frac1q\left[24(1-\chi(p))+\chi(p)\frac{2-\beta(r)\tanh(k\theta)}{\sqrt p}\right]. \end{equation} The second term is defined as zero where $\chi=0$. These data are smooth on a neighborhood of $\Delta_d$, and $1/q\le G_k\le34/q$ there on the simplex.

For each admissible $r$, let $0<y<L(r)$ be the interval corresponding to the simplex interior, where \[ L(r)^2=1-\sum_{j=2}^{d-1}r_j+(d-1)\min(0,r_2,\ldots,r_{d-1})>0. \] Define $v_0(y,r)=qyG_k(y^2,r)$, extend it by zero to $y<0$, and extend it constantly to the right of $L(r)$. The construction gives $0\le v_0\le3$. Let $v(s,y,r)$ be the bounded nonnegative entropy solution of \begin{equation}\label{wsref-0-62-75-72-67-65-72-73} v_s+\partial_y(v^2/2)=0,\qquad v(0,y,r)=v_0(y,r). \end{equation} The value on $0<y<L(r)$ is independent of the chosen nonnegative right extension. Indeed, its Hopf--Lax minimizers lie in $[0,y]$: the initial primitive is nondecreasing, and the quadratic penalty increases for a footpoint to the right of $y$. The zero extension also gives the left trace zero. We use the usual bounded entropy uniqueness and vanishing-viscosity theorem for this scalar Cauchy problem [Kru].

Theorem: For each $d\ge3$ and the fixed cutoffs above, there is a $T_0>0$, independent of $k\ge1$, with the following properties. For $0<T\le T_0$ and the data (data), the class $\mathcal A(f,g,T)$ is nonempty. For every $\phi\in\mathcal A(f,g,T)$, \begin{equation}\label{wsref-0-63-6f-6e-76-65-72-67-65-6e-63-65} U^{\varepsilon,\phi}\longrightarrow U^*(t,m)=\left(\frac{v(T-t,\sqrt p,r)}{q\sqrt p},0,\ldots,0\right) \quad\text{in }L^1_{\rm loc}((0,T)\times\operatorname{int}\Delta_d). \end{equation} Consequently all selected feedback fields coincide, and any two admissible families have feedback distance tending to zero on every interior compact set.

Among bounded feedback fields $Z=(Z_1,\ldots,Z_{d-1})$, an intrinsic admissibility condition is that $Z_2=\cdots=Z_{d-1}=0$, $Z_1\ge0$, and that for almost every $r$ the function $q\sqrt p\,Z_1$, expressed in time $s=T-t$ and in $y=\sqrt p$, is an entropy solution of (burgers) on $0<y<L(r)$ with initial trace $v_0(\cdot,r)$: its $L^1$ distance to $v_0(\cdot,r)$ on each compact subinterval of $(0,L(r))$ tends to zero as $s\downarrow0$ outside a null set of times. This condition determines the selected field uniquely and agrees with the classical master field whenever the latter exists.

The data are nonpotential even modulo a common component. If $kT>1$, the selected field has an interior shock for $1/k<s\le T$. Thus the assertion includes a nonclassical regime, while retaining every family in $\mathcal A$.

Bounds that do not use derivatives uniform in the noise

The following estimates also justify the reduction to one active value. All constants may depend on the fixed data and on the compact set, but not on $\varepsilon$ or on derivatives of $\phi_\varepsilon$.
Lemma: Suppose $f=0$, $g=(G,0,\ldots,0)$ and $0<\gamma\le G\le M$. Every admissible solution satisfies $0\le U_i\le M$. For $D_\eta=\{m_i\ge\eta\ \forall i\}$, put \[ C=(d-1)M+1,\quad \delta=\tfrac12\eta e^{-CT},\quad a_\varepsilon=\sup_{[\delta,1]}\phi_\varepsilon,\quad e_\varepsilon=d\exp\left(-\frac{\eta^2}{4\varepsilon T e^{2CT}}\right). \] For all sufficiently small $\varepsilon$, on $[0,T]\times D_\eta$, \begin{equation}\label{wsref-0-70-61-73-73-69-76-65} 0\le U_i\le M(Ta_\varepsilon+e_\varepsilon)\quad(i>1),\qquad U_1\ge\gamma e^{-[qM+(d-1)a_\varepsilon]T}(1-e_\varepsilon). \end{equation}
Proof: First fix $\varepsilon$. The ordinary maximum principle extends to the degenerate boundary as follows. Add $hB_a(m)=h\sum_jm_j^a$, $0<a<1$, to a component being maximized. Bounded first derivatives force the maximum into the interior: adding a small empty coordinate gains order $h\rho^a$, whereas the other changes are $O(\rho)$. At this maximum the Hamiltonian and value-switching terms have the appropriate nonpositive signs. The drift without $R^\phi$ has coordinate lower bounds $-C_\varepsilon m_j$, since the solution is bounded for this fixed $\varepsilon$. The repulsion contributes \[ -hR^\phi\cdot DB_a=-ha\sum_j\phi_\varepsilon(m_j)\{m_j^{a-1}-B_a(m)\}. \] Because $B_a\le d^{1-a}$, each summand with $m_j\le1/d$ is nonpositive. The remaining coordinates lie in $[1/d,1]$, where $\phi_\varepsilon$ is bounded. Thus this contribution is $O_\varepsilon(h)$ as well, regardless of the size of the repulsion close to zero.

If $m_l\ge1/d$, the tangent first-order condition bounds $ha m_j^{a-1}$ by $L_\varepsilon+ha d^{1-a}$, where $L_\varepsilon$ bounds the first derivatives. The diffusion error is at most \[ \varepsilon d(1-a)(L_\varepsilon+ha d^{1-a}), \] and the drift error is $O_\varepsilon(h)$. A strict time barrier, followed by $h\downarrow0$ and then $a\uparrow1$, gives $U_i\le M$. Applied to $-U$ it gives $U_i\ge0$; at a minimum component $H_i=0$. No uniform bound on $L_\varepsilon$ was used.

