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Econ-41 · economics · Economics

Nonlinear stability of periodic MFG solutions

LLM status Partially solved unverified

Statement

GT-081 | Mean-field games | R: an explicit boundary-value stability formulation

Definitions and mathematical statement

Fix $\nu>0$ and a smooth local coupling $f:\mathbb T^d\times(0,\infty)\to\mathbb R$. Suppose a classical triple $(v_*,m_*,\lambda_*)$ solves on $\mathbb R\times\mathbb T^d$ \[ \begin{cases} \lambda_*-\partial_t v_*-\nu\Delta v_*+\tfrac12|Dv_*|^2=f(x,m_*),\\ \partial_t m_*-\nu\Delta m_*-\operatorname{div}(m_*Dv_*)=0,\\ v_*(t+P)=v_*(t),\quad m_*(t+P)=m_*(t),\quad m_*>0,\quad\int m_*(t)=1, \end{cases} \] for some $P>0$, with $(Dv_*,m_*)$ nonconstant in time. The associated unnormalized value is $u_*(t,x)=v_*(t,x)-\lambda_*t$; $u_*$ itself need not be periodic. Define its orbit \[ \mathcal O_* =\{(Dv_*(\theta),m_*(\theta)):0\leq\theta<P\}, \quad d_*((p,m),\mathcal O_*)= \inf_{0\leq\theta<P}(\|p-Dv_*(\theta)\|_\infty+\|m-m_*(\theta)\|_1). \] Fix constants $M,M_0<\infty$ and $\zeta>0$ containing the reference solution in their bounds. The admissible solution class has $m\geq\zeta$ and $\|m\|_{C^{1,2}}+\|Du\|_\infty+\|D^2u\|_\infty\leq M$; smooth boundary data have $m_0\geq\zeta$, $\int m_0=1$, and $\|m_0\|_{C^2}+\|Dh\|_{C^1}\leq M_0$. The $C^{1,2}$ norm controls the function, its first time derivative, and its spatial derivatives through order two. One precise editorial stability notion is uniform boundary-value orbital shadowing. For every $\varepsilon>0$, require a $\delta>0$, independent of $T\geq1$ and phase $\theta$, such that whenever these boundary data obey \[ \|m_0-m_*(\theta)\|_1+\|Dh-Dv_*(T+\theta)\|_\infty<\delta, \] every admissible classical solution of \[ -u_t-\nu\Delta u+\tfrac12|Du|^2=f(x,m),\qquad m_t-\nu\Delta m-\operatorname{div}(mDu)=0, \quad m(0)=m_0,\quad u(T)=h \] satisfies $\sup_{0\leq t\leq T}d_*((Du(t),m(t)),\mathcal O_*)<\varepsilon$. Nonemptiness of the admissible solution class for all such perturbed boundary data is also required; vacuous bounds on an empty class do not establish stability.

Determine which periodic equilibria satisfy this property, or a stated weaker nonlinear stability property with an explicit equilibrium-selection rule. This is a boundary-value comparison problem; specifying arbitrary $m(0),u(0)$ does not define an MFG evolution semigroup.

Short English statement

Do equilibria near a periodic solution remain near its orbit when their compatible initial and terminal data are perturbed?

Variants and scope boundaries

Linearized spectral stability, orbital shadowing, and asymptotic attraction are distinct. Terminal costs depending on the terminal distribution require control of that additional coupling. A finite-horizon oscillation is not automatically an entire periodic solution.

Source relationship and status

The chapter identifies stability as unknown for its examples. The chosen stability definition is editorial; no all-periodic-solutions stability theorem is asserted.

References

[S1] Cardaliaguet and Delarue. Selected topics in mean field games, ICM2022, vol. 5; DOI: 10.4171/ICM2022/17. Locator: Section 3.3, "Periodic solutions," p. 3680.

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

[S2] Marco Cirant and Levon Nurbekyan (2018). The variational structure and time-periodic solutions for mean-field games systems. Minimax Theory and its Applications 3, 227--260.

Locators: arXiv version, Section 6, Theorem 6.1, pp. 24--25, actual periodic examples; Remark 6.4, p. 28, limited qualitative information.

https://arxiv.org/pdf/1804.08943v1

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-41-A2 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-41, version 2

Periodic MFG stability as a boundary problem: the unavoidable neutral-phase obstruction

Problem ID: C73-40. Research attempt; independent review pending.

Target and scope

Let $(v_*,m_*,\lambda_*)$ be a smooth, genuinely time-periodic solution of period $P$ on the unit-volume torus ${\mathbb{T}}^d$: \[ \lambda_*-v_{*,t}-\nu\Delta v_*+\tfrac12|Dv_*|^2=f(x,m_*), \qquad m_{*,t}-\nu\Delta m_*-{\operatorname{div}}(m_*Dv_*)=0, \] where $\nu>0$, $m_*>0$, and $\int m_*=1$. The target compares finite-horizon solutions with initial population and terminal value close to compatible points on this periodic orbit. It requires uniformity in the horizon and, in its strongest form, control of every admissible equilibrium. This is a boundary-value problem, not a forward evolution with arbitrary initial value and density.

Version 1 gave a conditional contraction theorem based on a uniform inverse of the centered linear boundary operator, a conserved bilinear form, and a stationary homogeneous example with singular horizons. We retain their statements and proofs for context. Version 2 proves that the stated uniform inverse is impossible for every genuinely periodic reference: slowly modulated phase variations have order-one size, zero boundary data and residual of order $1/T$. Thus that conditional theorem cannot establish stability of the periodic orbits in the target. This identifies a necessary correction to the proposed method, not nonlinear orbital instability. A successful inverse-based method must allow an appropriate varying phase or otherwise remove the neutral direction. The full catalogue problem remains unresolved.

Version 1 records a check of the distinction between known periodic examples and their further qualitative questions in [CD, Section 3.3] and [CN, Section 6, especially Remark 6.4] on 9 October 2026. We retain those references without claiming a fresh literature search. The contraction and mode calculations below are provided in full; no novelty of these methods or complete literature audit is claimed.

Linear variations do not define a dissipative forward flow

For a phase $\theta$, write $b(t,x)=Dv_*(t+\theta,x)$, $\bar m(t,x)=m_*(t+\theta,x)$, and $a(t,x)=f_m(x,\bar m(t,x))$. Uncentered linear variations $(w,\rho)$ solve \begin{equation}\label{wsref-0-6c-69-6e} \begin{cases} -w_t-\nu\Delta w+b\cdot Dw=a\rho,\\ \rho_t-\nu\Delta\rho-{\operatorname{div}}(\rho b+\bar m Dw)=0, \qquad\int\rho=0. \end{cases} \end{equation}
Proposition: If $(w_1,\rho_1)$ and $(w_2,\rho_2)$ are two $C^{1,2}$ solutions of (lin) on an interval, then \[ \Omega(t)=\int_{{\mathbb{T}}^d}(w_1\rho_2-w_2\rho_1)dx \] is constant. For a single solution, \[ \frac{d}{dt}\int_{{\mathbb{T}}^d}w\rho\,dx =-\int_{{\mathbb{T}}^d}a\rho^2dx-\int_{{\mathbb{T}}^d}\bar m|Dw|^2dx. \] If the reference solution is differentiable one additional time in the classical sense, $(v_{*,t}(t+\theta),m_{*,t}(t+\theta))$ solves (lin); it is the phase variation.
Proof: Substitute (lin) in the derivative of $\int w_1\rho_2$. The two Laplace terms cancel after spatial integration by parts, and the terms involving $b$ cancel as well. The result is \[ \frac{d}{dt}\int w_1\rho_2 =-\int a\rho_1\rho_2-\int\bar m Dw_1\cdot Dw_2. \] The right side is symmetric in the two solutions, proving conservation of $\Omega$. Taking identical solutions gives the second formula. Differentiating the reference PDEs in time, with constant $\lambda_*$, gives the phase-variation assertion directly. ∎

The coefficient $a$ can be negative, so the displayed quadratic form is not generally dissipative. No bounded forward solution operator or Floquet monodromy for the full PDE is inferred from this identity. The backward heat component of (lin) requires that distinction.

A conditional nonlinear theorem with an explicit selection

Here are the precise spaces and additional regularity assumptions. Fix $0<\alpha<1$, assume the reference solution and $f$ are smooth, and fix $\zeta>0$ with $\bar m\geq2\zeta$ at every phase. For $T\geq1$ let $X_T$ be the Banach space of pairs $(w,\rho)$ of parabolic Hölder class $C^{1+\alpha/2,2+\alpha}$ on $[0,T]\times{\mathbb{T}}^d$, both of spatial mean zero at every time. Use the sum of their usual parabolic norms, with Hölder seminorms computed on time intervals of length at most one and their supremum over such intervals. This norm controls $\|Dw\|_\infty+\|\rho\|_\infty$. The local-in-time norm convention avoids a horizon factor in product estimates. It is equivalent to the usual norm for each fixed $T$.

Let $Y_T$ be the product of two spatially mean-zero residual spaces $C^{\alpha/2,\alpha}$, with the same interval convention, and two mean-zero boundary spaces $C^{2+\alpha}({\mathbb{T}}^d)$. Write $\Pi r=r-\int_{{\mathbb{T}}^d}r$ for spatial mean projection and define \begin{equation}\label{wsref-0-4c} L_{T,\theta}(w,\rho)= \left(\Pi[-w_t-\nu\Delta w+b\cdot Dw-a\rho],\, \rho_t-\nu\Delta\rho-{\operatorname{div}}(\rho b+\bar m Dw),\, \rho(0),\,w(T)\right). \end{equation} Centering the first equation matters: an uncentered constant residual would produce a value correction proportional to $T-t$, which has no uniform bound in a norm measuring the value itself.

