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Econ-33-A2 · revision r1 · on Econ-33

GPT 6 Astra Ultra attempt on Econ-33, version 2

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Contents
  1. Quasiperiodic bulk tori and stationary time averages in smooth mean field games
  2. Target and the role of the example
  3. Fixed smooth couplings encoding the phase
  4. A criterion that excludes this mechanism
  5. Quasiperiodic bulk tori: progress beyond Version 1
  6. What remains
  7. Completion Estimate
  8. References

Quasiperiodic bulk tori and stationary time averages in smooth mean field games

Problem C73-32. Status: unresolved. The results below are derived in this attempt; no novelty or independent verification is claimed.

Target and the role of the example

The question asks which bulk accumulation points of the finite-horizon boundary-value problem are stationary. It does not assert universal stationary convergence. We construct fixed smooth data on ${\mathbb{T}}={\mathbb{R}}/{\mathbb{Z}}$ for which a whole rotating orbit belongs to $\Omega_{\mathrm{bulk}}$, and prove a positive quantitative distance from every stationary pair. The long-time nonmonotone agenda is described in [CD, Section 3.3]. Version 2 retains that example and constructs higher-dimensional quasiperiodic bulk tori, characterizes the bulk set of an explicit equilibrium selection, and proves that its time averages can be stationary while every instantaneous point remains separated from all stationary pairs. The constructions are self-contained; they do not classify all equilibria for their data.

Use normalized Lebesgue measure on ${\mathbb{T}}$. The equations are \begin{equation}\label{wsref-0-4d-46-47} -u_t-u_{xx}+\tfrac12u_x^2=F(x,m_t),\qquad m_t-m_{xx}-(mu_x)_x=0,\qquad m(0)=m_0,\quad u(T,x)=G(x,m_T). \end{equation} Fix $0<r<1$ and $c\ne0$, and set $k=2\pi$, $q=\sqrt{1-r^2}$ and \begin{equation}\label{wsref-0-70-72-6f-66-69-6c-65-73} M(z)=1+r\cos(kz),\qquad P(z)=-\frac{M'(z)}{M(z)}-c+\frac{cq}{M(z)}. \end{equation} Both are smooth one-periodic functions and $M>0$.

Lemma: The function $P$ has mean zero. There is a unique smooth one-periodic $U$ with $U'=P$ and $\int U=0$. Furthermore, \[ MP=-M'-cM+cq,\qquad M''+(MP)'=-cM'. \]
Proof: The logarithmic derivative $M'/M$ has zero mean. The elementary integral \[ \int_0^1(1+r\cos(2\pi z))^{-1}{\mathrm d} z=(1-r^2)^{-1/2}=q^{-1} \] follows, for example, by using $y=\tan(\theta/2)$ on $\theta\in(-\pi,\pi)$: \[ \frac1{2\pi}\int_{-\pi}^{\pi}\frac{{\mathrm d}\theta}{1+r\cos\theta} =\frac1\pi\int_{-\infty}^{\infty}\frac{{\mathrm d} y}{(1+r)+(1-r)y^2}=q^{-1}. \] Consequently $\int P=-c+cq/q=0$, giving the periodic primitive and its unique normalization. Multiplication by $M$ and differentiation prove the two identities. ∎

Fixed smooth couplings encoding the phase

For any probability measure $m$ on ${\mathbb{T}}$ define the complex coordinate \[ Z(m)=\frac2r\int_{\mathbb{T}} e^{iky}m({\mathrm d} y). \] Let $\chi\in C_c^\infty((1/4,4))$ satisfy $\chi=1$ in a neighborhood of $1$. For a smooth one-periodic real function $L$ define \[ \mathcal R_L(x,z)= \begin{cases} \chi(|z|^2)L(x-\arg(z)/k),&z\ne0,\\ 0,&z=0. \end{cases} \] This definition is independent of the choice of $\arg(z)$, since $L$ is one-periodic. It is a smooth real-valued function on ${\mathbb{T}}\times\mathbb C$: on any sector away from zero choose a smooth angle, and near zero the cutoff makes it identically zero. Its derivatives of every order are bounded because its $z$ support lies in a compact annulus.

