A Priori Fourth-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 fluctuation processes $\mathfrak{s}_t=\sqrt{N}(\Sigma_t-S_t)$ and $\mathfrak{a}_t=\sqrt{N}(\alpha_t-A_t)$. Assume that $\beta$ admits a \reftext{def:c2-transition-rate-extension-2026a}{twice continuously differentiable extension} $(U,\bar{\beta})$ with derivative bound $K$; the bound $\Lambda$ below is warranted by the \reftext{lem:extended-drift-regularity-2026a}{regularity and derivative bounds of the extended aggregate state drift}. Let $b$ be the \reftext{def:aggregate-state-drift-2026a}{aggregate state drift} of $\beta$, let $M_t=(M^\gamma_t)_{\gamma\in\{1,\dots,l\}}$ be as in the \reftext{thm:n-agent-martingale-decomposition-2026a}{martingale decomposition}, write $\mathbb{E}$ for the \reftext{def:expectation-variance-2026a}{expectation}, $|\cdot|$ for the Euclidean norm (\reftext{def:euclidean-distance-rn-2026a}{Euclidean distance} to the origin), and $\mathbf{1}_{\Omega_0}$ for the function equal to $1$ on $\Omega_0$ and $0$ off $\Omega_0$, set $g_s=\sqrt{N}\,(b(\Sigma_s,\alpha_s)-b(S_s,A_s))$ as in the \reftext{lem:fluctuation-state-moment-bound-2026a}{a priori second-moment bound}, set

and let $c_M=6\,l^2\,\big(2(l-1)\big)^4$ be the constant of the \reftext{lem:n-agent-counter-fourth-moment-2026a}{moment bounds for the aggregate compensated counters}.

\textbf{(a) (Well-definedness.)} The maps $(t,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)|\mathfrak{s}_t(\omega)|^4$, $(t,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)|\mathfrak{a}_t(\omega)|^4$, and $(t,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)|g_t(\omega)|^4$ 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} together with \reftext{lem:continuous-composition-measurable-2026a}{measurability of sequentially continuous functions of measurable Euclidean maps}. Consequently, by the \reftext{thm:tonelli-fubini-2026a}{Tonelli theorem}, the functions $t\mapsto\mathbb{E}[|\mathfrak{s}_t|^4]$, $t\mapsto\mathbb{E}[|\mathfrak{a}_t|^4]$, and $t\mapsto\mathbb{E}[|g_t|^4]$ (unchanged by the $\mathbf{1}_{\Omega_0}$ modification, $\Omega_0$ having probability $1$) are measurable $[0,\infty]$-valued functions on $[0,T]$, with $\mathbb{E}[|\mathfrak{s}_t|^4]\le16\,N^2$ and $\mathbb{E}[|g_t|^4]\le256\,l^2\,N^2\,(l-1)^4\,B^4$ finite for every $t$, and the \reftext{lem:interval-lebesgue-toolkit-2026a}{Lebesgue integral}

is well defined with value in $[0,\infty]$ (the subscripted $\mathcal{A}_4$ is distinct from the control energy $\mathcal{A}$ of the second-moment bound).

\textbf{(b) (Three-term estimate.)} For every $t\in[0,T]$, with the right-hand side valued in $[0,\infty]$,

\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 $\mathcal{A}_4=+\infty$. In particular, if $\mathcal{A}_4<\infty$, then $\sup\{\mathbb{E}[|\mathfrak{s}_t|^4]:t\in[0,T]\}$ is finite, with a bound depending on $N$ only through $\mathbb{E}[|\mathfrak{s}_0|^4]$ and $\mathcal{A}_4$.