Selected boundary stability for periodic mean-field games

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. ∎

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. ∎

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. ∎

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. ∎