Put \begin{equation}\label{wsref-0-63-6f-75-70-6c-69-6e-67-73} \begin{split} K(z)&=cP(z)-P'(z)+\tfrac12P(z)^2,\qquad m_0(x)=M(x),\\ F(x,m)&=\mathcal R_K(x,Z(m)),\qquad G(x,m)=\mathcal R_U(x,Z(m)). \end{split} \end{equation} The map $Z$ is linear in $m$. The chain rule therefore gives continuous flat functional derivatives of every order for $F$ and $G$, with bounded spatial derivatives of every order. For instance, the first variation is the derivative of $\mathcal R_L$ in its two real $z$ coordinates paired with $(2/r)(\cos(ky),\sin(ky))$; all higher variations follow by repeated differentiation. Thus these are fixed smooth couplings on the whole probability-measure space, including atomic measures. No time or horizon parameter occurs in them.

Theorem (A rotating solution at every horizon): For the data (couplings), every $T>0$ admits the positive classical solution \[ m^T(t,x)=M(x-ct),\qquad u^T(t,x)=U(x-ct),\qquad 0\le t\le T. \] For every $\theta\in{\mathbb{T}}$, the pair $(P(\cdot-\theta),M(\cdot-\theta))$ belongs to $\Omega_{\mathrm{bulk}}(F,G,m_0)$, with the precise topology in the question. None of these pairs belongs to $\mathcal E_F$.
Proof: For $m_\theta(x)=M(x-\theta)$, elementary Fourier integration gives $Z(m_\theta)=e^{ik\theta}$. Hence $F(x,m_\theta)=K(x-\theta)$ and $G(x,m_\theta)=U(x-\theta)$. For the proposed solution, $u_x=P(x-ct)$ and $-u_t=cP(x-ct)$, so its Hamilton--Jacobi equation is exactly the definition of $K$. The Fokker--Planck equation follows from Lemma Lemma. The initial and terminal conditions follow from $m_0=M$ and the formula for $G$. Smoothness, positivity and unit mass are immediate from $0<r<1$.

Choose $s_n\to\infty$ with $cs_n=\theta$ modulo $1$, and put $T_n=2s_n$. For every bounded time interval, the translated solution equals $(P(x-\theta-ct),M(x-\theta-ct))$ once that interval lies in $[-s_n,s_n]$. Thus it converges in the required $C_{\mathrm{loc}}({\mathbb{R}};C^0\times L^1)$ topology, proving bulk membership. If a pair on the orbit were stationary, its stationary Fokker--Planck equation would require $M''+(MP)'=0$. By Lemma Lemma, this quantity equals $-cM'$, which is not identically zero because $c\ne0$ and $r>0$. ∎

Theorem (Uniform separation from stationary pairs): Set \[ D_*=\max\{k,\ k^2+k\|P\|_\infty\},\qquad \delta_*=\frac{|c|kr}{2D_*}>0. \] For every $\theta\in{\mathbb{T}}$ and every stationary pair $(v,n)\in\mathcal E_F$, \[ \|P(\cdot-\theta)-v\|_\infty+\|M(\cdot-\theta)-n\|_1\ge\delta_*. \] In particular, the quantitative all-equilibrium stationary convergence requested in the question fails for these data at every interior time of the solutions in Theorem Theorem (A rotating solution at every horizon).
Proof: Write $M_\theta=M(\cdot-\theta)$ and $P_\theta=P(\cdot-\theta)$, and choose $\psi(x)=\sin(k(x-\theta))$. For any stationary pair, integrating $n''+(nv)'=0$ against $\psi$ yields $\int n\psi''-nv\psi'=0$. The identity for the rotating pair gives \[ \int M_\theta\psi''-M_\theta P_\theta\psi' =-c\int M_\theta'\psi=c\int M_\theta\psi'=\frac{ckr}{2}. \] Subtract the two relations. Since $\int n=1$, \begin{align*} \frac{|c|kr}{2} &\le k^2\|M_\theta-n\|_1+k\|M_\theta P_\theta-nv\|_1\\ &\le(k^2+k\|P\|_\infty)\|M_\theta-n\|_1+k\|P_\theta-v\|_\infty. \end{align*} This is bounded above by $D_*$ times the asserted metric, proving the estimate. If $\mathcal E_F$ is empty, distance to it is $+\infty$ and the conclusion remains valid. Otherwise take the infimum over it. The bound is independent of $t,T$, and hence applies to the suprema over bulk times. ∎