Lemma (Uniform quadratic remainder): Under the smoothness and positivity assumptions just stated, there are $r_0>0$ and $C<\infty$, independent of $T\geq1$ and $\theta$, such that $r_0\leq\zeta$ and the nonlinear residual \[ N_{T,\theta}(w,\rho)= \left(\Pi\left[\tfrac12|Dw|^2- \{f(x,\bar m+\rho)-f(x,\bar m)-a\rho\}\right], -{\operatorname{div}}(\rho Dw),0,0\right) \] satisfies, whenever $\|x\|_{X_T},\|y\|_{X_T}\leq r_0$, \[ \|N(x)\|_{Y_T}\leq C\|x\|_{X_T}^2, \qquad \|N(x)-N(y)\|_{Y_T} \leq C(\|x\|_{X_T}+\|y\|_{X_T})\|x-y\|_{X_T}. \] Here $x$ and $y$ denote pairs of perturbations, not spatial points.
Proof: On the indicated ball, $\bar m+\rho\geq\zeta$ and all densities lie in a fixed compact positive interval. The derivatives of $f$ needed below, through third order in $m$ and the corresponding spatial Hölder bounds, are uniformly bounded on that interval. Reference norms are uniform in phase and horizon because the reference is smooth and periodic. Taylor's integral formula gives \[ f(x,\bar m+\rho)-f(x,\bar m)-a\rho =\rho^2\int_0^1(1-s)f_{mm}(x,\bar m+s\rho)ds. \] The multiplier on the right has uniformly bounded Hölder norm on $r_0$ balls and is Lipschitz in $\rho$ in that norm, by the bounded third derivative and the composition rule obtained by the mean-value formula. For scalar functions the elementary product estimate $[uv]_\alpha\leq\|u\|_\infty[v]_\alpha+ \|v\|_\infty[u]_\alpha$ applies to space and time differences. Together with $u^2-v^2=(u-v)(u+v)$, it proves both estimates for the Taylor remainder and $|Dw|^2$. For the second component expand ${\operatorname{div}}(\rho Dw)=D\rho\cdot Dw+\rho\Delta w$ and use the same product estimate, followed by subtraction of the two products for the Lipschitz bound. The $X_T$ norm controls every derivative used. Projection $\Pi$ is bounded in each residual norm. All estimates are local on time intervals of length at most one, so taking their supremum introduces no $T$ factor. Increasing one finite constant $C$ gives the announced bounds. ∎
Theorem (Selected orbital shadowing from a uniform inverse): In addition to the preceding assumptions, suppose every $L_{T,\theta}:X_T\to Y_T$ is a bounded linear isomorphism and \begin{equation}\label{wsref-0-69-6e-76-65-72-73-65} \sup_{T\geq1,\,\theta\in[0,P)}\|L_{T,\theta}^{-1}\|\leq K<\infty. \end{equation} Let $C,r_0$ be as in Lemma Lemma (Uniform quadratic remainder). For boundary data define \[ d_0=m_0-m_*(\theta),\qquad d_T=\Pi(h-v_*(T+\theta)),\qquad \delta=\|d_0\|_{C^{2+\alpha}}+\|d_T\|_{C^{2+\alpha}}, \] with $\int m_0=1$. If \begin{equation}\label{wsref-0-73-6d-61-6c-6c} \delta\leq\min\{r_0/(2K),\,1/(8CK^2)\}, \end{equation} then there is exactly one centered perturbation $(w,\rho)$ in the closed $X_T$ ball of radius $2K\delta$ producing a classical finite-horizon MFG solution with $m(0)=m_0$, $u(T)=h$. It satisfies \[ \|w\|_{X_T\text{ component}}+\|\rho\|_{X_T\text{ component}} \leq2K\delta,\qquad m\geq\zeta, \] and, for the orbit distance in the catalogue, \[ \sup_{0\leq t\leq T} \inf_{s\in[0,P)}\left(\|Du(t)-Dv_*(s)\|_\infty +\|m(t)-m_*(s)\|_1\right) \leq2K\delta. \] Selection is defined by this unique small fixed point. For two boundary perturbations satisfying a common smallness bound, their selected centered solutions differ by at most $2K$ times the $Y_T$ norm of the difference of the boundary vectors.
Proof: Put $d=(0,0,d_0,d_T)$ and consider the map $\mathcal A(x)=L_{T,\theta}^{-1}(d-N_{T,\theta}(x))$. For $R=2K\delta\leq r_0$ and $\|x\|\leq R$, \[ \|\mathcal A(x)\|\leq K\delta+KCR^2 \leq K\delta+\tfrac12K\delta<R \] when $\delta>0$, by (small); for $\delta=0$ the zero vector is the sole point of the stated ball. For $\|x\|,\|y\|\leq R$, \[ \|\mathcal A(x)-\mathcal A(y)\| \leq2KCR\|x-y\|\leq\tfrac12\|x-y\|. \] The complete closed ball therefore has a unique fixed point by the contraction theorem (whose proof is the convergence of iterates under this geometric distance bound). It solves $Lx+N(x)=d$. The density $m=\bar m+\rho$ has mass one and is at least $2\zeta-r_0\geq\zeta$.

To recover the actual value, set $u_{\rm ref}(t,x)=v_*(t+\theta,x)-\lambda_*t$. The first residual of $u_{\rm ref}+w$ with density $m$ has zero spatial projection and is therefore a scalar function $r(t)$. Choose a scalar $c(t)$ by $c'(t)=r(t)$ and $c(T)=\int_{{\mathbb{T}}^d}(h-u_{\rm ref}(T))dx$. Then $u=u_{\rm ref}+w+c$ satisfies the unprojected HJB equation because its residual is $r-c'=0$, and its terminal value is $h$. The Fokker--Planck equation and initial condition already follow from the fixed-point equation. This also proves nonemptiness for every allowed perturbation; no vacuous stability bound is used.

At time $t$, use orbit phase $s=t+\theta$ modulo $P$. The difference of gradients is $Dw$ and the difference of densities is $\rho$. The stated $X_T$ norm and unit torus volume bound their required norms by $\|x\|\leq2K\delta$, proving shadowing. For two fixed points, subtract their fixed-point equations and use the same $1/2$ Lipschitz bound for the nonlinear part to obtain $\|x-x'\|\leq K\|d-d'\|+\tfrac12\|x-x'\|$. This proves the final assertion and completes the proof. ∎

This theorem measures perturbations in a stronger boundary topology than the catalogue's $L^1$ and gradient-uniform topology and selects the nearby branch. It does not control far-away equilibria with the same data. More strongly, Theorem Theorem (Linear growth of any phase-fixed inverse) below proves that its hypothesis (inverse) fails for every genuine periodic reference under the present norms. The theorem remains a valid conditional implication, but is not an applicable stability result for those orbits.

Exact singular horizons in the homogeneous linear model

Proposition: Linearize the homogeneous quadratic MFG around $v=0,m=1$, with $a=f'(1)$. Let $q>0$ be a spatial Laplace eigenvalue and suppose $a<-\nu^2q$. Put $\omega=\sqrt{-q(\nu^2q+a)}$. For the corresponding mode amplitudes $(U,R)$, impose $R(0)=0$ and $U(T)=d$. Define \[ D(T)=\cos(\omega T)+\frac{\nu q}{\omega}\sin(\omega T),\qquad T_j=\frac{j\pi-\arctan(\omega/(\nu q))}{\omega},\quad j=1,2,\ldots. \] If $D(T)\ne0$, the unique mode solution has $U(0)=d/D(T)$. At every $T_j$, zero boundary data have a nonzero linear solution, and boundary data with $d\ne0$ have no linear solution. Furthermore $|D'(T_j)|=\sqrt{-aq}>0$, so the boundary response gain is unbounded as $T\to T_j$. Hence a bound of the form (inverse) over all $T\geq1$ fails for this homogeneous reference.
Proof: The mode system is \[ \binom{U'}{R'}= A\binom U R,\qquad A=\begin{pmatrix}\nu q&-a\\-q&-\nu q\end{pmatrix}, \qquad A^2=-\omega^2 I. \] Thus $e^{tA}=I\cos(\omega t)+(A/\omega)\sin(\omega t)$. Starting at $(U(0),0)$ gives \[ U(t)=D(t)U(0),\qquad R(t)=-\frac q\omega\sin(\omega t)U(0). \] The terminal condition therefore is the scalar equation $D(T)U(0)=d$. The listed $T_j$ are exactly the positive zeros of $D$. At a zero, $\cos(\omega T_j)=-(\nu q/\omega)\sin(\omega T_j)$, and $\sin(\omega T_j)\ne0$. Direct differentiation gives \[ |D'(T_j)|=\sqrt{\omega^2+\nu^2q^2}=\sqrt{-aq}. \] This proves both the singularity and the divergent inverse gain. There are arbitrarily large $T_j$, so restricting to $T\geq1$ does not remove the obstruction. A smooth spatial eigenfunction embeds each mode in all the Hölder spaces used above, so it obstructs a bounded inverse for the PDE operator as well. ∎