Stop an interior diffusion, with the Kimura covariance and drift $b(m,U)+R^\phi+2\varepsilon(e_I-m)$, on leaving $D_\delta$ or reaching $T$. The label $I$ may switch in any adapted way. Once $(d-1)a_\varepsilon+2\varepsilon\le1$, each drift coordinate is at least $-Cm_j$. After multiplication by $e^{C(s-t)}$, its finite-variation part is nondecreasing. Its martingale part has quadratic variation at most $\varepsilon T e^{2CT}/2$, because the coordinate covariance rate is $2\varepsilon m_j(1-m_j)\le\varepsilon/2$. An exit requires one such martingale to fall by at least $\eta/2$. The exponential martingale inequality and a union bound give exit probability at most $e_\varepsilon$.

For a passive component, write \[ -H_i(U)=\sum_{j\ne i}c_{ij}(U_j-U_i),\qquad c_{ij}=\tfrac12(U_i-U_j)_+. \] Its $j=1$ term is nonpositive and can be dropped for an upper comparison. The repulsion term involving $U_1$ is at most $M\phi_\varepsilon(m_1)$. Keep the nonnegative switching rates between passive labels. The stopped cooperative-system representation has terminal value zero, exit value at most $M$, and source at most $Ma_\varepsilon$. This proves the first estimate in (passive). All coefficients are bounded on the stopped interior domain for fixed $\varepsilon$.

Finally $H_1\le qU_1^2\le qMU_1$, and its repulsion value term is at least $-(d-1)a_\varepsilon U_1$. The stopped scalar comparison with killing rate $qM+(d-1)a_\varepsilon$, terminal value $\gamma$ and exit value zero gives the second estimate. ∎

We also record a local consistency consequence used at the boundary of the shock region. Write $A=(m_j\delta_{jl}-m_jm_l)_{j,l}$, so that the second-order part of (master) is $\varepsilon\operatorname{tr}(AD^2U_i)$, and $|x|_1=\sum_l|x_l|$ for $x\in\mathbb R^d$.

Lemma: Suppose $f=0$, $g=(G,0,\ldots,0)$, $0<\gamma\le G\le M$ and $\phi\in\mathcal A(f,g,T)$. Let $\mathcal O\subset\operatorname{int}\Delta_d$ be open, and let $U^{\rm cl}$, once continuously differentiable in $t$ and twice in $m$, solve the deterministic master equation \[ \partial_tU^{\rm cl}_i+b(m,U^{\rm cl})\cdot DU^{\rm cl}_i-H_i(U^{\rm cl})=0,\qquad U^{\rm cl}(T)=g, \] on a neighborhood of $[0,T]\times(\operatorname{int}\Delta_d\setminus\mathcal O)$ in $[0,T]\times\operatorname{int}\Delta_d$. Let $\zeta:\Delta_d\to[0,1]$ be smooth with $\zeta=1$ on $\mathcal O$, put $|D\zeta|=\sup_{\Delta_d}\max_l|(\partial_{m_l}-\partial_{m_d})\zeta|$ and $M_1=2(d-1)M+1$, and assume $T(M_1|D\zeta|+1)<1$. Let $\mathcal K\subset\operatorname{int}\Delta_d$ be compact with $\zeta=0$ on $\mathcal K$, choose $\eta>0$ with $\mathcal K\subset D_\eta$, and put $C=(d-1)M+1$, $\delta'=\frac12\eta e^{-(C+1)T}$ and $a'_\varepsilon=\sup_{[\delta',1]}\phi_\varepsilon$. Then there is $c>0$ such that \begin{equation}\label{wsref-0-6c-6f-63-61-6c-63-6c-61-73-73-69-63-61-6c} \sup_{[0,T]\times\mathcal K}|U^{\varepsilon,\phi}-U^{\rm cl}|=O\big(\varepsilon+a'_\varepsilon+e^{-c/\varepsilon}\big). \end{equation}
Proof: Put $N=D_{\delta'}\setminus\mathcal O$, a compact subset of the interior near which $U^{\rm cl}$ is defined, so $U^{\rm cl}$ and its derivatives of the orders above are bounded on $[0,T]\times N$; also $\mathcal K\subset N$. Let $M_0=M+\sup_{[0,T]\times N}|U^{\rm cl}|$, and take $\varepsilon$ so small that $(d-1)a'_\varepsilon+2\varepsilon\le1$, $2da'_\varepsilon+4\varepsilon\le1$ and $\varepsilon\le\delta'/2$. Write $U=U^{\varepsilon,\phi}$, $E=U-U^{\rm cl}$, $b^{(i)}=b(m,U)+R^\phi+2\varepsilon(e_i-m)$ and $\mathcal P_i=\partial_t+\varepsilon\operatorname{tr}(AD^2\,\cdot\,)+b^{(i)}\cdot D$. Let $\vartheta_{ij}\in[0,1]$ be the difference quotient of $x\mapsto x_+$ between $U_i-U_j$ and $U^{\rm cl}_i-U^{\rm cl}_j$ (say $\vartheta_{ij}=1$ where these coincide), so that $(U_i-U_j)_+-(U^{\rm cl}_i-U^{\rm cl}_j)_+=\vartheta_{ij}(E_i-E_j)$, and similarly for the squares with the factor $(U_i-U_j)_++(U^{\rm cl}_i-U^{\rm cl}_j)_+$. Subtracting the two equations gives \begin{equation}\label{wsref-0-65-72-72-6f-72-73-79-73-74-65-6d} \mathcal P_iE_i+\sum_{j\ne i}(c_{ij}+\phi_\varepsilon(m_j))(E_j-E_i)+(b(U)-b(U^{\rm cl}))\cdot DU^{\rm cl}_i+\rho_i=0,\qquad E(T)=0, \end{equation} with $c_{ij}=\frac12\{(U_i-U_j)_++(U^{\rm cl}_i-U^{\rm cl}_j)_+\}\vartheta_{ij}\ge0$ and \[ \rho_i=\varepsilon\operatorname{tr}(AD^2U^{\rm cl}_i)+(R^\phi+2\varepsilon(e_i-m))\cdot DU^{\rm cl}_i+\sum_{j\ne i}\phi_\varepsilon(m_j)(U^{\rm cl}_j-U^{\rm cl}_i). \] On $N$ every $\phi_\varepsilon(m_j)\le a'_\varepsilon$, so $|R^\phi|_1\le2da'_\varepsilon$ and $\bar\rho=\sup_{[0,T]\times N}\max_i|\rho_i|=O(\varepsilon+a'_\varepsilon)$. Also $b_l(U)-b_l(U^{\rm cl})=\sum_{j\ne l}(m_j\vartheta_{jl}+m_l\vartheta_{lj})(E_j-E_l)$, so $|b(U)-b(U^{\rm cl})|_1\le4(d-1)\max_j|E_j|$. Since $b(U)-b(U^{\rm cl})$ is tangent, the coupling term in (errorsystem), which may have either sign, is at most $D_0\max_j|E_j|$ in absolute value, where $D_0=4(d-1)\sup_N\max_{i,l}|(\partial_{m_l}-\partial_{m_d})U^{\rm cl}_i|$. By Lemma Lemma, $0\le U_i\le M$. Hence $b_l(m,U)\ge-(d-1)Mm_l$, and $\sum_lb_l=0$ gives $|b(m,U)|_1\le2(d-1)M$, so $|b^{(i)}|_1\le M_1$; as in the proof of Lemma Lemma, $b^{(i)}_j\ge-Cm_j$ on $N$.