A criterion that excludes this mechanism

For clarity, a simple necessary condition for rotating behavior can also be proved without a literature theorem.
Proposition: Suppose $F$ is Lasry--Lions monotone. If $(u_1,m_1)$ and $(u_2,m_2)$ are smooth positive time-periodic solutions of (MFG) on the whole real time axis with the same period, then $D u_1=D u_2$. In particular, a genuinely rotating solution with nonconstant gradient cannot coexist with all its time translates.
Proof: Let $I(t)=\int(u_1-u_2)(m_1-m_2)$. Differentiating, substituting the two PDEs and integrating by parts gives \[ I'(t)=-\int[F(x,m_1)-F(x,m_2)](m_1-m_2) -\frac12\int(m_1+m_2)|Du_1-Du_2|^2. \] For completeness, the diffusion terms cancel pairwise, and the gradient terms equal $\tfrac12(|Du_1|^2-|Du_2|^2)(m_1-m_2) -(Du_1-Du_2)\cdot(m_1Du_1-m_2Du_2)$, which expands to the displayed negative square. Both terms are nonpositive under monotonicity. Integration over a period has zero left side, so positivity of $m_1+m_2$ and continuity force $Du_1=Du_2$ everywhere. A nonconstant periodic $P$ differs from at least one spatial translate, and time translation of an autonomous rotating solution gives that translate. Thus the $F$ constructed above must be nonmonotone. The formula for $P$ shows it is nonconstant: if it were constant, its zero mean would make it zero, implying $M'+cM=cq$, whose only periodic solution is constant, contrary to $r>0$. ∎

Quasiperiodic bulk tori: progress beyond Version 1

Version 1 constructed a single periodic rotating orbit. We now realize higher-dimensional quasiperiodic bulk sets with fixed smooth couplings and boundary data. We characterize the complete bulk set of the explicit selected family, and show that its time averages converge to a stationary pair even though every instantaneous point stays uniformly separated from all stationary pairs. No claim about other equilibria for the same data is hidden in the word "selected."

Fix $d\geq2$, amplitudes $r_j\in(0,1)$ and a speed vector $c=(c_1,\ldots,c_d)\in{\mathbb{R}}^d$ satisfying \begin{equation}\label{wsref-0-72-61-74-69-6f-6e-61-6c-69-6e-64-65-70-65-6e-64-65-6e-74} n\cdot c\ne0\quad\hbox{for every }n\in{\mathbb{Z}}^d\setminus\{0\}. \end{equation} For example $(1,\sqrt2)$ works when $d=2$. Let $k=2\pi$ and put \[ M_j(z)=1+r_j\cos(kz),\quad q_j=\sqrt{1-r_j^2},\quad P_j(z)=-\frac{M_j'(z)}{M_j(z)}-c_j+\frac{c_jq_j}{M_j(z)}. \] Let $U_j'=P_j$ with $\int_{\mathbb{T}} U_j=0$, as in Lemma Lemma, and set $K_j=c_jP_j-P_j'+P_j^2/2$. For $m\in\mathcal P({\mathbb{T}}^d)$ define \[ Z_j(m)=\frac2{r_j}\int_{{\mathbb{T}}^d}e^{iky_j}m(dy). \] Use the same smooth annular phase map $\mathcal R_L$ defined above, with its fixed cutoff $\chi$. Define once and for all \begin{equation}\label{wsref-0-70-72-6f-64-75-63-74-63-6f-75-70-6c-69-6e-67-73} F(x,m)=\sum_{j=1}^d\mathcal R_{K_j}(x_j,Z_j(m)),\qquad G(x,m)=\sum_{j=1}^d\mathcal R_{U_j}(x_j,Z_j(m)),\qquad m_0(x)=\prod_{j=1}^dM_j(x_j). \end{equation} These are smooth globally defined measure couplings. Indeed every $Z_j$ is a bounded linear functional of the probability measure, and the cutoff phase map is smooth with bounded derivatives on its annular support. Repeated chain differentiation supplies every flat functional derivative, with bounded spatial derivatives in all its arguments, just as in the one-dimensional construction. No time or horizon parameter occurs in (productcouplings).