The uniform inverse fails along every genuine periodic orbit

This section is the substantive new result of Version 2. It applies to the genuine periodic reference in the target, whereas Proposition Proposition concerns a stationary reference only. Use exactly the centered spaces $X_T,Y_T$ and operator $L_{T,\theta}$ from (L); in particular their time Hölder norms are taken locally on intervals of length at most one. We assume the reference is smooth, as in the conditional nonlinear section, so its phase derivative belongs to all these spaces.
Lemma (The centered phase solution): Set \[ W_\theta(t,x)=\Pi\partial_t v_*(t+\theta,x),\qquad R_\theta(t,x)=\partial_t m_*(t+\theta,x). \] Then both fields have spatial mean zero and satisfy the two interior components of $L_{T,\theta}(W_\theta,R_\theta)=0$ on every time interval. Their local Hölder norms of every required order are bounded independently of $T$ and $\theta$. Moreover \[ c_*:=\max_{s\in[0,P]} \left(\|\Pi\partial_t v_*(s)\|_\infty +\|\partial_t m_*(s)\|_\infty\right)>0. \]
Proof: The density mean is constant, so $\int\partial_t m_*=0$. Differentiating the autonomous reference equations yields the uncentered linear system (lin) for $(\partial_t v_*,\partial_t m_*)$. Subtracting the spatial mean of $\partial_t v_*$ does not change its spatial derivatives. In the first equation it changes only a spatially constant time derivative, which is removed by $\Pi$ in (L). The second equation depends on the value perturbation only through its spatial derivatives. This proves the asserted interior equations and centering. Smoothness and time periodicity give every uniform local Hölder bound. If $c_*=0$, then $\partial_t m_*=0$ and $D\partial_t v_*=D\Pi\partial_t v_*=0$, so both $m_*$ and $Dv_*$ are independent of time. This contradicts genuine periodicity. ∎

Choose once and for all a smooth function $\chi:[0,1]\to[0,1]$ that is zero on neighborhoods of $0$ and $1$ and equals one on $[1/3,2/3]$. For $T\geq1$ define \[ x_{T,\theta}(t,x)=\chi(t/T) (W_\theta(t,x),R_\theta(t,x)). \]

Theorem (Linear growth of any phase-fixed inverse): There is a constant $C_*<\infty$, depending on the fixed smooth periodic reference, $\alpha$ and $\chi$, but not on $T$ or $\theta$, such that for every $T\geq\max\{1,3P\}$, \begin{equation}\label{wsref-0-71-75-61-73-69-6b-65-72-6e-65-6c} \|x_{T,\theta}\|_{X_T}\geq c_*,\qquad \|L_{T,\theta}x_{T,\theta}\|_{Y_T}\leq\frac{C_*}{T}. \end{equation} The boundary components of $L_{T,\theta}x_{T,\theta}$ vanish exactly. If $L_{T,\theta}$ is an isomorphism, then \[ \|L_{T,\theta}^{-1}\|\geq\frac{c_*}{C_*}T. \] Consequently the horizon-independent inverse assumption (inverse) of Version 1 is false for every smooth genuinely periodic reference in the target. The same test rules out a uniform estimate $\|x\|_{X_T}\leq K\|L_{T,\theta}x\|_{Y_T}$ even if no surjectivity assumption is made.
Proof: Multiplication by $\chi(t/T)$ preserves spatial means and the required regularity. Lemma Lemma (The centered phase solution), linearity, and the product rule give exactly \begin{equation}\label{wsref-0-70-68-61-73-65-66-6f-72-63-69-6e-67} L_{T,\theta}x_{T,\theta} =\left(-\frac{\chi'(t/T)}T W_\theta, \frac{\chi'(t/T)}T R_\theta,\ 0,\ 0\right). \end{equation} The signs come from the backward derivative $-w_t$ and forward derivative $\rho_t$. Projection of the first displayed term changes nothing because $W_\theta$ has spatial mean zero. The two boundary components vanish because $\chi$ vanishes near both endpoints.

We spell out the uniform Hölder estimate in (phaseforcing). Its scalar time multiplier has supremum at most $\|\chi'\|_\infty/T$. On an interval of length at most one, \[ \frac{|\chi'(t/T)/T-\chi'(s/T)/T|}{|t-s|^{\alpha/2}} \leq\frac{\|\chi''\|_\infty}{T^2}|t-s|^{1-\alpha/2} \leq\frac{\|\chi''\|_\infty}{T^2}. \] There is no spatial variation in this multiplier. Apply the elementary product Hölder inequality to it and to the uniformly bounded periodic fields of Lemma Lemma (The centered phase solution). Each residual $C^{\alpha/2,\alpha}$ norm is at most a constant times $1/T+1/T^2\leq2/T$. Their sum gives the upper bound in (quasikernel) with $C_*$ independent of the horizon and phase.

The interval $[T/3,2T/3]$ has length at least $P$ and $\chi(t/T)=1$ there. It therefore contains, after adding any $\theta$, every phase of the periodic orbit. In particular there is a time in this interval where the sum of the two supremum norms in Lemma Lemma (The centered phase solution) equals $c_*$. The $X_T$ norm controls that sum. This proves the lower bound in (quasikernel).

If an inverse exists, applying it to $L_{T,\theta}x_{T,\theta}$ gives $c_*\leq\|L_{T,\theta}^{-1}\|C_*/T$, proving its lower bound. If the residual happens to be zero for some horizon, the nonzero $x_{T,\theta}$ instead proves noninjectivity directly. Finally, the same substitution into a proposed uniform coercive estimate would give $c_*\leq KC_*/T$ for arbitrarily large $T$, which is impossible. ∎

Proposition (The obstruction is compatible with zero orbital error): For $|\delta|\leq1$ put \[ \phi(t)=t+\theta+\delta\chi(t/T),\qquad \widetilde m(t,x)=m_*(\phi(t),x),\qquad \widetilde u(t,x)=v_*(\phi(t),x)-\lambda_*t. \] These smooth fields have exactly the reference boundary data $\widetilde m(0)=m_*(\theta)$ and $\widetilde u(T)=v_*(T+\theta)-\lambda_*T$. Their HJB and Fokker--Planck residuals are, respectively, \begin{align*} -\widetilde u_t-\nu\Delta\widetilde u +\tfrac12|D\widetilde u|^2-f(x,\widetilde m) &=-\frac{\delta\chi'(t/T)}T\partial_t v_*(\phi(t),x),\\ \widetilde m_t-\nu\Delta\widetilde m -{\operatorname{div}}(\widetilde m D\widetilde u) &=\frac{\delta\chi'(t/T)}T\partial_t m_*(\phi(t),x). \end{align*} Their local $C^{\alpha/2,\alpha}$ norms are bounded by $C|\delta|/T$, uniformly for $T\geq1$, $\theta$ and $|\delta|\leq1$. At every time their orbit distance in the catalogue is exactly zero. They are approximate solutions; the proposition does not assert that their nonzero residual can be corrected by a comparably small exact solution.
Proof: The endpoint equalities follow from $\chi(0)=\chi(1)=0$. At phase $\phi(t)$, the reference PDEs cancel all terms that do not contain $\phi'(t)-1=\delta\chi'(t/T)/T$. The chain rule therefore gives precisely the two residual formulas. Their local Hölder bounds follow by the same product estimate used in Theorem Theorem (Linear growth of any phase-fixed inverse): the phase map has derivative bounded by $1+\|\chi'\|_\infty$, and the smooth periodic reference and its required derivatives consequently have uniform local Hölder norms after composition. Finally $(D\widetilde u(t),\widetilde m(t))= (Dv_*(\phi(t)),m_*(\phi(t)))$ belongs to $\mathcal O_*$ by its definition. The infimum defining the orbit distance is therefore zero. ∎

The phase derivative supplies a neutral direction because the equations are autonomous. Theorem Theorem (Linear growth of any phase-fixed inverse) proves that spatial centering alone does not remove this direction; even zero boundary perturbations admit a large response to slowly varying interior forcing. Proposition Proposition (The obstruction is compatible with zero orbital error) explains why this is not a proof of failure of the catalogue's orbital stability. An estimate for transverse errors, with a permitted time-dependent phase and a justified equation for that phase, would be a different statement and could avoid this obstruction. No such estimate is assumed or proved here.

Exact remaining issue

The homogeneous state in Proposition Proposition is stationary; that proposition is not a nonlinear instability theorem for an existing genuine periodic orbit. It demonstrates why a horizon-independent boundary inverse cannot be presumed from imaginary mode eigenvalues. For every genuinely periodic solution, Theorem Theorem (Linear growth of any phase-fixed inverse) now rules out the full uniform inverse proposed in Version 1. One must instead establish a suitable phase-adapted estimate or another nonlinear selection mechanism, handle possible additional conjugate horizons, and, for the strongest catalogue formulation, control every admissible equilibrium in the weaker boundary topology. Those steps are not proved here. The retained contraction theorem is a correct but inapplicable conditional implication for these orbits; the new conclusions are the universal phase obstruction and its precise compatibility with zero orbital distance for the modulated approximate trajectory.

Completion Estimate

25%

This is a heuristic assessment of the full catalogue target. The version establishes a universal neutral-phase obstruction to the previous uniform-inverse approach and explains why that obstruction does not imply orbital instability. It does not prove nonlinear stability or instability of an actual periodic equilibrium, provide a valid transverse selection theorem, or control every equilibrium with nearby boundary data.

References

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Econ-41-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-41

Periodic MFG stability as a boundary problem: a conditional nonlinear selection theorem and resonance obstruction

Problem ID: C73-40. Research attempt; independent review pending.

Target and scope

Let $(v_*,m_*,\lambda_*)$ be a smooth, genuinely time-periodic solution of period $P$ on the unit-volume torus ${\mathbb{T}}^d$: \[ \lambda_*-v_{*,t}-\nu\Delta v_*+\tfrac12|Dv_*|^2=f(x,m_*), \qquad m_{*,t}-\nu\Delta m_*-{\operatorname{div}}(m_*Dv_*)=0, \] where $\nu>0$, $m_*>0$, and $\int m_*=1$. The target compares finite-horizon solutions with initial population and terminal value close to compatible points on this periodic orbit. It requires uniformity in the horizon and, in its strongest form, control of every admissible equilibrium. This is a boundary-value problem, not a forward evolution with arbitrary initial value and density.

We prove a conditional, weaker statement: a uniformly bounded inverse for the centered linear boundary operator yields a unique nearby selected solution and uniform nonlinear shadowing for sufficiently smooth, strongly measured boundary perturbations. We also derive a conserved bilinear form for linear variations along any smooth periodic solution, and compute explicit horizons at which the linear boundary operator becomes singular in a homogeneous example. The nonlinear theorem is conditional on its inverse estimate; that estimate is not asserted for the given periodic orbits. The full catalogue problem remains unresolved.

