Weighted Second-Moment Evolution of the State Fluctuation Process

Adopt the setting of the \reftext{def:n-agent-fluctuation-processes-2026a}{fluctuation processes of the controlled $N$-agent dynamics}: a \reftext{def:transition-rate-family-2026a}{transition-rate family} $\beta$ with rate bound $B$ on $l$ states with control dimension $m$, an \reftext{def:observation-rate-family-2026a}{observation-rate family} $\tilde{\beta}$, a horizon $T>0$, an \reftext{def:n-agent-driving-system-2026a}{$N$-agent driving system} $(\Omega,\mathcal{F},P)$, an \reftext{def:observation-driven-control-policy-2026a}{observation-driven control policy} $h$, a \reftext{def:n-agent-controlled-dynamics-2026a}{solution} on $[0,T]$ with regular event $\Omega_0$, empirical state measure $\Sigma_t$, control $\alpha_t$, and system filtration $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$, a \reftext{def:mean-field-trajectory-pair-2026a}{mean-field trajectory pair} $(S,A)$ for $\beta$ with horizon $T$, and the state fluctuation process $\mathfrak{s}_t=\sqrt{N}(\Sigma_t-S_t)$. Let $b$ be the \reftext{def:aggregate-state-drift-2026a}{aggregate state drift} of $\beta$, let $\Theta$ be the \reftext{def:aggregate-fluctuation-covariance-2026a}{aggregate fluctuation covariance} of $\beta$, write $\mathbb{E}$ for the \reftext{def:expectation-variance-2026a}{expectation}, and set

Let $Z$ be a family of real $l\times l$ matrices $Z_t=(Z^{\gamma\delta}_t)_{\gamma,\delta\in\{1,\dots,l\}}$, $t\in[0,T]$, that is symmetric ($Z^{\gamma\delta}_t=Z^{\delta\gamma}_t$ for all $t,\gamma,\delta$) and continuously differentiable in integral form: there are \reftext{def:continuity-closed-interval-c54-2026b}{continuous} functions $\dot{z}^{\gamma\delta}:[0,T]\to\mathbb{R}$ with

the integral being the \reftext{def:riemann-integrable-closed-interval-c54-2026b}{Riemann integral} (which exists by \reftext{lem:continuous-implies-riemann-integrable-c54-2026b}{continuity}; it is $0$ for $t=0$). Write $\dot{z}(s)$ for the matrix $(\dot{z}^{\gamma\delta}(s))_{\gamma,\delta\in\{1,\dots,l\}}$, and for vectors $x,y\in\mathbb{R}^l$ and a real $l\times l$ matrix $M$ write $x\cdot My=\sum_{\gamma,\delta=1}^{l}M^{\gamma\delta}x^\gamma y^\delta$. Then:

\textbf{(a) (Well-definedness.)} For every $s\in[0,T]$ the expectations $\mathbb{E}[\mathfrak{s}_s\cdot\dot{z}(s)\mathfrak{s}_s]$, $\mathbb{E}[\mathfrak{s}_s\cdot Z_sg_s]$, and $\mathbb{E}[\Theta^{\gamma\delta}(\Sigma_s,\alpha_s)]$ are finite, and as functions of $s$ they are bounded and \reftext{def:measurable-function-2026a}{measurable} on $[0,T]$, so the \reftext{lem:interval-lebesgue-toolkit-2026a}{Lebesgue integrals} below exist.

\textbf{(b) (Evolution identity.)} For every $t\in[0,T]$,