Put $C_\zeta=1+\sup_{\Delta_d}\{|\operatorname{tr}(AD^2\zeta)|+D\zeta\cdot AD\zeta\}$, $\kappa_1=\delta'/(4\varepsilon)$, $\kappa_2=1/(2\varepsilon C_\zeta)$, and \[ \psi_1=\sum_j\psi_{1,j},\quad \psi_{1,j}=\Big(\frac{\delta'}{m_j}\Big)^{\kappa_1}e^{\kappa_1(C+1)(T-t)},\qquad \psi_2=\exp\big(\kappa_2\{\zeta-1+(M_1|D\zeta|+1)(T-t)\}\big). \] Since $A_{jj}=m_j(1-m_j)$, $b^{(i)}_j\ge-Cm_j$ and $m_j\ge\delta'$ on $N$, \[ \mathcal P_i\psi_{1,j}=\kappa_1\psi_{1,j}\Big\{-(C+1)+\varepsilon(\kappa_1+1)\frac{1-m_j}{m_j}-\frac{b^{(i)}_j}{m_j}\Big\} \le\kappa_1\psi_{1,j}\Big\{-\frac34+\frac\varepsilon{\delta'}\Big\}\le0. \] Since $b^{(i)}$ is tangent, $|b^{(i)}\cdot D\zeta|\le|b^{(i)}|_1|D\zeta|\le M_1|D\zeta|$, and therefore \[ \mathcal P_i\psi_2=\kappa_2\psi_2\big\{-M_1|D\zeta|-1+\varepsilon\operatorname{tr}(AD^2\zeta)+\varepsilon\kappa_2D\zeta\cdot AD\zeta+b^{(i)}\cdot D\zeta\big\} \le\kappa_2\psi_2\big\{-\tfrac12+\varepsilon C_\zeta\big\}\le0 \] once $\varepsilon C_\zeta\le1/2$. Moreover $\psi_1\ge1$ where some $m_j=\delta'$, $\psi_2\ge1$ on the closure of $\mathcal O$, and on $\mathcal K$, where all $m_j\ge\eta$ and $\zeta=0$, \[ \psi_1+\psi_2\le d\,2^{-\kappa_1}+e^{-\kappa_2\{1-T(M_1|D\zeta|+1)\}}. \]

Fix $\Lambda\ge D_0$ and $\varpi>0$, and put $\Gamma=\bar\rho(T-t)+M_0(\psi_1+\psi_2)+\varpi(1+T-t)$. We claim that $\pm E_ie^{-\Lambda(T-t)}\le\Gamma$ on $[0,T]\times N$ for every $i$ and both signs. Otherwise the maximum of $\pm E_ie^{-\Lambda(T-t)}-\Gamma$ over $t$, $m\in N$, $i$ and the sign is positive; let it be attained at $(t_*,m_*)$, for the index $i$ and the sign $\pm$, and put $e=\pm E_ie^{-\Lambda(T-t)}$. Since $E(T)=0$, and $|E|\le M_0\le M_0(\psi_1+\psi_2)$ at points of $N$ where some $m_j=\delta'$ or which lie in the closure of $\mathcal O$, we have $t_*<T$, and $m_*$ is an interior point of $N$. At $(t_*,m_*)$ we have $\partial_t(e-\Gamma)\le0$, $D(e-\Gamma)=0$ and $D^2(e-\Gamma)\le0$ on tangent vectors, so $\mathcal P_ie\le\mathcal P_i\Gamma$. There also $e>0$ and $e\ge e^{-\Lambda(T-t)}|E_j|$ for every $j$, so the switching sum in (errorsystem), multiplied by $\pm e^{-\Lambda(T-t)}$, is nonpositive, and (errorsystem) gives \[ \mathcal P_ie=\Lambda e\mp e^{-\Lambda(T-t)}\Big[\sum_{j\ne i}(c_{ij}+\phi_\varepsilon(m_j))(E_j-E_i)+(b(U)-b(U^{\rm cl}))\cdot DU^{\rm cl}_i+\rho_i\Big]\ge(\Lambda-D_0)e-\bar\rho\ge-\bar\rho. \] But $\mathcal P_i\Gamma=-\bar\rho+M_0\mathcal P_i(\psi_1+\psi_2)-\varpi\le-\bar\rho-\varpi$, a contradiction. Letting $\varpi\downarrow0$ gives $|E|\le e^{\Lambda T}\{\bar\rho T+M_0(\psi_1+\psi_2)\}$ on $[0,T]\times N$, which proves (localclassical) with $c=\min\{\delta'\log2/4,\ (1-T(M_1|D\zeta|+1))/(2C_\zeta)\}$. No bound on derivatives of $U^{\varepsilon,\phi}$ uniform in $\varepsilon$ was used. ∎