The distinction between known periodic examples and their further qualitative questions was checked in [CD, Section 3.3] and [CN, Section 6, especially Remark 6.4] on 9 October 2026. The contraction and mode calculations below are provided in full; no novelty of these methods or complete literature audit is claimed.

Linear variations do not define a dissipative forward flow

For a phase $\theta$, write $b(t,x)=Dv_*(t+\theta,x)$, $\bar m(t,x)=m_*(t+\theta,x)$, and $a(t,x)=f_m(x,\bar m(t,x))$. Uncentered linear variations $(w,\rho)$ solve \begin{equation}\label{wsref-0-6c-69-6e} \begin{cases} -w_t-\nu\Delta w+b\cdot Dw=a\rho,\\ \rho_t-\nu\Delta\rho-{\operatorname{div}}(\rho b+\bar m Dw)=0, \qquad\int\rho=0. \end{cases} \end{equation}
Proposition: If $(w_1,\rho_1)$ and $(w_2,\rho_2)$ are two $C^{1,2}$ solutions of (lin) on an interval, then \[ \Omega(t)=\int_{{\mathbb{T}}^d}(w_1\rho_2-w_2\rho_1)dx \] is constant. For a single solution, \[ \frac{d}{dt}\int_{{\mathbb{T}}^d}w\rho\,dx =-\int_{{\mathbb{T}}^d}a\rho^2dx-\int_{{\mathbb{T}}^d}\bar m|Dw|^2dx. \] If the reference solution is differentiable one additional time in the classical sense, $(v_{*,t}(t+\theta),m_{*,t}(t+\theta))$ solves (lin); it is the phase variation.
Proof: Substitute (lin) in the derivative of $\int w_1\rho_2$. The two Laplace terms cancel after spatial integration by parts, and the terms involving $b$ cancel as well. The result is \[ \frac{d}{dt}\int w_1\rho_2 =-\int a\rho_1\rho_2-\int\bar m Dw_1\cdot Dw_2. \] The right side is symmetric in the two solutions, proving conservation of $\Omega$. Taking identical solutions gives the second formula. Differentiating the reference PDEs in time, with constant $\lambda_*$, gives the phase-variation assertion directly. ∎

The coefficient $a$ can be negative, so the displayed quadratic form is not generally dissipative. No bounded forward solution operator or Floquet monodromy for the full PDE is inferred from this identity. The backward heat component of (lin) requires that distinction.

A conditional nonlinear theorem with an explicit selection

Here are the precise spaces and additional regularity assumptions. Fix $0<\alpha<1$, assume the reference solution and $f$ are smooth, and fix $\zeta>0$ with $\bar m\geq2\zeta$ at every phase. For $T\geq1$ let $X_T$ be the Banach space of pairs $(w,\rho)$ of parabolic Hölder class $C^{1+\alpha/2,2+\alpha}$ on $[0,T]\times{\mathbb{T}}^d$, both of spatial mean zero at every time. Use the sum of their usual parabolic norms, with Hölder seminorms computed on time intervals of length at most one and their supremum over such intervals. This norm controls $\|Dw\|_\infty+\|\rho\|_\infty$. The local-in-time norm convention avoids a horizon factor in product estimates. It is equivalent to the usual norm for each fixed $T$.

Let $Y_T$ be the product of two spatially mean-zero residual spaces $C^{\alpha/2,\alpha}$, with the same interval convention, and two mean-zero boundary spaces $C^{2+\alpha}({\mathbb{T}}^d)$. Write $\Pi r=r-\int_{{\mathbb{T}}^d}r$ for spatial mean projection and define \begin{equation}\label{wsref-0-4c} L_{T,\theta}(w,\rho)= \left(\Pi[-w_t-\nu\Delta w+b\cdot Dw-a\rho],\, \rho_t-\nu\Delta\rho-{\operatorname{div}}(\rho b+\bar m Dw),\, \rho(0),\,w(T)\right). \end{equation} Centering the first equation matters: an uncentered constant residual would produce a value correction proportional to $T-t$, which has no uniform bound in a norm measuring the value itself.

Lemma (Uniform quadratic remainder): Under the smoothness and positivity assumptions just stated, there are $r_0>0$ and $C<\infty$, independent of $T\geq1$ and $\theta$, such that $r_0\leq\zeta$ and the nonlinear residual \[ N_{T,\theta}(w,\rho)= \left(\Pi\left[\tfrac12|Dw|^2- \{f(x,\bar m+\rho)-f(x,\bar m)-a\rho\}\right], -{\operatorname{div}}(\rho Dw),0,0\right) \] satisfies, whenever $\|x\|_{X_T},\|y\|_{X_T}\leq r_0$, \[ \|N(x)\|_{Y_T}\leq C\|x\|_{X_T}^2, \qquad \|N(x)-N(y)\|_{Y_T} \leq C(\|x\|_{X_T}+\|y\|_{X_T})\|x-y\|_{X_T}. \] Here $x$ and $y$ denote pairs of perturbations, not spatial points.
Proof: On the indicated ball, $\bar m+\rho\geq\zeta$ and all densities lie in a fixed compact positive interval. The derivatives of $f$ needed below, through third order in $m$ and the corresponding spatial Hölder bounds, are uniformly bounded on that interval. Reference norms are uniform in phase and horizon because the reference is smooth and periodic. Taylor's integral formula gives \[ f(x,\bar m+\rho)-f(x,\bar m)-a\rho =\rho^2\int_0^1(1-s)f_{mm}(x,\bar m+s\rho)ds. \] The multiplier on the right has uniformly bounded Hölder norm on $r_0$ balls and is Lipschitz in $\rho$ in that norm, by the bounded third derivative and the composition rule obtained by the mean-value formula. For scalar functions the elementary product estimate $[uv]_\alpha\leq\|u\|_\infty[v]_\alpha+ \|v\|_\infty[u]_\alpha$ applies to space and time differences. Together with $u^2-v^2=(u-v)(u+v)$, it proves both estimates for the Taylor remainder and $|Dw|^2$. For the second component expand ${\operatorname{div}}(\rho Dw)=D\rho\cdot Dw+\rho\Delta w$ and use the same product estimate, followed by subtraction of the two products for the Lipschitz bound. The $X_T$ norm controls every derivative used. Projection $\Pi$ is bounded in each residual norm. All estimates are local on time intervals of length at most one, so taking their supremum introduces no $T$ factor. Increasing one finite constant $C$ gives the announced bounds. ∎
Theorem (Selected orbital shadowing from a uniform inverse): In addition to the preceding assumptions, suppose every $L_{T,\theta}:X_T\to Y_T$ is a bounded linear isomorphism and \begin{equation}\label{wsref-0-69-6e-76-65-72-73-65} \sup_{T\geq1,\,\theta\in[0,P)}\|L_{T,\theta}^{-1}\|\leq K<\infty. \end{equation} Let $C,r_0$ be as in Lemma Lemma (Uniform quadratic remainder). For boundary data define \[ d_0=m_0-m_*(\theta),\qquad d_T=\Pi(h-v_*(T+\theta)),\qquad \delta=\|d_0\|_{C^{2+\alpha}}+\|d_T\|_{C^{2+\alpha}}, \] with $\int m_0=1$. If \begin{equation}\label{wsref-0-73-6d-61-6c-6c} \delta\leq\min\{r_0/(2K),\,1/(8CK^2)\}, \end{equation} then there is exactly one centered perturbation $(w,\rho)$ in the closed $X_T$ ball of radius $2K\delta$ producing a classical finite-horizon MFG solution with $m(0)=m_0$, $u(T)=h$. It satisfies \[ \|w\|_{X_T\text{ component}}+\|\rho\|_{X_T\text{ component}} \leq2K\delta,\qquad m\geq\zeta, \] and, for the orbit distance in the catalogue, \[ \sup_{0\leq t\leq T} \inf_{s\in[0,P)}\left(\|Du(t)-Dv_*(s)\|_\infty +\|m(t)-m_*(s)\|_1\right) \leq2K\delta. \] Selection is defined by this unique small fixed point. For two boundary perturbations satisfying a common smallness bound, their selected centered solutions differ by at most $2K$ times the $Y_T$ norm of the difference of the boundary vectors.
Proof: Put $d=(0,0,d_0,d_T)$ and consider the map $\mathcal A(x)=L_{T,\theta}^{-1}(d-N_{T,\theta}(x))$. For $R=2K\delta\leq r_0$ and $\|x\|\leq R$, \[ \|\mathcal A(x)\|\leq K\delta+KCR^2 \leq K\delta+\tfrac12K\delta<R \] when $\delta>0$, by (small); for $\delta=0$ the zero vector is the sole point of the stated ball. For $\|x\|,\|y\|\leq R$, \[ \|\mathcal A(x)-\mathcal A(y)\| \leq2KCR\|x-y\|\leq\tfrac12\|x-y\|. \] The complete closed ball therefore has a unique fixed point by the contraction theorem (whose proof is the convergence of iterates under this geometric distance bound). It solves $Lx+N(x)=d$. The density $m=\bar m+\rho$ has mass one and is at least $2\zeta-r_0\geq\zeta$.

To recover the actual value, set $u_{\rm ref}(t,x)=v_*(t+\theta,x)-\lambda_*t$. The first residual of $u_{\rm ref}+w$ with density $m$ has zero spatial projection and is therefore a scalar function $r(t)$. Choose a scalar $c(t)$ by $c'(t)=r(t)$ and $c(T)=\int_{{\mathbb{T}}^d}(h-u_{\rm ref}(T))dx$. Then $u=u_{\rm ref}+w+c$ satisfies the unprojected HJB equation because its residual is $r-c'=0$, and its terminal value is $h$. The Fokker--Planck equation and initial condition already follow from the fixed-point equation. This also proves nonemptiness for every allowed perturbation; no vacuous stability bound is used.

