Fluctuation LQG Data of a Stationary Mean-Field Triple

Let $l$, $m$, $\beta$ with rate bound $B$, $(U,\bar{\beta})$ with derivative bound $K$, $(L,G)$, $(V,\bar{L},\bar{G})$ with second-derivative bound $K_c$, $T>0$, $(S,A)$, and the \reftext{def:stationary-mean-field-triple-2026a}{stationary co-state} $P$ be as in the definition of the \reftext{def:fluctuation-lqg-cost-2026b}{fluctuation linear-quadratic cost functional}, with $\bar{b}$ the \reftext{def:extended-aggregate-state-drift-2026a}{extended aggregate state drift} of $(U,\bar{\beta})$, and let $H_{ij}(t)$ and $F_{\gamma\delta}$ be the \reftext{def:fluctuation-lqg-cost-2026b}{fluctuation Hessian coefficients} of these data. Let moreover $\tilde{l}\ge1$ be a \reftext{def:natural-numbers-2026a}{natural number}, let $\tilde{\beta}$ be an \reftext{def:observation-rate-family-2026a}{observation-rate family} on $l$ states with $\tilde{l}$ observation channels and rate bound $\tilde{B}$, let $(\tilde{U},\bar{\tilde{\beta}})$ be a \reftext{def:c2-observation-rate-extension-2026a}{twice continuously differentiable extension} of $\tilde{\beta}$ with derivative bound $\tilde{K}$, let $\bar{\tilde{b}}$ be its \reftext{def:extended-aggregate-observation-drift-2026a}{extended aggregate observation drift}, let $\tilde{b}$ be the \reftext{def:aggregate-observation-drift-2026a}{aggregate observation drift} of $\tilde{\beta}$, and let $\Theta$ be the \reftext{def:aggregate-fluctuation-covariance-2026a}{aggregate fluctuation covariance} of $\beta$. Adopt the coordinate and partial-derivative notation of the \reftext{def:c2-transition-rate-extension-2026a}{transition-rate extension definition}, so that $\partial_\gamma$ with $\gamma\le l$ differentiates in the $\gamma$-th state coordinate and $\partial_{l+j}$ with $j\le m$ in the $j$-th control coordinate, together with the notation $\partial_\gamma$ ($\gamma\in\{1,\dots,l\}$) of the \reftext{def:c2-observation-rate-extension-2026a}{observation-rate extension definition}; the partial derivatives of $\bar{b}$ and of $\bar{\tilde{b}}$ used below exist by the \reftext{lem:extended-drift-regularity-2026a}{regularity of the extended aggregate state drift} and the \reftext{lem:extended-observation-drift-regularity-2026a}{regularity of the extended aggregate observation drift}. Matrices are indexed as in the definition of the \reftext{def:matrix-vector-product-2026a}{matrix-vector product}: the first index labels rows and the second labels columns. We write $\mathbf{1}_{\{\cdot\}}$ for the indicator equal to $1$ when the subscripted condition holds and $0$ otherwise.

The \textbf{fluctuation LQG data} of the stationary mean-field triple $(S,A,P)$ relative to the chosen extensions and the observation-rate family consist of the following matrices, defined for each $t\in[0,T]$:

\textbf{1. (State matrix.)} $\mathcal{E}_t$, with $l$ rows and $l$ columns and entries $(\mathcal{E}_t)_{\gamma\delta}=\partial_\delta\bar{b}^\gamma(S_t,A_t)$ for $\gamma,\delta\in\{1,\dots,l\}$.

\textbf{2. (Control matrix.)} $\mathcal{B}_t$, with $l$ rows and $m$ columns and entries $(\mathcal{B}_t)_{\gamma j}=\partial_{l+j}\bar{b}^\gamma(S_t,A_t)$ for $\gamma\in\{1,\dots,l\}$ and $j\in\{1,\dots,m\}$.

\textbf{3. (Observation matrix.)} $\tilde{\mathcal{E}}_t$, with $\tilde{l}$ rows and $l$ columns and entries $(\tilde{\mathcal{E}}_t)_{\upsilon\gamma}=\partial_\gamma\bar{\tilde{b}}^\upsilon(S_t)$ for $\upsilon\in\{1,\dots,\tilde{l}\}$ and $\gamma\in\{1,\dots,l\}$.

\textbf{4. (Hessian blocks.)} $H^{SS}_t$ with $l$ rows and $l$ columns, $H^{SA}_t$ with $l$ rows and $m$ columns, $H^{AS}_t$ with $m$ rows and $l$ columns, and $H^{AA}_t$ with $m$ rows and $m$ columns, with entries

for $\gamma,\delta\in\{1,\dots,l\}$ and $j,k\in\{1,\dots,m\}$.

\textbf{5. (Terminal matrix.)} $F^\star$, with $l$ rows and $l$ columns and entries $(F^\star)_{\gamma\delta}=F_{\gamma\delta}$ for $\gamma,\delta\in\{1,\dots,l\}$ (independent of $t$).

\textbf{6. (State noise covariance.)} $\Theta^\star_t$, with $l$ rows and $l$ columns and entries $(\Theta^\star_t)_{\gamma\delta}=\Theta^{\gamma\delta}(S_t,A_t)$ for $\gamma,\delta\in\{1,\dots,l\}$.

\textbf{7. (Observation noise covariance.)} $\tilde{\Theta}^\star_t$, with $\tilde{l}$ rows and $\tilde{l}$ columns and entries $(\tilde{\Theta}^\star_t)_{\upsilon\upsilon'}=\mathbf{1}_{\{\upsilon=\upsilon'\}}\,\tilde{b}^\upsilon(S_t)$ for $\upsilon,\upsilon'\in\{1,\dots,\tilde{l}\}$.