Reduction and the regular collars

Write $u=U_1$. On an interior compact cylinder, (passive) makes all passive components tend uniformly to zero and leaves $u$ uniformly positive. Elementary algebra in the rates and Hamiltonians gives \begin{equation}\label{wsref-0-73-63-61-6c-61-72-70-65-72-74-75-72-62} u_s+q\partial_p(pu^2) =\varepsilon\mathcal L u+E_\varepsilon\cdot D_{p,r}u+F_\varepsilon, \qquad \|E_\varepsilon\|_\infty+\|F_\varepsilon\|_\infty\longrightarrow0. \end{equation} Here $\mathcal L$ is the scalar Kimura operator including the extra drift $2(e_1-m)\cdot D$, and the errors contain no derivatives of passive values. Indeed, $b_1=-2qpu+O(\max_{i>1}U_i)$, $b_j-b_d=O(\max_{i>1}U_i)$, $H_1=qu^2+O(\max_{i>1}U_i)$, and the repulsion terms have uniformly small coefficients on the compact set. Treat these error functions, obtained from the actual solution, as fixed coefficients in (scalarperturb).

Under $V=qyu$, the unperturbed part of (scalarperturb) becomes $V_s+VV_y$. In $(y,r)$ the second-order coefficients are \begin{equation}\label{wsref-0-63-6f-65-66-66-69-63-69-65-6e-74-73} A_{yy}=\frac{1-y^2}{4},\qquad A_{yr_j}=-\frac{yr_j}{2},\qquad A_{r_jr_l}=m_j\delta_{jl}+m_d-r_jr_l. \end{equation} The first-order coefficients are $(1-y^2)/(4y)$ and $-r_j$, and the zeroth-order coefficient is $-(1-y^2)/(4y^2)$. After the fixed change $\theta=y-1/2-\alpha r_2$, (scalarperturb) is exactly \begin{equation}\label{wsref-0-56-65-71} V_s+VV_\theta=\varepsilon\mathcal D V+B_\varepsilon\cdot D_{\theta,r}V+C_\varepsilon, \qquad \|B_\varepsilon\|_\infty+\|C_\varepsilon\|_\infty\longrightarrow0. \end{equation} On interior compact sets, $\mathcal D$ has fixed smooth coefficients and a positive definite second-order matrix. Denote its $\partial_{\theta\theta}$ coefficient by $a(\theta,r)>0$. The conversion of the errors is harmless because $y$ is bounded away from zero.

Consider the rectangles \[ \Omega=\{(\theta,r):|\theta|<1/8,\ |r|_\infty<R_d\},\qquad \Omega^\pm=\{(\theta,r):|\theta|<1/8\pm1/64,\ |r|_\infty<(1\pm1/8)R_d\}, \] and the collar $\mathcal C=\overline{\Omega^+}\setminus\Omega^-$, a compact neighborhood of $\partial\Omega$. On $\overline{\Omega^+}$ one has $11/32<y<2/3$ and $L(r)^2\ge1-(2d-3)\cdot9R_d/8>81/100$, so $\overline{\Omega^+}$ lies in the simplex interior. Since $\chi=1$ wherever $y\ge1/8$, the initial value for (Veq) is exactly $2-\beta(r)\tanh(k\theta')$ at $(\theta',r)$ whenever $(\theta,r)\in\overline{\Omega^+}$ and $\theta-1/32\le\theta'\le\theta$.

We now choose $T_0$, before choosing $k$. All derivatives through any fixed order of the initial data are bounded uniformly in $k$ outside the smaller strip \[ \mathcal S=\{|\theta|\le1/16,\ |r|_\infty\le R_d/2\}. \] This follows from $\sup_{k\ge1}k^j\operatorname{sech}^2(k/16)<\infty$; in the remaining region $\beta$ vanishes. The cutoff near $p=0$ is fixed and stays away from this strip. Put $\mathcal S_1=\{|\theta|\le3/32,\ |r|_\infty\le3R_d/4\}$. Then $\mathcal S\subset\mathcal S_1\subset\Omega^-$, so $\mathcal S_1$ does not meet $\mathcal C$. Fix a smooth $\zeta:\Delta_d\to[0,1]$ with $\zeta=1$ on $\mathcal S_1$ and $\zeta=0$ outside $\Omega^-$, for instance a product of cutoff functions of $\theta$ and of $r_2,\ldots,r_{d-1}$; it depends only on $d$.

Let $C_1\ge\sup_{k\ge1}k\operatorname{sech}^2(k/16)$ bound the negative part of $\partial_yv_0$ outside $\mathcal S$, uniformly in $k$. With $M=34/q$, and with $M_1$ and $|D\zeta|$ as in Lemma Lemma, choose $T_0>0$ with \[ T_0C_1<1/2,\qquad 3T_0<1/32,\qquad T_0(M_1|D\zeta|+1)<1. \] These are finitely many positive requirements independent of $k$.