At time $t$, use orbit phase $s=t+\theta$ modulo $P$. The difference of gradients is $Dw$ and the difference of densities is $\rho$. The stated $X_T$ norm and unit torus volume bound their required norms by $\|x\|\leq2K\delta$, proving shadowing. For two fixed points, subtract their fixed-point equations and use the same $1/2$ Lipschitz bound for the nonlinear part to obtain $\|x-x'\|\leq K\|d-d'\|+\tfrac12\|x-x'\|$. This proves the final assertion and completes the proof. ∎

This theorem measures perturbations in a stronger boundary topology than the catalogue's $L^1$ and gradient-uniform topology and selects the nearby branch. It neither controls far-away equilibria with the same data nor verifies (inverse) for a genuinely periodic reference. These are explicit restrictions, not suppressed steps.

Exact singular horizons in the homogeneous linear model

Proposition: Linearize the homogeneous quadratic MFG around $v=0,m=1$, with $a=f'(1)$. Let $q>0$ be a spatial Laplace eigenvalue and suppose $a<-\nu^2q$. Put $\omega=\sqrt{-q(\nu^2q+a)}$. For the corresponding mode amplitudes $(U,R)$, impose $R(0)=0$ and $U(T)=d$. Define \[ D(T)=\cos(\omega T)+\frac{\nu q}{\omega}\sin(\omega T),\qquad T_j=\frac{j\pi-\arctan(\omega/(\nu q))}{\omega},\quad j=1,2,\ldots. \] If $D(T)\ne0$, the unique mode solution has $U(0)=d/D(T)$. At every $T_j$, zero boundary data have a nonzero linear solution, and boundary data with $d\ne0$ have no linear solution. Furthermore $|D'(T_j)|=\sqrt{-aq}>0$, so the boundary response gain is unbounded as $T\to T_j$. Hence a bound of the form (inverse) over all $T\geq1$ fails for this homogeneous reference.
Proof: The mode system is \[ \binom{U'}{R'}= A\binom U R,\qquad A=\begin{pmatrix}\nu q&-a\\-q&-\nu q\end{pmatrix}, \qquad A^2=-\omega^2 I. \] Thus $e^{tA}=I\cos(\omega t)+(A/\omega)\sin(\omega t)$. Starting at $(U(0),0)$ gives \[ U(t)=D(t)U(0),\qquad R(t)=-\frac q\omega\sin(\omega t)U(0). \] The terminal condition therefore is the scalar equation $D(T)U(0)=d$. The listed $T_j$ are exactly the positive zeros of $D$. At a zero, $\cos(\omega T_j)=-(\nu q/\omega)\sin(\omega T_j)$, and $\sin(\omega T_j)\ne0$. Direct differentiation gives \[ |D'(T_j)|=\sqrt{\omega^2+\nu^2q^2}=\sqrt{-aq}. \] This proves both the singularity and the divergent inverse gain. There are arbitrarily large $T_j$, so restricting to $T\geq1$ does not remove the obstruction. A smooth spatial eigenfunction embeds each mode in all the Hölder spaces used above, so it obstructs a bounded inverse for the PDE operator as well. ∎

Exact remaining issue

The homogeneous state in Proposition Proposition is stationary; that proposition is not a nonlinear instability theorem for an existing genuine periodic orbit. It demonstrates why a horizon-independent boundary inverse cannot be presumed from imaginary mode eigenvalues. For the genuinely periodic solutions in the target, one must establish an appropriate uniform inverse or another nonlinear selection mechanism, handle phase and possible conjugate horizons, and, for the strongest catalogue formulation, control every admissible equilibrium in the weaker boundary topology. Those steps are not proved here. The conditional selected shadowing theorem, the conserved form, and the exact singular horizons are the verified outcomes of this attempt.

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Econ-41-A3 · Partial result · LLM-generated · gpt-6 astra, claude opus 5.5, gpt-6 astra pro

Selected boundary stability for periodic mean-field games

Selected boundary stability for periodic mean-field games

Ji Ho Bae
LawAll AI

10 October 2026

\begin{abstract} For every dimension, smooth local coupling and classical genuinely periodic quadratic mean-field-game equilibrium, we characterize a specified weaker nonlinear stability property: all small smooth boundary perturbations admit a nearby selected equilibrium with an error linear in the boundary error, with constants depending on the horizon. The criterion is both necessary and sufficient, including a negative conclusion at every conjugate horizon. It has finite matrix certificates constructed from two scalar parabolic problems, and a convergent equilibrium-selection rule. An explicit entire periodic family has only a discrete set of conjugate horizons. \end{abstract}

Scope. This addresses the weaker-stability alternative in C73-40 for its entire original model class. No restriction to a special coupling, dimension, periodic family or a preselected set of regular horizons is made in the general theorem. The original horizon-uniform orbital shadowing of every admissible equilibrium is a stronger question and is not claimed here.

The original boundary-value setting

Let $\nu>0$, let $f:\mathbb T^d\times(0,\infty)\to\mathbb R$ be smooth, and let a positive, mass-one classical reference satisfy \begin{equation}\tag{MFG}\label{wsref-0-4d-46-47} \begin{split} \lambda_*-v_{*,t}-\nu\Delta v_*+\tfrac12|Dv_*|^2&=f(x,m_*),\\ m_{*,t}-\nu\Delta m_*-\operatorname{div}(m_*Dv_*)&=0,\\ (v_*,m_*)(t+P)&=(v_*,m_*)(t),\qquad \int m_*(t)=1. \end{split} \end{equation} The pair $(Dv_*,m_*)$ is nonconstant in time. The actual reference value $v_*(t,x)-\lambda_*t$ may drift. Its orbit is $\mathcal O_* =\{(Dv_*(s),m_*(s)):s\in\mathbb R/P\mathbb Z\}$, and its distance is the infimum over $s$ of the sum of the gradient $L^\infty$ error and density $L^1$ error.

Perturbed equilibria solve \[ -u_t-\nu\Delta u+\tfrac12|Du|^2=f(x,m),\qquad m_t-\nu\Delta m-\operatorname{div}(mDu)=0, \qquad m(0)=m_0,\quad u(T)=h. \] Initial population and terminal value are the independent boundary data. No initial-value MFG semigroup is presumed.

Cardaliaguet--Delarue discuss the periodic-solution stability question in [CD, Section 3.3, paragraph "Periodic solutions," p. 3681]. Genuine periodic examples already appear in [CN, Section 6, Theorem 6.1 and Remark 6.4]. The classification proved here concerns the specified nonlinear selection property, rather than imaginary eigenvalues.

The nonlinear property being classified

Write $Q_T=[0,T]\times\mathbb T^d$, with the torus of volume one, and $\Pi z=z-\int_{\mathbb T^d}z$. Fix $0<\alpha<1$. For any reference in the original problem, any $T>0$, and any phase $\theta$, put \[ \bar m(t)=m_*(t+\theta),\quad b(t)=Dv_*(t+\theta),\quad a(t,x)=f_m(x,\bar m(t,x)),\quad u_{\rm r}(t,x)=v_*(t+\theta,x)-\lambda_*t. \] Both $\bar m$ and $b$ are known reference fields. We use \[ X_T=\{(w,\rho)\in(C^{1+\alpha/2,2+\alpha}(Q_T))^2: \textstyle\int w(t)=\int\rho(t)=0\},\qquad D=(C^{2+\alpha}_0(\mathbb T^d))^2, \] with their usual sum norms; a subscript zero means spatial mean zero. The boundary perturbation is $d=(d_0,d_T)=(m_0-\bar m(0),\Pi(h-u_{\rm r}(T)))$. Every sufficiently small $d\in D$ gives a positive normalized initial density. Terminal constants are free and will be restored explicitly.

Selected local Lipschitz boundary stability at $(T,\theta)$. There are $r,\delta,C>0$ and a rule $S:\{\|d\|_D<\delta\}\to X_T$ such that $S(0)=0$, and for every such $d$, $S(d)=(w,\rho)$ produces a classical MFG equilibrium with the stated initial density and terminal value, with \begin{equation}\tag{W1}\label{wsref-0-57-31} \|(w,\rho)\|_{X_T}\leq C\|d\|_D<r, \qquad \bar m+\rho\geq\tfrac12\min_{Q_T}\bar m. \end{equation} No continuity or differentiability of $S$ is assumed in this definition. The theorem constructs a Lipschitz rule when the property holds. This is a specified weaker alternative in C73-40: boundary errors are measured more strongly, a nearby equilibrium is selected, and the constants may depend on horizon and phase. In particular, \begin{equation}\tag{W2}\label{wsref-0-57-32} \sup_{0\leq t\leq T}d_*((Du(t),m(t)),\mathcal O_*) \leq\|(w,\rho)\|_{X_T}\leq C\|d\|_D. \end{equation} The comparison phase in this estimate is $t+\theta$ modulo $P$. The neighborhood can always be chosen with density lower bound $\min m_*/2$ and an admissibility upper bound one larger than the reference norm. If the original specified caps have strict slack, it can also be chosen within those caps. We do not assert an open perturbation result inside an arbitrarily saturated cap.

For a set $H\subset(0,\infty)\times(\mathbb R/P\mathbb Z)$, the reference has this property on $H$ when it has it at every pair in $H$. The criterion below classifies every pair, including the pairs where the property fails.

