The Ising Population Model Instantiates the Data of the Fluctuation Theory

Fix parameters $\chi$, $\psi$, $q$, $q_{0}$, $\mu$, $T$, $\underline{a}$, $\bar{a}$ as in The Ising Population Data §parameters and adopt the Ising population data with those parameters: the control set $\mathcal{A}$, the open sets $U$, $V$, $U_{c}$, $\tilde{U}$, the transition-rate coefficients $(\beta_{0},\beta_{1})$ with the rates $\beta$ and their extension $\bar{\beta}$, the observation rates $\tilde{\beta}$ with their extension $\bar{\tilde{\beta}}$, the costs $(L,G)$ with their extension $(\bar{L},\bar{G})$, the function $\phi$, and the vectors $v$, $\mathsf{n}$. Write $\Lambda$ for the Lipschitz constant of an affine-controlled transition-rate family, $B$ for the rate bound of a transition-rate family, $K$ for the derivative bound of a twice continuously differentiable extension of such a family, $\tilde{B}$ and $\tilde{K}$ for the corresponding constants of an observation-rate family and of its twice continuously differentiable extension, and $K_{c}$ for the second-derivative bound of a twice continuously differentiable extension of population cost data. Adopt the coordinate and partial-derivative notation $\partial_{i}$ of the extension definitions, so that $\partial_{1},\partial_{2}$ differentiate in the state coordinates $x^{1},x^{2}$ and $\partial_{3},\partial_{4}$ in the control coordinates $a^{1},a^{2}$. Then the following hold.

1. (The control set.)¶ $\mathcal{A}$ is a nonempty convex subset of $\mathbb{R}^{2}$, compact for the topology determined by the Euclidean distance; thus hypothesis (A) of Quadratic Growth of the Mean-Field Hamiltonian in the Control along a Stationary Mean-Field Triple holds. Moreover $\sup_{a\in\mathcal{A}}|a|=\sqrt{2}\,\bar{a}$, and, with

every $a\in\mathbb{R}^{2}$ with $|a-(1,1)|\le\varrho_{\mathcal{A}}$ lies in $\mathcal{A}$.

2. (The transition-rate family and its drift.)¶ The pair $(\beta_{0},\beta_{1})$ is an affine-controlled transition-rate family on $2$ states with control set $\mathcal{A}$ and Lipschitz constant $\Lambda=0$. The transition-rate family that The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data associates with it is the family $\beta$ of clause The Ising Population Data §rates, and it is a transition-rate family on $2$ states with control set $\mathcal{A}$ and rate bound $B=\bar{a}$. Its aggregate state drift is

3. (The aggregate fluctuation covariance.)¶ The aggregate fluctuation covariance of $\beta$ is

4. (The rate extension.)¶ The triple $(U,V,\bar{\beta})$ is a twice continuously differentiable extension of $\beta$ with derivative bound $K=1$. Its only nonvanishing partial derivatives of order one or two are $\partial_{3}\bar{\beta}(1,2,\cdot)=1$ and $\partial_{4}\bar{\beta}(2,1,\cdot)=1$ on $U\times V$. Its extended aggregate state drift is

5. (The observation-rate family.)¶ The family $\tilde{\beta}$ is an observation-rate family on $2$ states with $2$ observation channels and rate bound $\tilde{B}=q+q_{0}$. Its aggregate observation drift is $\tilde{b}^{\upsilon}(\Sigma)=q\,\Sigma^{\upsilon}+q_{0}$ for $\Sigma\in\Delta^{2}$ and $\upsilon\in\{1,2\}$, and satisfies $\tilde{b}^{\upsilon}(\Sigma)\ge q_{0}>0$ for all such $\Sigma$ and $\upsilon$; thus hypothesis (OC) of Injection Certificates on the Trimmed Synthetic Copy for the Family of N-Agent Solutions: the Law-Transported Van Trees Certificate Hypothesis Holds under Deterministic Initial States, Uniform Observation Positivity and a C^2 Observation Extension holds with $\underline{b}=q_{0}$.

6. (The observation-rate extension.)¶ The pair $(\tilde{U},\bar{\tilde{\beta}})$ is a twice continuously differentiable extension of $\tilde{\beta}$ with derivative bound $\tilde{K}=0$; all its partial derivatives of order one and two vanish. Its extended aggregate observation drift is

7. (The population cost data.)¶ The pair $(L,G)$ is population cost data on $2$ states with control dimension $2$, with lower bounds $C_{L}=C_{G}=0$. Moreover $L(x,a)\ge0$ for every $(x,a)\in\Delta^{2}\times\mathbb{R}^{2}$, and $L(x,a)=0$ holds if and only if $x=\bigl(\tfrac{1}{2},\tfrac{1}{2}\bigr)$ and $a=(1,1)$. For every $x\in\Delta^{2}$ the map $a\mapsto L(x,a)$ is convex on $\mathbb{R}^{2}$; in particular $L$ is convex in the control on $\mathcal{A}$. Finally, hypothesis (LipC) of Pointwise-in-Time Tracking of the Mean-Field Flow and Cost along the Realized Control of the Controlled N-Agent Dynamics holds with

8. (The cost extension.)¶ The triple $(U_{c},\bar{L},\bar{G})$ is a twice continuously differentiable extension of $(L,G)$ with second-derivative bound

All partial derivatives of $\bar{G}$ vanish, and the partial derivatives of $\bar{L}$ on $U_{c}\times\mathbb{R}^{2}$ are

and, for the second order,

all evaluated at the point $(x,a)$.