For the deterministic ansatz $(u,0,\ldots,0)$ the equations reduce exactly to (burgers), with $r$ constant along characteristics. Its characteristic map is $y=z+sv_0(z,r)$. Let $0<T\le T_0$ and $0<s\le T$. Since $0\le v_0\le3$, every Hopf--Lax minimizer $z$ at $(s,y,r)$ satisfies $y-z=sv_0(z,r)\in[0,3s]$. The set $\mathcal S'=\{-1/16\le\theta\le1/16+3T,\ |r|_\infty\le R_d/2\}$ lies in the interior of $\mathcal S_1$, since $3T<1/32$. If $(\theta,r)\notin\mathcal S'$, the footpoint segment $\{(\theta',r):\theta-3s\le\theta'\le\theta\}$ does not meet $\mathcal S$: this is clear if $|r|_\infty>R_d/2$, and the segment lies to the left of $\mathcal S$ if $\theta<-1/16$ and to its right if $\theta>1/16+3T$. On this segment the derivative of the action $z\mapsto\int_0^zv_0+(y-z)^2/(2s)$ increases at rate at least $1/s-C_1\ge1/(2s)$, so the minimizer is unique and nondegenerate, and $\partial_z(z+sv_0)\ge1-sC_1\ge1/2$. By the implicit function theorem $v$ is smooth near each such point, up to $s=0$. Hence $U^*$ is a classical solution of the deterministic master equation with terminal value $g$ on $[0,T]\times(\operatorname{int}\Delta_d\setminus\mathcal S')$, a neighborhood of $[0,T]\times(\operatorname{int}\Delta_d\setminus\operatorname{int}\mathcal S_1)$.

Apply Lemma Lemma with $\mathcal O=\operatorname{int}\mathcal S_1$, $U^{\rm cl}=U^*$, the function $\zeta$, and any compact $\mathcal K\subset\operatorname{int}\Delta_d\setminus\Omega^-$; its hypotheses hold by the choice of $T_0$. Thus $U^{\varepsilon,\phi}\to U^*$ uniformly on $[0,T]\times\mathcal K$. In particular the actual boundary values on the collar $\mathcal C$ converge uniformly to the classical field. No classical field is assumed across the shock.

A uniform Cole--Hopf estimate

Let $w(s,x;\nu,\lambda)$ solve viscous Burgers on $\mathbb R$, with viscosity $\nu>0$ and initial value $2-\lambda\tanh(kx)$, where $0\le\lambda\le1$. Write \begin{equation}\label{wsref-0-61-63-74-69-6f-6e} S(z)=2z-\frac\lambda k\log\cosh(kz)+\frac{(x-z)^2}{2s},\qquad w=\frac{x-\mathbb E_\nu z}{s},\qquad d\mu_\nu(z)=\frac{e^{-S(z)/(2\nu)}dz}{\int e^{-S/(2\nu)}}. \end{equation} This is the Cole--Hopf formula; differentiating the heat kernel verifies the equation and initial value. In this section expectations and covariances use $\mu_\nu$. All estimates are uniform for $x$ in a fixed compact interval, $s$ in a fixed bounded interval, and $\lambda\in[0,1]$. The parameter $k$ is fixed.
Lemma: Put $Q=-w_x\ge0$. There is a modulus $\omega(\nu)\to0$ such that \begin{align} |w|&\le3,\qquad |\nu w_{xx}|+|w_\lambda|\le C(1+Q),\label{wsref-0-43-48-31}\\ |\nu w_\nu|+|\nu^2w_{x\nu}|+|\nu^3w_{\nu\nu}| +|\nu^2w_{\nu\lambda}|+|\nu w_{x\lambda}|+|\nu w_{\lambda\lambda}| &\le\omega(\nu)(1+Q).\label{wsref-0-43-48-32} \end{align} The estimates hold from $s=0$ by continuous extension.
Proof: The maximum principle gives $1\le w\le3$ and $w_x\le0$. For $s\ge s_0>0$, put $Z=z-\mathbb E z$, $A=S-\mathbb E S$, and $R=\partial_\lambda S=-k^{-1}\log\cosh(kz)$. Differentiation under the integral gives \begin{align*} \operatorname{Var}(z)&=2\nu s^2(Q+1/s),& w_{xx}&=-\frac{\mathbb E Z^3}{4\nu^2s^3},\\ w_\nu&=-\frac{\mathbb E ZA}{2\nu^2s},& w_{x\nu}&=\frac{\operatorname{Var}(z)}{2\nu^2s^2}-\frac{\mathbb E Z^2A}{4\nu^3s^2},\\ w_{\nu\nu}&=\frac{\mathbb E ZA}{\nu^3s}-\frac{\mathbb E ZA^2}{4\nu^4s},& w_\lambda&=\frac{\operatorname{Cov}(z,R)}{2\nu s},\\ w_{x\lambda}&=\frac{\mathbb E Z^2(R-\mathbb E R)}{4\nu^2s^2},& w_{\lambda\lambda}&=-\frac{\mathbb E Z(R-\mathbb E R)^2}{4\nu^2s},\\ w_{\nu\lambda}&=-\frac{\operatorname{Cov}(z,R)}{2\nu^2s} +\frac{\mathbb E Z(R-\mathbb E R)A}{4\nu^3s}.&& \end{align*}

The moment estimates below are uniform, including the possibly degenerate transition at shock formation. On a bounded $z$ interval the derivatives $S'',S''',S''''$ do not vanish simultaneously: if $\lambda=0$ then $S''=1/s$; otherwise $S'''$ vanishes only at zero, where $S''''=2\lambda k^3>0$. Outside a sufficiently large interval $S''$ is uniformly positive. A finite cover consequently gives \[ |\{z:S(z)-\min S\le h\}|\le C h^{1/4}\quad(0<h\le1). \] The elementary sublevel estimate used here follows by choosing $j+1$ separated points of a set on which $|F|\le h$. Divided differences bound $\inf|F^{(j)}|$ by $C_jh/|\{|F|\le h\}|^j$. Apply this on each covering interval where one derivative of order $j\le4$ has a fixed nonzero sign. The quadratic tails of $S$ handle larger sublevels.