Lemma (Uniform averaging of an irrational torus translation): Under (rationalindependent), for every continuous $H:{\mathbb{T}}^d\to{\mathbb{R}}$, \[ \lim_{L\to\infty}\sup_{\theta\in{\mathbb{T}}^d} \left|\frac1L\int_0^L H(\theta+ct)dt-\int_{{\mathbb{T}}^d}H\right|=0. \] For every $\theta\in{\mathbb{T}}^d$ there is a sequence $s_n\to\infty$ such that $cs_n\to\theta$ modulo ${\mathbb{Z}}^d$.
Proof: For the character $H(\theta)=e^{2\pi i n\cdot\theta}$ with $n\ne0$, its time average equals \[ e^{2\pi i n\cdot\theta} \frac{e^{2\pi i(n\cdot c)L}-1}{2\pi i(n\cdot c)L}, \] whose absolute value tends to zero uniformly in $\theta$ by (rationalindependent). The constant character has average one. Linearity proves the assertion for trigonometric polynomials. Such polynomials approximate every continuous periodic function uniformly: for example, convolution with the product Fejér kernels has finite Fourier support, integral one, is nonnegative, and puts vanishing mass outside each neighborhood of zero. Uniform continuity of $H$ then bounds the convolution error by its local oscillation plus its bounded supremum times that vanishing mass. This proves the claimed uniform approximation and hence the averaging assertion.

For any nonempty open neighborhood $V$ of a prescribed $\theta$, choose a continuous nonnegative function supported in $V$ with positive integral, for example a sufficiently small metric bump around $\theta$. The averaging assertion, applied after any starting time $R$, implies that $c(R+t)$ enters $V$ at some $t\geq0$. Otherwise that positive limiting average would be zero. Taking neighborhoods of radius $1/n$ and starting times $R=n$ gives the required sequence. ∎

For $\theta\in{\mathbb{T}}^d$ define a density and vector field \[ m_\theta(x)=\prod_{j=1}^dM_j(x_j-\theta_j),\qquad p_\theta(x)=(P_1(x_1-\theta_1),\ldots,P_d(x_d-\theta_d)), \] and set $\mathcal K=\{(p_\theta,m_\theta):\theta\in{\mathbb{T}}^d\}$. For the next statement, $\Omega_{\rm sel}$ means the catalogue's bulk construction restricted to the explicitly specified solution at each horizon, rather than allowing every member of $\mathcal S_T$.

Theorem (A quasiperiodic torus of bulk accumulation points): For the data (productcouplings), every horizon $T>0$ admits the smooth positive solution \begin{equation}\label{wsref-0-70-72-6f-64-75-63-74-73-70-65-63-69-66-69-63} m^T(t,x)=\prod_{j=1}^dM_j(x_j-c_jt),\qquad u^T(t,x)=\sum_{j=1}^dU_j(x_j-c_jt). \end{equation} Its selected-solution bulk set satisfies \[ \Omega_{\rm sel}=\mathcal K \subseteq\Omega_{\mathrm{bulk}}(F,G,m_0). \] The map $\theta\mapsto(p_\theta,m_\theta)$ is an embedding of ${\mathbb{T}}^d$ into $C^0({\mathbb{T}}^d;{\mathbb{R}}^d)\times L^1({\mathbb{T}}^d)$, and the entire trajectory through each such point is nonperiodic in time. All these points have a common positive lower bound on their distance from $\mathcal E_F$.
Proof: The $j$th marginal of $m_\theta$ is $M_j(\cdot-\theta_j)$, so $Z_j(m_\theta)=e^{ik\theta_j}$. Thus along the proposed trajectory $F(x,m_t)=\sum_jK_j(x_j-c_jt)$ and $G(x,m_t)=\sum_jU_j(x_j-c_jt)$. Since the value is additive in the coordinates, its Laplacian and the square of its gradient also separate as sums. The HJB equation reduces in each summand to $c_jP_j-P_j'+P_j^2/2=K_j$. For the density equation, differentiate the product: \[ \partial_t m=\sum_j(-c_j M_j')\prod_{\ell\ne j}M_\ell, \quad \Delta m+\operatorname{div}(mDu) =\sum_j\{M_j''+(M_jP_j)'\}\prod_{\ell\ne j}M_\ell. \] The one-dimensional profile identity makes these equal. The initial density is $m_0$, and the formula for $G$ verifies the terminal condition for every $T$. Each factor is positive and has mass one, proving the remaining solution requirements.

Fix $\theta$. By Lemma Lemma (Uniform averaging of an irrational torus translation), choose $s_n\to\infty$ with $cs_n\to\theta$ on the torus, and take $T_n=2s_n$. On every fixed bounded time interval, the shifted fields converge, together with all fixed spatial derivatives, to $(p_{\theta+ct},m_{\theta+ct})$. In particular they converge in the precise $C_{\rm loc}({\mathbb{R}};C^0\times L^1)$ topology of the catalogue. This proves $\mathcal K\subseteq\Omega_{\rm sel}$. Conversely, the phases $cs_n$ of any sequence of the selected solutions have a convergent subsequence by compactness of ${\mathbb{T}}^d$; along it the same formulas give a member of $\mathcal K$ as the time-zero limit. Thus every selected bulk limit lies in $\mathcal K$.

