Proof of The Ising Population Model Instantiates the Data of the Fluctuation Theory

Throughout, $\phi$ denotes the regularised entropic rate cost with cut-offs $\underline{a}$ and $\bar{a}$ fixed in clause The Ising Population Data §cost, and we use freely its properties from claims The Regularised Entropic Rate Cost §profile, The Regularised Entropic Rate Cost §cost-function and The Regularised Entropic Rate Cost §sign-bounds of that lemma: $\phi$ is of class $C^{2}$ on $\mathbb{R}$ with $\phi''=\varpi$; $\phi\ge0$ with $\phi(u)=0$ only for $u=1$; $\phi$ is convex; $\phi(1)=\phi'(1)=0$; $|\phi'|\le2\bar{a}\,\underline{a}^{-1}$ and $0\le\phi''\le\underline{a}^{-1}$ everywhere; $\phi'$ is Lipschitz with constant $\underline{a}^{-1}$ and $\phi''$ with constant $2\,\underline{a}^{-2}$.

Claim 1. $\mathcal{A}$ contains $(1,1)$, since $\underline{a}<1<\bar{a}$, and is therefore nonempty. It is the closed box $[\underline{a},\bar{a}]\times[\underline{a},\bar{a}]$, hence compact by Closed Box in $\mathbb{R}^n$ is Compact. If $a,a'\in\mathcal{A}$ and $\lambda\in[0,1]$ then for each $j$ we have $\underline{a}=\lambda\underline{a}+(1-\lambda)\underline{a}\le\lambda a^{j}+(1-\lambda)a'^{j}\le\lambda\bar{a}+(1-\lambda)\bar{a}=\bar{a}$, so $\mathcal{A}$ is convex. Compactness and convexity are exactly hypothesis (A) of Quadratic Growth of the Mean-Field Hamiltonian in the Control along a Stationary Mean-Field Triple.

For $a\in\mathcal{A}$ we have $|a|^{2}=(a^{1})^{2}+(a^{2})^{2}\le2\bar{a}^{2}$, with equality at the point $(\bar{a},\bar{a})\in\mathcal{A}$; so $\sup_{a\in\mathcal{A}}|a|=\sqrt{2}\,\bar{a}$. Since $\underline{a}<1<\bar{a}$, the number $\varrho_{\mathcal{A}}$ is positive. If $|a-(1,1)|\le\varrho_{\mathcal{A}}$ then $|a^{j}-1|\le|a-(1,1)|\le\varrho_{\mathcal{A}}$ for each $j$, so $a^{j}\ge1-(1-\underline{a})=\underline{a}$ and $a^{j}\le1+(\bar{a}-1)=\bar{a}$, that is, $a\in\mathcal{A}$.

Claim 2. We check the two clauses of the definition of an affine-controlled transition-rate family for the two ordered pairs $(1,2)$ and $(2,1)$. Nonnegativity: for $\Sigma\in\Delta^{2}$ and $\alpha\in\mathcal{A}$, $\beta_{0}(1,2,\Sigma)+\beta_{1}(1,2,\Sigma)\cdot\alpha=\alpha^{1}\ge\underline{a}>0$ and likewise $\beta_{0}(2,1,\Sigma)+\beta_{1}(2,1,\Sigma)\cdot\alpha=\alpha^{2}\ge\underline{a}>0$. Lipschitz dependence: all four maps $\beta_{0}(\sigma,\gamma,\cdot)$, $\beta_{1}(\sigma,\gamma,\cdot)$ are constant on $\Delta^{2}$, hence Lipschitz with constant $0$. So $(\beta_{0},\beta_{1})$ is an affine-controlled transition-rate family with control set $\mathcal{A}$ (nonempty, convex and compact by claim 1) and $\Lambda=0$.

The family associated with it in The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data is $\beta(\sigma,\gamma,\Sigma,\alpha)=\beta_{0}(\sigma,\gamma,\Sigma)+\beta_{1}(\sigma,\gamma,\Sigma)\cdot\alpha$, which equals $\alpha^{1}$ for $(\sigma,\gamma)=(1,2)$ and $\alpha^{2}$ for $(\sigma,\gamma)=(2,1)$; this is the family displayed in clause The Ising Population Data §rates. Its constant $B=\sup\beta$ of that lemma equals $\sup_{\alpha\in\mathcal{A}}\max\{\alpha^{1},\alpha^{2}\}=\bar{a}$, and claim 2 of that lemma states that $\beta$ is a transition-rate family on $2$ states with control set $\mathcal{A}$ and rate bound $B$.