At a global minimizer, the uniform upper bound on $S''$ gives a Gaussian lower bound $c\sqrt\nu$ for the partition integral after subtraction of $\min S$. Integrating the sublevel estimate, or summing energy strips of width $\nu$, gives \begin{equation}\label{wsref-0-61-63-74-69-6f-6e-6d-6f-6d} \mathbb E(S-\min S)^j+\mathbb E|A|^j\le C\nu^{j-1/4}\qquad(1\le j\le4). \end{equation} Moreover $\mathbb E z\in[x-3s,x-s]$, by integration by parts in (action). Gaussian far tails, together with $\operatorname{Var}(z)\ge2\nu s$, imply \begin{equation}\label{wsref-0-7a-6d-6f-6d} \mathbb E|Z|^3+\mathbb E Z^4\le C\operatorname{Var}(z). \end{equation} Indeed, on a fixed bounded interval the higher powers are bounded by a constant times $Z^2$; the complementary moments are exponentially small compared with $\nu$. Equations (actionmom)--(zmom) and Cauchy--Schwarz prove the viscosity-derivative bounds in (CH2), for example with factor $C\nu^{3/8}$, and the bound on $\nu w_{xx}$. Since $R$ is $1$-Lipschitz, $|\operatorname{Cov}(z,R)|\le\operatorname{Var}(z)$ and $\mathbb E[Z^2(R-\mathbb E R)^2]\le C\operatorname{Var}(z)$. This also proves the bounds on $w_\lambda$ and $\nu^2w_{\nu\lambda}$.

The remaining two derivatives need a sharper observation. Set $h=x-2s$. Apart from an additive constant, \[ S(z)=z^2/(2s)-hz/s-(\lambda/k)\log\cosh(kz). \] If $h\ne0$, there is a unique global minimizer, of the sign of $h$. To see this for $h>0$, compare $z$ with $-z$ and then use that $S''$ is increasing on the positive half-line and $S'(0)<0$. If $h=0$, the minimizer is zero or a symmetric pair. Hence the even function $R$ has the same value at every global minimizer, for every parameter.

Compactness now implies that the oscillation of $R$ on the sublevel $S-\min S\le a$ tends uniformly to zero as $a\downarrow0$. Outside a fixed such sublevel the Gibbs moments are exponentially small. On the sublevel use this small oscillation for one factor and the Lipschitz covariance bound for the other factors. Since $\operatorname{Var}(z)\ge c\nu$, it follows that \[ \mathbb E[Z^2(R-\mathbb E R)]=o(\operatorname{Var}(z)),\qquad \mathbb E[Z(R-\mathbb E R)^2]=o(\operatorname{Var}(z)), \] uniformly in the parameters. For the second estimate one uses $\mathbb E|Z||R-\mathbb E R|\le\operatorname{Var}(z)$; centering by $\mathbb E R$ changes the sublevel oscillation only by the same vanishing modulus and an exponentially small tail. The displayed derivative identities prove the desired bounds on $\nu w_{x\lambda}$ and $\nu w_{\lambda\lambda}$.

Finally choose $s_0\le1/(2k)$. Before this time the differentiated Burgers equation gives $-2k\le w_x\le0$. Successive spatial derivatives satisfy linear parabolic equations with bounded coefficients and already bounded lower derivatives. Differentiating in $\nu$ or $\lambda$ gives the same type of equations, with initial data zero or derivatives of $2-\lambda\tanh(kx)$; the source for $w_\nu$, for example, is $w_{xx}$. Induction through the spatial order four and parameter order two gives bounds independent of $\nu$ on this initial interval. Each scaled derivative in (CH2) then tends to zero there as well. This completes the estimates through $s=0$. ∎

Comparison through the shock

On $\Omega^+$, set \[ W_\varepsilon(s,\theta,r)=w(s,\theta;\varepsilon a(\theta,r),\beta(r)). \] The chain rule and Lemma Lemma give, with a modulus $\eta_\varepsilon\to0$, \begin{align} 1-W_\theta&\asymp1+Q,\qquad W_\theta\le\eta_\varepsilon, \qquad |DW|+\varepsilon|D^2W|\le C(1-W_\theta),\label{wsref-0-77-65-69-67-68-74-65-64}\\ |W_s+WW_\theta-\varepsilon\mathcal D W|&\le\eta_\varepsilon(1-W_\theta).\label{wsref-0-72-65-73-69-64-75-61-6c} \end{align} The potentially large derivatives cause no omitted term. Since $\beta$ is independent of $\theta$, $W_\theta=w_x+O(\nu w_\nu)$ with $\nu=\varepsilon a$. The main normal diffusion cancels $w_s+ww_x=\nu w_{xx}$. The remaining normal terms involve $\nu w_\nu$, $\nu^2w_{x\nu}$ and $\nu^3w_{\nu\nu}$. Mixed and tangential diffusion adds $\nu w_{x\lambda}$, $\nu w_{\lambda\lambda}$ and $\nu^2w_{\nu\lambda}$, all small in the weighted sense. Terms with a single $w_\lambda$ carry a factor $\varepsilon$. The fixed lower-order coefficients are bounded. This proves (weighted)--(residual).