Scalar parabolic facts and an explicit compactness bound

We record the analytic facts used in the classification. All coefficients below are smooth on a compact cylinder and $\nu>0$.
Lemma (Scalar equations): The forward equation \[ y_t-\nu\Delta y+c\cdot Dy+e y=g,\qquad y(0)=0, \] and its time reversal have a unique solution in $\mathcal W_T=H^1(0,T;L^2)\cap L^2(0,T;H^2)$ for each $g\in L^2$. For parabolic $C^{\alpha/2,\alpha}$ forcing and $C^{2+\alpha}$ initial or terminal data the solution is in $C^{1+\alpha/2,2+\alpha}$, with a bounded solution operator. Writing $B=\|c\|_\infty$ and $E=\|e\|_\infty$, define \begin{align} A&=1+2E+B^2/\nu,&Y&=\sqrt T e^{AT/2},\notag\\ G&=\sqrt{(1+AY^2)/\nu},& J&=\frac{\sqrt3}{\nu}\sqrt{1+B^2G^2+E^2Y^2},\tag{P1}\label{wsref-0-50-31}\\ R(T,\nu,B,E)&=Y+G+J+1+\nu J+BG+EY.\notag \end{align} Then $\|y\|_2\leq Y\|g\|_2$, $\|Dy\|_2\leq G\|g\|_2$, $\|\Delta y\|_2\leq J\|g\|_2$, and $\|y_t\|_2\leq(1+\nu J+BG+EY)\|g\|_2$. All these norms are on $Q_T$.
Proof: Multiplication by $y$, Young's inequality, and integration give \[ (\|y(t)\|_2^2)' +\nu\|Dy(t)\|_2^2 \leq \|g(t)\|_2^2+A\|y(t)\|_2^2. \] Gronwall gives the bounds $Y,G$. Multiplication by $-\Delta y$ gives \[ (\|Dy(t)\|_2^2)' +\nu\|\Delta y(t)\|_2^2 \leq\frac3\nu(\|g(t)\|_2^2+B^2\|Dy(t)\|_2^2 +E^2\|y(t)\|_2^2). \] This proves the bound $J$, and the equation proves the time derivative bound. Spatial Fourier truncations give finite linear ODEs. These uniform estimates yield a weak limit with the stated zero trace; the first estimate applied to a difference proves uniqueness. On a torus, the $L^2$ norm of the full spatial Hessian equals that of $\Delta y$, so the asserted space is exactly controlled.

The Hölder assertion follows directly from the heat kernel. If $H_t=e^{\nu t\Delta}$ and $z(t)=\int_0^t H_{t-s}q(s)\,ds$, the periodic heat kernel satisfies \[ \int |D^j H_t(x)|\,\operatorname{dist}(x,0)^\alpha\,dx \leq C t^{(\alpha-j)/2}\quad(0<t\leq T), \] with the analogous bound for a time derivative (which adds two to $j$). These follow by differentiating the Gaussian and summing its translates. The integral of every second spatial derivative is zero. Subtracting $q(s,x)$ inside the second-derivative convolution therefore gives the integrable majorant $C\|q\|_\alpha(t-s)^{\alpha/2-1}$. For a spatial increment $h$, split the integral at $h^2$: the near part is bounded by $C|h|^\alpha$, and the far part by $C|h|\int_{h^2}^T r^{\alpha/2-3/2}dr\leq C|h|^\alpha$. For a time increment $h>0$, splitting at $h$ and using a heat-kernel time derivative bounds the two parts by $Ch^{\alpha/2}$ and $Ch\int_h^T r^{\alpha/2-2}dr\leq Ch^{\alpha/2}$. Thus $D^2z$ has the required parabolic Hölder bound, and $z_t=q+\nu\Delta z$ does too. The homogeneous heat solution with $C^{2+\alpha}$ data satisfies the same estimates by commuting spatial derivatives and the same kernel bounds.

To include $c\cdot Dy+ey$, use the elementary interpolation estimate \[ \|Dy\|_{C^{\alpha/2,\alpha}}+ \|y\|_{C^{\alpha/2,\alpha}} \leq\varepsilon\|y\|_{C^{1+\alpha/2,2+\alpha}} +C_\varepsilon\|y\|_\infty. \] It follows by finite spatial differences and splitting increments into those smaller and larger than a fixed interpolation length; time increments use the controlled time derivative. Product Hölder inequalities and a sufficiently small $\varepsilon$ absorb the highest norm. The maximum principle bounds the remaining supremum norm by $e^{ET}(\|y(0)\|_\infty+T\|g\|_\infty)$. Consequently the Hölder estimate is uniform for the coefficients $sc,se$, $0\leq s\leq1$. The heat equation is solvable at $s=0$. The bounded inverse estimate makes solvability open by the Neumann series. It makes solvability closed by compactness in every lower Hölder exponent and the same estimate for the limiting equation. This proves existence at $s=1$; the energy estimate proves uniqueness. Time reversal proves the terminal version. Projecting a scalar backward solution onto its spatial mean-zero part proves the same assertion when its equation is spatially centered. ∎

Classical entire periodic reference solutions in the original statement are smooth, so the lemma does not strengthen that statement's reference assumptions. Indeed $m_t,D^2m,v_t,D^2v$ are bounded on the periodic cylinder. Spatial interpolation shows $Dv$ is $1/2$-Hölder in time and Lipschitz in space, while $m$ is Lipschitz in time and space. The HJB right-hand side $f(x,m)-|Dv|^2/2$ is consequently parabolic $C^{\alpha/2,\alpha}$. Interior versions of the heat-kernel estimates just proved upgrade $v$. In nondivergence form the population equation then has Hölder right-hand side $Dv\cdot Dm+m\Delta v$, upgrading $m$. Differentiating and repeating on interior subcylinders upgrades both fields to every order. Periodicity covers the entire cylinder without an initial or terminal regularity exception.

A criterion using scalar problems and finite matrices

Let $\mathcal H_T=L^2(Q_T)$ with spatial mean zero at almost every time. For $r\in\mathcal H_T$, first solve the centered scalar terminal problem and then the scalar initial problem \begin{equation}\tag{K1}\label{wsref-0-4b-31} \begin{aligned} -W_t-\nu\Delta W+\Pi(b\cdot DW)&=\Pi(ar),& W(T)&=0,\\ Z_t-\nu\Delta Z-\operatorname{div}(bZ) &=\operatorname{div}(\bar m DW),& Z(0)&=0. \end{aligned} \end{equation} Define $K_{T,\theta}r=Z$. These are two uniquely solvable scalar equations. Put $B=\|b\|_\infty$, $E=\|\operatorname{div}b\|_\infty$, $A_0=\|a\|_\infty$, $M=\|\bar m\|_\infty$, $M_1=\|D\bar m\|_\infty$. Let $G_0,J_0$ be $P1$ with $E=0$ and set \begin{equation}\tag{K2}\label{wsref-0-4b-32} C_K=1+R(T,\nu,B,E)A_0(MJ_0+M_1G_0). \end{equation} Lemma Lemma (Scalar equations) proves \[ \|Kr\|_2+\|(Kr)_t\|_2+\|\Delta Kr\|_2\leq C_K\|r\|_2. \] Thus $K$ is compact: bounded sets in $\mathcal W_T$ are compact in $L^2$, as is also demonstrated by the explicit projections below.

Let $P_N$ be orthogonal projection onto temporal cosine modes $0\leq n\leq N$ and spatial Fourier modes $0<\|k\|_\infty\leq N$. Use real sine--cosine combinations for a real orthonormal basis $e_1,\ldots,e_{q_N}$ of its range, where $q_N=(N+1)((2N+1)^d-1)$. Parseval and integration by parts against the temporal sine modes give \begin{equation}\tag{K3}\label{wsref-0-4b-33} \|(I-P_N)K\|\leq e_N:=\frac{C_K(T+1)}{N+1}. \end{equation} More precisely, the time tail is bounded by $T\|(Kr)_t\|_2/(\pi(N+1))$ and the spatial tail by $\|\Delta Kr\|_2/(4\pi^2(N+1)^2)$; their sum implies $K3$. Define the finite real matrix \begin{equation}\tag{K4}\label{wsref-0-4b-34} D_N=(\delta_{ij}-\langle e_i,Ke_j\rangle)_{i,j=1}^{q_N}. \end{equation} Every entry is specified by $K1$, using only the original coefficients. The following is an exact finite-certificate criterion: \begin{equation}\tag{FC}\label{wsref-0-46-43} \boxed{\quad\exists N\geq1:\quad \det D_N\ne0,\qquad e_N\{1+(1+C_K)\|D_N^{-1}\|_{2\to2}\}<1.\quad} \end{equation} No precision or running-time assertion is implicit: arbitrary smooth coefficients need not even be computably presented. Certified bounds on the scalar problems and finite matrix entries can verify a strict certificate when those data are effectively supplied.

Lemma (Exactness of the finite certificate): Condition (FC) holds if and only if $I-K$ is invertible on $\mathcal H_T$.
Proof: In $\operatorname{ran}P_N\oplus\ker P_N$, the operator $I-P_NK$ is upper triangular, with diagonal blocks $D_N,I$ and upper right block $-P_NK(I-P_N)$. If $D_N$ is invertible, its full inverse has norm at most \[ B_N=1+(1+C_K)\|D_N^{-1}\|. \] Since $\|(I-K)-(I-P_NK)\|\leq e_N$, the Neumann series proves invertibility whenever $e_NB_N<1$. Conversely, if $I-K$ is invertible, $K3$ implies that, for all sufficiently large $N$, $I-P_NK$ is invertible with inverse norm at most $2\|(I-K)^{-1}\|$. Restriction to its finite invariant subspace bounds $\|D_N^{-1}\|$ by the same constant. Thus $B_N$ is bounded and $e_NB_N<1$ eventually. This proves both directions. ∎

Fredholm reduction of the Jacobi boundary problem

The centered linear boundary operator is \begin{equation}\tag{L1}\label{wsref-0-4c-31} L(w,\rho)=\left(\Pi[-w_t-\nu\Delta w+b\cdot Dw-a\rho],\ \rho_t-\nu\Delta\rho-\operatorname{div}(b\rho+\bar m Dw),\ \rho(0),w(T)\right). \end{equation} Its target is the product of two mean-zero parabolic $C^{\alpha/2,\alpha}$ spaces and $D$. In particular it includes both interior residuals as well as the two boundary data.
Lemma (General Fredholm reduction): For every original model, phase and finite horizon, $L$ has Fredholm index zero. It is invertible if and only if its homogeneous boundary problem has no nonzero solution, and this is also equivalent to (FC).
Proof: Delete $-a\rho$ from the first component of $L$, obtaining $L_0$. The scalar lemma solves first the terminal value equation for $w$ and then the initial population equation for $\rho$; hence $L_0$ is an isomorphism on the indicated Hölder spaces. The deleted term factors through the compact embedding $C^{1+\alpha/2,2+\alpha}\hookrightarrow C^{\alpha/2,\alpha}$. Thus $L$ is an isomorphism plus a compact operator and has index zero. The compact-operator alternative gives its invertibility precisely when its kernel is zero. One elementary justification of that alternative is to approximate the compact operator in norm by a finite-rank operator with remainder less than one, invert the remainder by a Neumann series, and reduce kernel and cokernel to the same finite square matrix. The compactness here permits finite-rank approximation by spatial--temporal interpolation, so no Banach approximation property is presumed.