By claim 3 of the same lemma the aggregate state drift is $b^{\gamma}(\Sigma,\alpha)=b^{\gamma}_{0}(\Sigma)+b^{\gamma}_{1}(\Sigma)\cdot\alpha$ with $b^{\gamma}_{0}$ and $b^{\gamma}_{1}$ as defined there. Since $\beta_{0}$ vanishes identically, $b^{\gamma}_{0}=0$. For $\gamma=1$ the only $\sigma\neq\gamma$ is $\sigma=2$, so $b^{1}_{1}(\Sigma)=\Sigma^{2}\beta_{1}(2,1,\Sigma)-\Sigma^{1}\beta_{1}(1,2,\Sigma)=\Sigma^{2}(0,1)-\Sigma^{1}(1,0)=(-\Sigma^{1},\Sigma^{2})$ and $b^{1}(\Sigma,\alpha)=\Sigma^{2}\alpha^{2}-\Sigma^{1}\alpha^{1}$. Symmetrically $b^{2}(\Sigma,\alpha)=\Sigma^{1}\alpha^{1}-\Sigma^{2}\alpha^{2}=-b^{1}(\Sigma,\alpha)$. Since $v=(1,-1)^{\top}$, this is the displayed formula.

Claim 3. With $l=2$ the definition of the aggregate fluctuation covariance gives, for $\Sigma\in\Delta^{2}$ and $\alpha\in\mathcal{A}$,

and the same value for $\Theta^{22}$ by exchanging the roles of the indices, while

The matrix with entries $1,-1,-1,1$ in this order is $vv^{\top}$, which gives the displayed formula.

Claim 4. The set $U$ is the open Euclidean ball of radius $2$ about the origin: it is open, it is bounded, and it is convex because $|\lambda x+(1-\lambda)y|\le\lambda|x|+(1-\lambda)|y|<2$ for $x,y\in U$ and $\lambda\in[0,1]$, by the triangle inequality and the homogeneity of the Euclidean norm. It contains $\Delta^{2}$, since $|x|\le x^{1}+x^{2}=1<2$ for $x\in\Delta^{2}$. The set $V$ is an open box, hence open, bounded, and convex by the argument of claim 1; and $\mathcal{A}\subseteq V$ because $\tfrac{1}{2}\underline{a}<\underline{a}\le a^{j}\le\bar{a}<\bar{a}+1$ for $a\in\mathcal{A}$. The product $U\times V$ is open in $\mathbb{R}^{4}$ by Products of Euclidean Open Sets are Open.

The function $\bar{\beta}(1,2,\cdot,\cdot)$ is the restriction to $U\times V$ of the third coordinate function of $\mathbb{R}^{4}$, and $\bar{\beta}(2,1,\cdot,\cdot)$ the restriction of the fourth; by Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map coordinate functions are of class $C^{2}$, with $\partial_{i}$ of the $k$-th coordinate function equal to $1$ if $i=k$ and to $0$ otherwise, and with all second-order partial derivatives equal to $0$. This gives the stated derivatives, the derivative bound $K=1$, and the uniform continuity of the second-order derivatives required by Twice Continuously Differentiable Extension of a Transition-Rate Family, since those derivatives are constant and any $\delta>0$ serves. The extension requirement holds because $\bar{\beta}(1,2,\Sigma,\alpha)=\alpha^{1}=\beta(1,2,\Sigma,\alpha)$ and $\bar{\beta}(2,1,\Sigma,\alpha)=\alpha^{2}=\beta(2,1,\Sigma,\alpha)$ for $(\Sigma,\alpha)\in\Delta^{2}\times\mathcal{A}$.

The extended aggregate state drift is $\bar{b}^{1}(x,a)=x^{2}\bar{\beta}(2,1,x,a)-x^{1}\bar{\beta}(1,2,x,a)=x^{2}a^{2}-x^{1}a^{1}$ and $\bar{b}^{2}(x,a)=-\bar{b}^{1}(x,a)$, which is the displayed formula.

Claim 5. For each pair $(\sigma,\upsilon)$ the map $\tilde{\beta}(\sigma,\upsilon,\cdot)$ is constant with value $q\,\mathbf{1}_{\{\sigma=\upsilon\}}+q_{0}$, which lies in $[q_{0},q+q_{0}]\subseteq[0,q+q_{0}]$; so the bounds clause holds with $\tilde{B}=q+q_{0}$, and the continuity clause holds because constant maps are continuous. For $\Sigma\in\Delta^{2}$,

using $\Sigma^{1}+\Sigma^{2}=1$. Since $\Sigma^{\upsilon}\ge0$ and $q>0$, this is at least $q_{0}>0$, which is hypothesis (OC) with $\underline{b}=q_{0}$.