On $[0,T]\times\mathcal C$ the reference converges uniformly to the classical field $v$. If $|r|_\infty\ge7R_d/8$, then $\beta(r)=0$ and $W_\varepsilon=2=v$. Otherwise $|\theta|\ge7/64$; take $x=\theta$, $\lambda=\beta(r)$ and $0<s\le T$ in (action). Since $1\le2-\lambda\tanh(kz)\le3$, we have $(z-x+s)/s\le S'(z)\le(z-x+3s)/s$, so $S$ decreases on $(-\infty,x-3s]$ and increases on $[x-s,\infty)$, while on $[x-3s,x-s]$ one has $|z|>1/16$, because $3s<1/32$, and hence $S''\ge1/s-C_1\ge1/(2s)$. Thus $S$ has a unique minimizer $z_*$, the classical value is $(x-z_*)/s=v(s,\theta,r)$, the data agreeing on $[\theta-3s,\theta]$ by the choice of $\Omega^+$, and integrating these bounds gives $S(z)-S(z_*)\ge(z-z_*)^2/(6s)$ for all $z$. Also $S''\le1/s$ and $|S'''|\le k^2$. Pairing the Gibbs weights at $z_*\pm\xi$ and using $|e^{-a}-e^{-b}|\le|a-b|e^{-\min(a,b)}$ and Taylor's formula, we get \[ |\mathbb E_\nu z-z_*|\le\frac{\int_0^\infty\xi\cdot\frac{k^2\xi^3}{6\nu}\,e^{-\xi^2/(12\nu s)}\,d\xi}{\int_{\mathbb R}e^{-\xi^2/(4\nu s)}\,d\xi}\le16k^2\nu s^2, \] so $|W_\varepsilon-v|\le16k^2\varepsilon s\,a(\theta,r)$. Let $b_\varepsilon\to0$ be the sum of the uniform errors of $W_\varepsilon$ and of $V_\varepsilon$ with respect to $v$ on $[0,T]\times\mathcal C$, the second tending to zero by Section 3, and let $L$ bound $|\partial_\theta v|$ there. Let $\Delta_\varepsilon\to0$ bound the two error coefficients in (Veq) on the rectangle.

Choose nonnegative functions $a_0(s),h(s)$ by \begin{equation}\label{wsref-0-6f-64-65-62-61-72-72-69-65-72} K=C(h+a_0+\eta_\varepsilon+\Delta_\varepsilon),\quad h'=a_0+K,\quad a_0'=Lh'+K,\quad h(0)=0,\quad a_0(0)=b_\varepsilon, \end{equation} where $C$ is a sufficiently large fixed constant. They tend uniformly to zero. For small $\varepsilon$ we have $h<1/64$ on $[0,T]$; then $(\theta\mp h(s),r)\in\Omega^+$ for $(\theta,r)\in\overline\Omega$, and for $(\theta,r)\in\partial\Omega$ the segment from $(\theta,r)$ to $(\theta\mp h(s),r)$ lies in $\mathcal C$. The functions \[ W^+(s,\theta,r)=W_\varepsilon(s,\theta-h(s),r)+a_0(s),\qquad W^-(s,\theta,r)=W_\varepsilon(s,\theta+h(s),r)-a_0(s) \] are respectively a supersolution and a subsolution of (Veq). Indeed, translation of the fixed smooth coefficients changes the residual by at most $Ch(1-W_\theta)$, using the Hessian bound in (weighted). The other errors, including the zeroth-order term on $a_0$, are at most $K(1-W_\theta)$. The upper residual is therefore at least \[ (-h'+a_0)W_\theta+a_0'-K(1-W_\theta)=a_0'-K\ge0; \] the lower residual is at most $K-a_0'\le0$.

The initial inequalities hold because $W_\varepsilon(0)=V_\varepsilon(0)$ on the rectangle. On its boundary, (odebarrier) gives $a_0-Lh\ge b_\varepsilon$, which proves the boundary inequalities. At each fixed $\varepsilon$ the scalar equation is uniformly parabolic on this interior rectangle, so the comparison principle yields $W^-\le V_\varepsilon\le W^+$. Only boundedness, not derivatives, of the error coefficients was used.

For each $r$, the reference tends to the Burgers entropy solution, except possibly on the shock plane $\theta=2s$ when $\beta(r)ks>1$. This is also the solution defined from the full data (data): both sets of minimizing footpoints lie within distance $3T_0$ of the observation point, where the initial functions coincide; their primitives differ only by an additive constant. Boundedness gives convergence in local $L^1$. Also $W_\theta\le\eta_\varepsilon$ and $1\le W\le3$ give a uniform bound on its total variation on each fixed $\theta$ interval. Its small normal shifts therefore tend to zero in $L^1$, uniformly in the other variables. The comparison just proved gives $V_\varepsilon\to v$ in $L^1$ on the rectangle. Outside it, Section 3 gives local uniform convergence, because for every compact $\mathcal K\subset\operatorname{int}\Delta_d$ the closure of $\mathcal K\setminus\Omega$ is a compact subset of $\operatorname{int}\Delta_d\setminus\Omega^-$. Lemma Lemma removes the passive components. This proves (convergence) for every admissible family.

Admissibility, nonemptiness and scope

The change $(p,r)\mapsto(y,r)$ has a smooth nonzero Jacobian on interior compact sets, so the convergence above is in the simplex measure. It implies compactness and pairwise independence of the regularizations. The entropy condition (burgers), imposed for almost every parameter $r$, is intrinsic to the limiting deterministic equation and has no reference to $\varepsilon$ or $\phi$. All selected feedback fields are bounded by the value estimate.

Entropy uniqueness also applies on the parameter-dependent interval used here. The local Kato inequality for two nonnegative entropy solutions has flux $\tfrac12(v+\widetilde v)|v-\widetilde v|\ge0$. Integrate with a cutoff that has compact support in $(0,L(r))$, increases near zero and decreases near $L(r)$. Bounded feedback gives $v,\widetilde v=O(y)$ near zero, so the left cutoff error vanishes. The right cutoff contribution is nonpositive and can be discarded. The initial $L^1$ distance is zero; hence the solutions agree. This is the usual entropy contraction argument [Kru, Theorem 1], with the inflow and outflow terms made explicit. It proves uniqueness under the stated intrinsic condition.

If a classical full master solution exists, follow one of its characteristics. The value vector solves $\dot u_i=H_i(u)$ backwards from $(G_k,0,\ldots,0)$. Uniqueness of this locally Lipschitz ODE keeps the passive values zero and the active value positive. The full classical solution therefore has the scalar reduction (burgers), and agrees with its entropy solution. This verifies classical consistency without assuming the ansatz for an arbitrary candidate classical field.