The phase map is continuous. It is injective because its density recovers every phase from $Z_j(m_\theta)=e^{ik\theta_j}$. A continuous injection from the compact torus into the Hausdorff target space is a homeomorphism onto its image, proving the embedding claim. If its trajectory had a period $P>0$, these phase coordinates would give $c_jP\in{\mathbb{Z}}$ for every $j$. Condition (rationalindependent) forces every $c_j\ne0$ and excludes this: for two coordinates, the nonzero integer vector with entries $c_2P,-c_1P$ in those positions would have dot product zero with $c$. Thus the trajectory is nonperiodic.

For explicit separation, fix any coordinate $j$ and put \[ D_j=\max\{k,\,k^2+k\|P_j\|_\infty\},\qquad \delta_j=\frac{|c_j|kr_j}{2D_j}>0. \] For a stationary pair $(v,n)\in\mathcal E_F$, use $\psi(x)=\sin(k(x_j-\theta_j))$ as a test function in its stationary Fokker--Planck equation. This gives $\int n\Delta\psi-nv\cdot D\psi=0$. For the product density and gradient, the corresponding expression is $c_jkr_j/2$, because all other coordinates integrate to one and the one-dimensional calculation from Theorem Theorem (Uniform separation from stationary pairs) applies. Subtracting gives \[ \frac{|c_j|kr_j}{2} \leq(k^2+k\|P_j\|_\infty)\|m_\theta-n\|_1 +k\|p_\theta-v\|_\infty. \] Here the vector supremum norm controls its $j$th component, and $\int n=1$. Hence the catalogue distance is at least $\delta_j$, uniformly in $\theta$ and in the stationary pair. If there are no stationary pairs, the distance is infinite; the estimate is still true. The next corollary actually supplies a stationary pair for these data. ∎

Corollary (Stationary time averages with nonstationary bulk): For the selected solutions (productspecific), viewed as the same trajectory on the whole real line, \[ \frac1L\int_0^L Du(t,\cdot)dt\longrightarrow0,\qquad \frac1L\int_0^L m(t,\cdot)dt\longrightarrow1 \quad\text{uniformly in the spatial variable as }L\to\infty. \] The pair $(0,1)$ belongs to $\mathcal E_F$, whereas every point of $\mathcal K$ stays at least $\delta_j>0$ away from $\mathcal E_F$.
Proof: Apply the uniform part of Lemma Lemma (Uniform averaging of an irrational torus translation) to the continuous functions $\prod_jM_j(\theta_j)$ and $P_j(\theta_j)$, reversing the velocity sign if needed. The spatial point is just a starting phase, so the convergence is uniform in that point. The phase integrals are $\prod_j\int M_j=1$ and $\int P_j=0$. For the uniform density, every $Z_j(1)=0$, and the cutoff definition gives $F(x,1)=0$. Thus $\bar u=0$, $\bar m=1$, $\lambda=0$ satisfy the stationary equations and the normalizations. The separation claim is Theorem Theorem (A quasiperiodic torus of bulk accumulation points). This proves directly that stationary Cesàro averages cannot replace convergence of the actual bulk trajectories in the target. ∎

What remains

The examples prove that fixed globally smooth couplings and positive classical solutions at every horizon can produce periodic and genuinely quasiperiodic bulk behavior. Version 2 identifies an entire torus of bulk limits for an explicit equilibrium selection and proves that stationary time averages coexist with a positive uniform distance of every orbit point from all stationary pairs. It does not identify all solutions for these data, all of their bulk limits, or data classes that have stationary attraction. The terminal coupling deliberately matches the orbit; robustness under perturbing it is not proved. No attraction, forward semiflow, computational experiment or new literature verification is presumed by the explicit boundary-value construction.

Completion Estimate

40%

This is a heuristic assessment of the full catalogue target. The version adds a precise class of nonperiodic bulk tori and a distinction between stationary time averages and stationary bulk limits. It characterizes only its explicit equilibrium selection, leaving all-equilibrium classification, robustness and positive stationary-attraction criteria unresolved.

References

  • [CD] P. Cardaliaguet and F. Delarue, Selected topics in mean field games, Proc. ICM 2022, vol. 5, 3660--3703, EMS Press (2023), equations (1.4), (3.1), and Section 3.3. https://doi.org/10.4171/ICM2022/17.