A Priori Second-Moment Bound for 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 $\Sigma_t$, and control $\alpha_t$, a \reftext{def:mean-field-trajectory-pair-2026a}{mean-field trajectory pair} $(S,A)$ for $\beta$ with horizon $T$, and the associated fluctuation processes $\mathfrak{s}_t=\sqrt{N}(\Sigma_t-S_t)$ and $\mathfrak{a}_t=\sqrt{N}(\alpha_t-A_t)$. Assume moreover that $\beta$ admits a \reftext{def:c2-transition-rate-extension-2026a}{twice continuously differentiable extension} $(U,\bar{\beta})$ with derivative bound $K$. Let $b$ be the \reftext{def:aggregate-state-drift-2026a}{aggregate state drift} of $\beta$, set $g_s=\sqrt{N}(b(\Sigma_s,\alpha_s)-b(S_s,A_s))$ with components $g^\gamma_s$, write $\mathbf{1}_{\Omega_0}$ for the function equal to $1$ on $\Omega_0$ and $0$ off $\Omega_0$, write $|\cdot|$ for the Euclidean norm (\reftext{def:euclidean-distance-rn-2026a}{Euclidean distance} to the origin), and set

\textbf{(a) (Well-definedness.)} The maps $(t,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)|\mathfrak{s}_t(\omega)|^2$ and $(t,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)|\mathfrak{a}_t(\omega)|^2$ are \reftext{def:measurable-function-2026a}{measurable} with respect to the \reftext{def:product-sigma-algebra-2026a}{product $\sigma$-algebra} of the \reftext{lem:interval-lebesgue-toolkit-2026a}{trace Borel $\sigma$-algebra} on $[0,T]$ and $\mathcal{F}$ (by the \reftext{lem:n-agent-joint-measurability-2026a}{joint measurability} of the state and control), and so is $(t,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)(g^\gamma_t(\omega))^2$ for each $\gamma$. Consequently, by the \reftext{thm:tonelli-fubini-2026a}{Tonelli theorem}, the functions $t\mapsto\mathbb{E}[|\mathfrak{s}_t|^2]$, $t\mapsto\mathbb{E}[|\mathfrak{a}_t|^2]$, and $t\mapsto\mathbb{E}[(g^\gamma_t)^2]$ (\reftext{def:expectation-variance-2026a}{expectations}, which are unchanged by the $\mathbf{1}_{\Omega_0}$ modification because $\Omega_0$ has probability $1$) are measurable on $[0,T]$ as $[0,\infty]$-valued functions, with $\mathbb{E}[|\mathfrak{s}_t|^2]\le4N$ and $\mathbb{E}[(g^\gamma_t)^2]\le16N(l-1)^2B^2$ finite for every $t$, and the \reftext{lem:interval-lebesgue-toolkit-2026a}{Lebesgue integral}

is well defined with value in $[0,\infty]$.

\textbf{(b) (Three-term estimate.)} For every $t\in[0,T]$,

\textbf{(c) (A priori bound.)} For every $t\in[0,T]$, with the \reftext{def:exponential-function-real-2026a}{exponential function},

where the right-hand side is interpreted as $+\infty$ when $\mathcal{A}=+\infty$.