The admissible class is nonempty. Bayraktar, Cecchin, Cohen and Delarue [BCCD, Theorem 3.4, equation $2.19$] provide a classical Wright--Fisher solution for smooth costs when the repulsion is sufficiently large near zero and vanishes beyond an arbitrarily small boundary layer. Their condition $2.19$ asks only that the repulsion be nonincreasing and Lipschitz, equal to a large constant on $[0,\delta]$ and to zero on $(2\delta,\infty)$, so it may be taken smooth. Their noise parameter is $\sqrt{2\varepsilon}$ in our normalization. Their first derivatives extend continuously to the boundary.

Uniqueness extends to the class used here. Let $U$ be this solution, let $\widetilde U$ be any solution of (master) with the regularity required in the definition of $\mathcal A$, and put $E=U-\widetilde U$. The subtraction leading to (errorsystem), with $U^{\rm cl}$ replaced by $\widetilde U$ and now on the whole cylinder, gives \[ \mathcal P_iE_i+\sum_{j\ne i}(c_{ij}+\phi_\varepsilon(m_j))(E_j-E_i)+(b(U)-b(\widetilde U))\cdot D\widetilde U_i=0,\qquad E(T)=0, \] with $c_{ij}\ge0$. The last term may have either sign; it is at most $D_0\max_j|E_j|$ in absolute value, where $D_0$ is $4(d-1)$ times a bound for the first derivatives of $\widetilde U$. Fix $\Lambda\ge D_0$, $0<a<1$, $h>0$ and $\kappa>0$, and maximize $\pm E_ie^{-\Lambda(T-t)}+hB_a(m)-\kappa(T-t)$ over $t$, $m\in\Delta_d$, $i$ and the sign. As in the proof of Lemma Lemma, bounded first derivatives of $E$ place the maximum in the interior. Suppose that it occurs at some $t<T$, and let $e=\pm E_ie^{-\Lambda(T-t)}$ for the maximizing index and sign. Then $e\ge e^{-\Lambda(T-t)}|E_j|$ for every $j$, so, as in the proof of Lemma Lemma, $\mathcal P_ie\ge(\Lambda-D_0)e\ge0$ there. On the other hand, at this maximum the first-order condition and the estimates of the terms of $hB_a$ in the proof of Lemma Lemma give $\mathcal P_ie\le-\kappa+\varepsilon d(1-a)(L_E+had^{1-a})+O(h)$, where $L_E$ bounds the first derivatives of $E$. For $\kappa$ larger than the last two terms this is impossible. Hence the maximum occurs at $t=T$, where $E=0$, and $\pm E_i\le e^{\Lambda T}(hd^{1-a}+\kappa T)$. Letting $h\downarrow0$ and $\kappa\downarrow\varepsilon d(1-a)L_E$, and then $a\uparrow1$, gives $E=0$. Taking the layer width to zero gives an admissible family. Only sufficiently small $\varepsilon$ matters for the limit; if a family is required at larger $\varepsilon$ too, the substitution $U=cV$, $\tau=ct$, $\phi=c\widetilde\phi$, with $c>2\varepsilon$, reduces it to the same theorem.

At $p=1/4,r=0$, the intrinsic terminal covector is $G_k\,dm_1$. Varying $m_2$ with $m_d$ compensating gives \[ \partial_{m_2}G_k=\frac{4\alpha k}{q}\ne0, \qquad \partial_{m_1}(g_2-g_d)=0. \] It is not a gradient, even after a common component is added to the costs. On the core where $\beta=1$, the initial Burgers derivative is $-k$ at $\theta=0$. Characteristics first cross at $s=1/k$; for later times the symmetric decreasing front has its entropy shock at $\theta=2s$. The construction of $T_0$ did not depend on $k$, so this shock occurs strictly inside the permitted cylinder whenever $kT>1$.

The theorem is a sufficient-condition result for the stated data family, not a claim that arbitrary nonpotential costs always have a unique vanishing-noise limit. No cost correction, prescribed rate for $\phi_\varepsilon$, or uniform bound on noisy derivatives is part of the theorem.

Corollary (Every finite horizon): For any $d\ge3$ and $T>0$, choose $\sigma>0$ with $\sigma T\le T_0$ and use $f=0$, $g=(\sigma G_k,0,\ldots,0)$. All the existence, independence, intrinsic admissibility and classical-consistency conclusions of Theorem Theorem hold. The entropy initial value is $\sigma v_0$, and an interior shock occurs when $\sigma kT>1$.
Proof: Set $\widehat U(\tau,m)=\sigma^{-1}U(\tau/\sigma,m)$, $\widehat\varepsilon=\varepsilon/\sigma$, and $\widehat\phi_{\widehat\varepsilon}=\sigma^{-1}\phi_{\sigma\widehat\varepsilon}$. The homogeneities of $H$ and $b$ show directly that (master) becomes the same equation on the horizon $\sigma T$, with terminal value $(G_k,0,\ldots,0)$. This transformation is a bijection of the respective admissible regularization classes; their regularity, uniqueness and local vanishing properties are preserved. Theorem Theorem applies, and scalar entropy solutions scale as $v^\sigma(s,y,r)=\sigma v(\sigma s,y,r)$. One may take $\sigma=\min(1,T_0/(2T))$ and then choose $k>1/(\sigma T)$ to obtain a shock on any prescribed horizon. ∎

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

  • [CD] P. Cardaliaguet and F. Delarue. Selected topics in mean field games. Proceedings of the International Congress of Mathematicians 2022, vol. 5, 3660--3703. DOI: 10.4171/ICM2022/17.
  • [BCCD] E. Bayraktar, A. Cecchin, A. Cohen and F. Delarue. Finite state mean field games with Wright--Fisher common noise. arXiv:1912.06701v2, 10 January 2021, Theorem 3.4 and §3.2.2.
  • [Kru] S. N. Kruzhkov. First order quasilinear equations in several independent variables. Mathematics of the USSR-Sbornik 10$2$ $1970$, 217--243. DOI: 10.1070/SM1970v010n02ABEH002156.

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