Claim 6. $\tilde{U}=U$ is open and contains $\Delta^{2}$ by claim 4. Each $\bar{\tilde{\beta}}(\sigma,\upsilon,\cdot)$ is constant on $\tilde{U}$, hence of class $C^{2}$ with all partial derivatives of orders one and two equal to $0$, by Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map; the derivative bound $\tilde{K}=0$ and the uniform continuity of the second-order derivatives (with any $\delta^{\circ}>0$) follow, and the extension requirement holds because the two families have the same constant values. The extended aggregate observation drift is $\bar{\tilde{b}}^{\upsilon}(x)=\sum_{\sigma=1}^{2}x^{\sigma}(q\,\mathbf{1}_{\{\sigma=\upsilon\}}+q_{0})=q\,x^{\upsilon}+q_{0}(x^{1}+x^{2})$ for $x\in\tilde{U}$; note that the sum $x^{1}+x^{2}$ is not $1$ off $\Delta^{2}$.

Claim 7. $L$ maps $\Delta^{2}\times\mathbb{R}^{2}$ into $\mathbb{R}$ and $G$ maps $\Delta^{2}$ into $\mathbb{R}$. Continuity: if $(\Sigma_{n},\alpha_{n})\to(\Sigma,\alpha)$ in the sense of the definition, then each coordinate converges, so $\phi(\alpha^{j}_{n})\to\phi(\alpha^{j})$ by continuity of $\phi$ and the sequential characterisation of continuity, and $L(\Sigma_{n},\alpha_{n})\to L(\Sigma,\alpha)$ by the arithmetic of limits of real sequences of Arithmetic of Limits of Real Sequences; $G$ is constant. Lower bounds: for $(x,a)\in\Delta^{2}\times\mathbb{R}^{2}$ we have $x^{1},x^{2}\ge0$, $\phi\ge0$ and $(x^{2}-x^{1})^{2}\ge0$, so $L(x,a)\ge0$, and $G=0$; hence $C_{L}=C_{G}=0$ serve.

If $L(x,a)=0$ then, all three summands being nonnegative, each vanishes. From $\tfrac{1}{2}\psi(x^{2}-x^{1})^{2}=0$ and $\psi>0$ we get $x^{1}=x^{2}$, hence $x^{1}=x^{2}=\tfrac{1}{2}$ because $x^{1}+x^{2}=1$. Then $\chi^{-1}x^{j}\phi(a^{j})=0$ with $x^{j}=\tfrac{1}{2}>0$ forces $\phi(a^{j})=0$, hence $a^{j}=1$ for $j=1,2$ by The Regularised Entropic Rate Cost §sign-bounds. Conversely $L\bigl((\tfrac{1}{2},\tfrac{1}{2}),(1,1)\bigr)=0$ because $\phi(1)=0$ and $x^{2}-x^{1}=0$.

Convexity in the control: fix $x\in\Delta^{2}$, let $a,a'\in\mathbb{R}^{2}$ and $\lambda\in[0,1]$. For each $j$, convexity of $\phi$ gives $\phi(\lambda a^{j}+(1-\lambda)a'^{j})\le\lambda\phi(a^{j})+(1-\lambda)\phi(a'^{j})$; multiplying by $\chi^{-1}x^{j}\ge0$, summing over $j$ and adding the term $\tfrac{1}{2}\psi(x^{2}-x^{1})^{2}$, which does not depend on the control, gives $L(x,\lambda a+(1-\lambda)a')\le\lambda L(x,a)+(1-\lambda)L(x,a')$.

Hypothesis (LipC): let $x,x'\in\Delta^{2}$ and $a\in\mathcal{A}$. For $j\in\{1,2\}$ we have $\underline{a}\le a^{j}\le\bar{a}$, so $|a^{j}-1|\le\max\{1-\underline{a},\bar{a}-1\}\le\bar{a}$; since $\phi(1)=0$ and $|\phi'|\le2\bar{a}\,\underline{a}^{-1}$, the mean value estimate obtained from $\phi(u)-\phi(1)=\int_{[1,u]}\phi'$ (respectively $-\int_{[u,1]}\phi'$) and monotonicity of the integral gives $0\le\phi(a^{j})\le2\bar{a}\,\underline{a}^{-1}|a^{j}-1|\le2\bar{a}^{2}\underline{a}^{-1}$. Hence

using $|u^{1}|+|u^{2}|\le\sqrt{2}\,|u|$ from claim 1 of Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals. For the interaction term, $|x^{2}-x^{1}|\le1$ and $|x'^{2}-x'^{1}|\le1$ on $\Delta^{2}$, so

Adding the two estimates gives $|L(x,a)-L(x',a)|\le K_{L}|x-x'|$ with the stated $K_{L}$, and $|G(x)-G(x')|=0$ gives $K_{G}=0$.