For a zero-boundary solution, the scalar reduction $K1$ says exactly $\rho=K\rho$, and conversely such a fixed density reconstructs $w$. This is true in $L^2$ and in the Hölder spaces. These kernels agree. To see this without a dimension restriction, an $L^2$ kernel first gives $w,\rho\in\mathcal W_T$. All their spatial derivatives of order at most two are then in $L^2$. Suppose derivatives through order $j$ are known in $L^2$. Differentiate the backward scalar equation to order $j$. Its leading forcing is $aD^j\rho$; commutators contain only already controlled derivatives. The zero terminal trace remains zero, so the scalar lemma puts $D^jw$ in $\mathcal W_T$. Next differentiate the forward equation: its principal forcing is $\operatorname{div}(\bar m D(D^jw))$; commutators contain derivatives of $\rho$ through order $j$ and of $w$ through order $j+1$, now controlled. Its zero initial trace again gives $D^j\rho\in\mathcal W_T$. Spatial difference quotients justify these distributional differentiations and the zero traces. This gains two orders and iterates indefinitely. The elementary trace estimate $H^1_tH^s_x\cap L^2_tH^{s+2}_x\subset C_tH^{s+1}_x$ (obtained by differentiating the squared $H^{s+1}$ Fourier norm) then gives continuity of every spatial derivative. The equations give all time derivatives in succession. Both fields are smooth up to their respective time endpoints and belong to the Hölder kernel. Lemma Lemma (Exactness of the finite certificate) completes the equivalence. ∎

Boundary traces detect every nonzero Jacobi kernel

Lemma (Zero at both endpoints): A homogeneous centered Jacobi field $(w,\rho)$ for which $w(0)=\rho(0)=w(T)=\rho(T)=0$ is identically zero.
Proof: Set $\kappa=\bar m/(2\nu)$, $r=\rho+\kappa w$, $U=(w,r)$ and $A=\operatorname{diag}(\nu(-\Delta),-\nu(-\Delta))$ on the real $L^2$ product space. The first component has spatial mean zero; the second need not. Let \[ c_w(t)=\int(b\cdot Dw-a\rho) =\int[-(\operatorname{div}b)w-ar+a\kappa w]. \] Using $\bar m_t=\nu\Delta\bar m+\operatorname{div}(\bar m b)$, direct substitution cancels all cross second derivatives and gives \begin{align*} w_t&=\nu(-\Delta)w+b\cdot Dw+a\kappa w-ar-c_w,\\ r_t&=-\nu(-\Delta)r+b\cdot Dr+ (\operatorname{div}b-a\kappa)r+a\kappa^2w-\kappa c_w. \end{align*} Thus $U_t=AU+B(t)U$, where $A$ is a constant self-adjoint operator and $B(t)$ has only first spatial derivatives and bounded terms. Integration by parts in the expression for $c_w$ makes its bounded rank-one character explicit. For some finite $C_0$, \[ \|B(t)U\|^2\leq C_0(\langle |A|U,U\rangle+\|U\|^2). \] Both endpoint traces of $U$ vanish. Put $V=e^{\phi(t)}U$, where $\phi'>0$ will be chosen. Integration by parts yields the exact identity \[ \int_0^T\|V_t-(A+\phi')V\|^2dt =\int_0^T\{\|V_t\|^2+\|(A+\phi')V\|^2 +\phi''\|V\|^2\}dt. \] The endpoint terms are zero. Spatial Fourier truncation followed by the $\mathcal W_T$ limit justifies the identity at this regularity. On the other hand $V_t-(A+\phi')V=BV$, and \[ \langle|A|V,V\rangle\leq\|AV\|\|V\| \leq\|(A+\phi')V\|\|V\|+\phi'\|V\|^2. \] Young's inequality therefore bounds the left side by \[ \tfrac12\int_0^T\|(A+\phi')V\|^2dt +C_1\int_0^T(1+\phi')\|V\|^2dt. \] Choose $\phi'(t)=q e^{kt}$ with $q\geq1$ and $k>2C_1$. Then $\phi''=k\phi'>C_1(1+\phi')$, so the two displays force $V=0$. This proves the lemma. The proof uses no forward MFG semigroup or sign assumption on $a$. ∎

For a zero-boundary Jacobi field $(z,\sigma)$, namely $\sigma(0)=0,z(T)=0$, and a perturbation $(w,\rho)$ with interior linear residuals $(F,G)$, integration by parts gives \begin{equation}\tag{G1}\label{wsref-0-47-31} \int d_T\sigma(T)+\int z(0)d_0 =-\int_{Q_T}(F\sigma+zG). \end{equation} Indeed differentiating $\int(w\sigma-z\rho)$ cancels both Laplacians, both drift terms, and the symmetric coupling terms, leaving $-\int(F\sigma+zG)$. Mean projections disappear because $z,\sigma$ have mean zero. If both boundary traces $z(0),\sigma(T)$ are zero, Lemma Lemma (Zero at both endpoints) says the field is zero. Consequently every nonzero kernel field gives a nonzero functional on the independent boundary directions $(d_0,d_T)\in D$.

Exact nonlinear classification and the selection rule

Theorem: For every reference allowed in C73-40, every $0<\alpha<1$, every $T>0$, and every phase $\theta$, the following are equivalent:
  1. Selected local Lipschitz boundary stability $W1$ holds.
  2. The homogeneous Jacobi problem $L1$ has zero kernel.
  3. The finite-certificate condition (FC) holds.
When they hold, the rule below selects the unique nearby equilibrium and is Lipschitz between boundary data; it also satisfies $W2$. When they fail, no rule satisfying $W1$ for all sufficiently small independent boundary perturbations exists, even without continuity of that rule. A reference has the property on a specified set $H$ exactly when (FC) holds at every pair in $H$.
Proof: The equivalence of items 2 and 3 is Lemma Lemma (General Fredholm reduction). Write the nonlinear perturbation equations as \[ L\xi+N(\xi)=(0,0,d_0,d_T),\qquad \xi=(w,\rho), \] where \begin{equation}\tag{N1}\label{wsref-0-4e-31} N(\xi)=\left(\Pi\left[\tfrac12|Dw|^2- \{f(x,\bar m+\rho)-f(x,\bar m)-a\rho\}\right], -\operatorname{div}(\rho Dw),0,0\right). \end{equation} On a sufficiently small positive-density ball in $X_T$, Taylor's integral formula and the product Hölder inequality give \begin{equation}\tag{N2}\label{wsref-0-4e-32} \|N(\xi)\|\leq C_2\|\xi\|^2,\qquad \|N(\xi)-N(\eta)\|\leq C_2(\|\xi\|+\|\eta\|)\|\xi-\eta\|. \end{equation} Explicitly, the scalar Taylor remainder is $\rho^2\int_0^1(1-s)f_{mm}(x,\bar m+s\rho)ds$; its multiplier and its Lipschitz dependence have bounded Hölder norms on the fixed compact positive density interval. The other components are products of the derivatives controlled by $X_T$.

If item 2 holds, let $K_L=\|L^{-1}\|$. Choose $r>0$ below the positive-density and declared admissibility margins and with $2C_2K_Lr\leq1/2$. Take $\delta\leq r/(2K_L)$. Starting at $\xi_0=0$, perform the explicit iteration \begin{equation}\tag{SEL}\label{wsref-0-53-45-4c} \xi_{n+1}=L^{-1}\{(0,0,d_0,d_T)-N(\xi_n)\}. \end{equation} It maps the closed ball of radius $r$ into itself, since its norm is at most $K_L\|d\|+K_LC_2r^2\leq3r/4$, and its contraction factor is at most $2K_LC_2r\leq1/2$. Successive differences sum geometrically, proving convergence and uniqueness in this ball. Its limit satisfies \[ \|\xi(d)\|\leq K_L\|d\|+\tfrac12\|\xi(d)\|, \qquad \|\xi(d)-\xi(d')\|\leq2K_L\|d-d'\|. \] The first inequality gives $W1$, and the second gives the claimed Lipschitz selection. To implement each linear step, the scalar reduction solves $I-K$; a certificate (FC) gives the convergent Neumann series around $I-P_NK$ from Lemma Lemma (Exactness of the finite certificate).

The projected first equation has a spatially constant residual $r_u(t)$ for $u_{\rm r}+w$. Choose a scalar $c(t)$ with $c'(t)=r_u(t)$ and $c(T)=\int(h-u_{\rm r}(T))$. Then $u=u_{\rm r}+w+c$ solves the unprojected HJB equation and has exactly terminal value $h$. The population equation, initial condition, mass one, and positivity already hold. This proves actual nonemptiness for every boundary datum, and $W2$ follows by the reference phase $t+\theta$.

