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 (vi), and we use freely its properties from claims The Regularised Entropic Rate Cost (i), The Regularised Entropic Rate Cost (iii) and The Regularised Entropic Rate Cost (iv) 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 Boxes in $\mathbb{R}^n$ Are 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 (iv). 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 $\alpha\in\mathcal{A}$,

$$\Theta^{11}=\Sigma^{2}\beta(2,1,\Sigma,\alpha)+\Sigma^{1}\beta(1,2,\Sigma,\alpha)=\Sigma^{1}\alpha^{1}+\Sigma^{2}\alpha^{2},$$

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

$$\Theta^{12}=\Theta^{21}=-\Sigma^{1}\beta(1,2,\Sigma,\alpha)-\Sigma^{2}\beta(2,1,\Sigma,\alpha)=-\bigl(\Sigma^{1}\alpha^{1}+\Sigma^{2}\alpha^{2}\bigr).$$

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 Basic Properties of Cᵏ Functions 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 $\Sigma\in\Delta^{2}$,

$$\tilde{b}^{\upsilon}(\Sigma)=\sum_{\sigma=1}^{2}\Sigma^{\sigma}\bigl(q\,\mathbf{1}_{\{\sigma=\upsilon\}}+q_{0}\bigr)=q\,\Sigma^{\upsilon}+q_{0}\sum_{\sigma=1}^{2}\Sigma^{\sigma}=q\,\Sigma^{\upsilon}+q_{0},$$

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 Basic Properties of Cᵏ Functions; 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 (iv). 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

$$\bigl|\chi^{-1}\bigl(x^{1}\phi(a^{1})+x^{2}\phi(a^{2})\bigr)-\chi^{-1}\bigl(x'^{1}\phi(a^{1})+x'^{2}\phi(a^{2})\bigr)\bigr|\le\frac{2\bar{a}^{2}}{\chi\,\underline{a}}\bigl(|x^{1}-x'^{1}|+|x^{2}-x'^{2}|\bigr)\le\frac{2\sqrt{2}\,\bar{a}^{2}}{\chi\,\underline{a}}\,|x-x'| ,$$

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

$$\tfrac{1}{2}\psi\bigl|(x^{2}-x^{1})^{2}-(x'^{2}-x'^{1})^{2}\bigr|\le\tfrac{1}{2}\psi\cdot2\cdot\bigl|(x^{2}-x'^{2})-(x^{1}-x'^{1})\bigr|\le\sqrt{2}\,\psi\,|x-x'| .$$

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 Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers, 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 Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers; 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 $\chi^{-1}x^{1}\phi''(a^{1})$,

$$\bigl|x^{1}\phi''(a^{1})-y^{1}\phi''(b^{1})\bigr|\le|x^{1}|\,\bigl|\phi''(a^{1})-\phi''(b^{1})\bigr|+\bigl|\phi''(b^{1})\bigr|\,|x^{1}-y^{1}|\le\bigl(2\cdot2\underline{a}^{-2}+\underline{a}^{-1}\bigr)d\bigl((x,a),(y,b)\bigr),$$

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 $K_{c}$.