Claim 8. $U_{c}=U$ is open and contains $\Delta^{2}$ by claim 4, and $U_{c}\times\mathbb{R}^{2}$ is open in $\mathbb{R}^{4}$ by Products of Euclidean Open Sets are Open. Extension: for $(x,a)\in\Delta^{2}\times\mathbb{R}^{2}$ we have $x^{1}+x^{2}-1=0$, so the term weighted by $\mu$ vanishes and $\bar{L}(x,a)=L(x,a)$; and $\bar{G}=0=G$ on $\Delta^{2}$.

Regularity: the partial derivative $\partial_{i}$ at a point is the derivative of the slice function obtained by letting the $i$-th coordinate vary and freezing the others. For $i=1$ the slice of $\bar{L}$ is $u\mapsto\chi^{-1}\bigl(u\,\phi(a^{1})+x^{2}\phi(a^{2})\bigr)+\tfrac{1}{2}\psi(x^{2}-u)^{2}+\mu(u+x^{2}-1)^{2}$, a polynomial in $u$ whose derivative, by Derivative of a Polynomial Function on the Real Line and Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, is $\chi^{-1}\phi(a^{1})-\psi(x^{2}-u)+2\mu(u+x^{2}-1)$; evaluating at $u=x^{1}$ gives the stated $\partial_{1}\bar{L}$, and the case $i=2$ is symmetric. For $i=3$ the slice is $u\mapsto\chi^{-1}x^{1}\phi(u)+c$ with $c$ independent of $u$, whose derivative is $\chi^{-1}x^{1}\phi'(u)$ by the constant-multiple and sum rules of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives; evaluating at $u=a^{1}$ gives the stated $\partial_{3}\bar{L}$, and the case $i=4$ is symmetric. Iterating the same computation on each of these four functions gives the ten displayed second-order partial derivatives. Every one of them is continuous on $U_{c}\times\mathbb{R}^{2}$: the first four are constants, and the others are products of a coordinate function with $\phi'$ or $\phi''$ evaluated at a coordinate, all continuous. Hence $\bar{L}$ is of class $C^{2}$; $\bar{G}$ is constant, hence of class $C^{2}$ with vanishing derivatives.

Second-derivative bounds: on $U_{c}$ we have $|x^{j}|\le|x|<2$. Therefore $|\partial_{1}\partial_{1}\bar{L}|=|\partial_{2}\partial_{2}\bar{L}|=\psi+2\mu$, $|\partial_{1}\partial_{2}\bar{L}|=|2\mu-\psi|\le\psi+2\mu$, $|\partial_{3}\partial_{1}\bar{L}|\le\chi^{-1}\cdot2\bar{a}\,\underline{a}^{-1}$, and $|\partial_{3}\partial_{3}\bar{L}|\le\chi^{-1}\cdot2\cdot\underline{a}^{-1}$, with the same bounds for the remaining entries; each of these is at most $K_{c}$, and all second derivatives of $\bar{G}$ vanish.

Uniform continuity of the second derivatives: we show that every second-order partial derivative of $\bar{L}$ is Lipschitz on $U_{c}\times\mathbb{R}^{2}$ with the constant $\Lambda_{c}=\chi^{-1}\bigl(4\underline{a}^{-2}+\underline{a}^{-1}\bigr)$, which yields that requirement by taking $\delta=\varepsilon/\Lambda_{c}$. Constants are Lipschitz with constant $0$. For $\chi^{-1}\phi'(a^{1})$, the Lipschitz constant $\underline{a}^{-1}$ of $\phi'$ and the estimate $|a^{1}-b^{1}|\le d\bigl((x,a),(y,b)\bigr)$ give the Lipschitz constant $\chi^{-1}\underline{a}^{-1}\le\Lambda_{c}$. For $\chi^{-1}x^{1}\phi''(a^{1})$,

using $|x^{1}|<2$, the Lipschitz constant $2\underline{a}^{-2}$ of $\phi''$ and the bound $\phi''\le\underline{a}^{-1}$; multiplying by $\chi^{-1}$ gives the constant $\Lambda_{c}$. The remaining entries are of one of these three shapes, and all second derivatives of $\bar{G}$ vanish. Hence $(U_{c},\bar{L},\bar{G})$ is a twice continuously differentiable extension of $(L,G)$ with second-derivative bound $K_{c}$.