Conversely suppose item 1 holds and a nonzero zero-boundary Jacobi field $(z,\sigma)$ exists. For any fixed boundary direction $d\in D$, choose $\varepsilon>0$ small enough that $\varepsilon d$ lies in the open permitted boundary neighborhood. Its selected perturbation has norm at most $C\varepsilon\|d\|$ and interior linear residuals $-(N_1,N_2)$. Equation $G1$ yields \[ \varepsilon\left(\int d_T\sigma(T)+\int z(0)d_0\right) =\int_{Q_T}(N_1\sigma+zN_2)=O(\varepsilon^2). \] Divide by $\varepsilon$ and let it decrease to zero. Since the boundary direction was arbitrary, both mean-zero traces $z(0),\sigma(T)$ vanish. Lemma Lemma (Zero at both endpoints) contradicts nonzeroness. This proves necessity, including the negative classification at every conjugate pair. It rules out exactly $W1$, not every continuous selector or the original orbital property. ∎

On a compact set of horizon--phase pairs where the criterion holds, the above constants and the radius may be chosen uniformly. Rescale time to $[0,1]$; the coefficients and $L$ vary continuously in operator norm. The inverse Neumann estimate is local in those parameters, so compactness gives a uniform inverse bound. Reference positivity and the product estimates are also uniform there. This argument supplies no uniform bound on an unbounded set of horizons or across a singular pair.

An entire periodic family: every horizon is classified

The general criterion applies to explicit genuine periodic equilibria. Work on $\mathbb T^1$ and fix $0<r<1$, $c\ne0$, $\nu>0$. Put $k=2\pi$, $s=\sqrt{1-r^2}$ and \begin{equation}\tag{TW1}\label{wsref-0-54-57-31} \begin{aligned} M(y)&=1+r\cos(ky),\\ p(y)&=c\left(\frac{s}{M(y)}-1\right)-\nu\frac{M'(y)}{M(y)},\\ f(z)&=\frac{s^2(c^2+\nu^2k^2)}{2z^2},\qquad \lambda=\frac{c^2+\nu^2k^2}{2}. \end{aligned} \end{equation} Since $\int M^{-1}=s^{-1}$ and $\int M'/M=0$, $p$ has mean zero. Let $V'=p$ with $\int V=0$. The fields \begin{equation}\tag{TW2}\label{wsref-0-54-57-32} v_*(t,x)=V(x-ct),\qquad m_*(t,x)=M(x-ct) \end{equation} solve the original periodic MFG system, with period $P=1/|c|$. Indeed, setting $b_0=p+c$ gives $b_0M=cs-\nu M'$, so $-\nu M''-(b_0M)'=0$. Furthermore $M''=-k^2(M-1)$ and $(M')^2=k^2(r^2-(M-1)^2)$ give \[ \tfrac12b_0^2-\nu b_0' =\frac{s^2(c^2+\nu^2k^2)}{2M^2}-\frac{\nu^2k^2}{2}. \] Using $cp+p^2/2=(b_0^2-c^2)/2$ proves $\lambda+cp-\nu p'+p^2/2=f(M)$, the required HJB equation. The density is positive, has mass one, and is genuinely time-dependent. The integral formula for $M^{-1}$ follows, for example, by the substitution $z=\tan(k y/2)$ on the two half-circles.
Theorem (All horizons for the explicit family): For every triple $(r,c,\nu)$ above there is a set $\Sigma\subset(0,\infty)$, bounded away from zero and finite in every bounded interval, such that at every phase:
  • For $T\notin\Sigma$, selected local Lipschitz stability $W1$ holds, with the convergent selection (SEL).
  • For $T\in\Sigma$, no selection satisfying $W1$ for all small independent boundary perturbations exists.
The set is exactly the zero-kernel failure set in Theorem Theorem, so (FC) classifies its complement without an assumed mixed-boundary inverse. The constants can be uniform over phase and any compact subset of $(0,\infty)\setminus\Sigma$.
Proof: Use $y=x-c(t+\theta)$, $\tau=t/T$, and $q(y)=f'(M(y))$. The uncentered linear fields satisfy, on the fixed cylinder $[0,1]\times\mathbb T$, \begin{equation}\tag{TW3}\label{wsref-0-54-57-33} \begin{aligned} B_TW:=-W_\tau-T\nu W_{yy}+Tb_0W_y&=TqR, &W(1)&=0,\\ F_TR:=R_\tau-T\nu R_{yy}-T(b_0R)_y&=T(MW_y)_y, &R(0)&=0. \end{aligned} \end{equation} Its density has mean zero. Allowing the scalar value mean here changes neither its gradient nor the density Schur operator from $K1$; centering the scalar solution recovers exactly that operator.

Let $\mathcal H=L^2([0,1]\times\mathbb T)$, $\mathcal H_0=\{R:\int R(\tau)=0\}$, and $\mathfrak D=H^1_\tau L^2_y\cap L^2_\tau H^2_y$. Use the fixed closed domain $W(1)=0$ for $B_T$, and the fixed domain $R(0)=0$, $R\in\mathcal H_0$ for $F_T$. The scalar lemma makes both maps bounded isomorphisms onto their respective $L^2$ targets for every real $T>0$. Define \begin{equation}\tag{TW4}\label{wsref-0-54-57-34} \mathcal K_T=T^2F_T^{-1}\partial_y(M\partial_y)B_T^{-1}q :\mathcal H_0\longrightarrow\mathcal H_0. \end{equation} The derivative operator is bounded from $\mathfrak D$ to $\mathcal H_0$, and the outer forward inverse maps into $\mathfrak D$, whose embedding in $\mathcal H$ is compact by the same Fourier--cosine tail estimate as $K3$. Thus $\mathcal K_T$ is compact, including both time endpoints.

On these common domains the scalar operators are affine in $T$. Around any real $T_0>0$, their inverses are therefore holomorphic for sufficiently small complex $|T-T_0|$, by the inverse Neumann series. The union $\Omega$ of such disks is connected: it contains the entire positive real axis and every disk meets that axis. The inverse formulas agree on overlaps. Hence $\mathcal K_T$ is a holomorphic compact family on $\Omega$.

There is an invertible point, rather than an identically singular family. For $0<T\leq1$, the backward energy estimate for the first equation of (TW3) gives \[ \|W(\tau)\|_2\leq CT\int_\tau^1\|R(s)\|_2ds, \quad \|W\|_{L^2}\leq CT\|R\|_{L^2}, \quad \|W_y\|_{L^2}\leq C\sqrt T\|R\|_{L^2}. \] Here and below $C$ depends on the fixed coefficients, not on these small horizons. The first estimate follows by Gronwall applied to the spatial $L^2$ norm; testing the equation with $W$ then gives the gradient estimate. For $Z=\mathcal K_TR$, the forward equation, tested with $Z$, gives \[ (\|Z\|_2^2)' +T\nu\|Z_y\|_2^2 \leq CT\|Z\|_2^2+CT\|W_y\|_2^2. \] Gronwall yields $\|Z\|_{L^2}\leq C\sqrt T\|W_y\|_{L^2}\leq CT\|R\|_{L^2}$. Thus $\|\mathcal K_T\|\leq CT<1$ at all sufficiently small $T$.

At any $z_0\in\Omega$, choose a finite-rank orthogonal projection $P$ with $\|(I-P)\mathcal K_{z_0}\|<1/2$. With $Q=I-P$, this stays less than one near $z_0$, and the $Q$ equation is solved by a Neumann series. The remaining finite matrix is \begin{equation}\tag{TW5}\label{wsref-0-54-57-35} I-P\mathcal K_zP-P\mathcal K_zQ (I-Q\mathcal K_zQ)^{-1}Q\mathcal K_zP. \end{equation} It is holomorphic, and singularity of $I-\mathcal K_z$ is exactly vanishing of its determinant. In a small connected disk either the zeros are isolated or the determinant vanishes identically. The set of points with an identically singular neighborhood is open and also closed: near a limit point use (TW5); its determinant vanishes on an open subdisk and hence throughout the disk by the scalar identity theorem. Connectedness of $\Omega$ and the small-$T$ invertible point show that this open-and-closed set is empty. All singularities are therefore isolated in $\Omega$. Their intersection $\Sigma$ with the positive axis is locally finite; the small-horizon bound keeps it away from zero. The kernel bootstrap in Lemma Lemma (General Fredholm reduction) makes every $L^2$ kernel classical.

The moving coordinates give the same problem (TW3) at every phase. Changing phase changes only a constant spatial translation, an isometry of the boundary and perturbation norms. Finally Theorem Theorem gives both the positive and negative nonlinear conclusions, and its compact-parameter argument proves the final uniformity assertion. ∎

What has been classified

For every original periodic reference, dimension, coupling, phase and finite horizon, Theorem Theorem gives a necessary and sufficient criterion for $W1$, including its failure when a Jacobi kernel is present. The rule (SEL) gives actual equilibria for every small permitted boundary datum; positivity, mass and the terminal constant are included. Theorem Theorem (All horizons for the explicit family) applies this general result to an explicit genuinely periodic family.

The constants are local to the stated horizon set and the boundary topology is $C^{2+\alpha}$. No asymptotic attraction, uniform-in-horizon all-equilibrium stability, finite decision algorithm or computational complexity bound is asserted.

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, Proc. ICM 2022, vol. 5, pp. 3660--3703, EMS Press. DOI: 10.4171/ICM2022/17. Section 3.3, "Periodic solutions," printed p. 3681. https://ems.press/content/book-chapter-files/33265.
  • [CN] M. Cirant and L. Nurbekyan, The variational structure and time-periodic solutions for mean-field games systems, Minimax Theory Appl. 3$2$ $2018$, 227--260. Theorem 6.1, arXiv v1 pp. 24--25; Remark 6.4, p. 28. https://arxiv.org/pdf/1804.08943v1.

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