Mean-Square Linearization Residual 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 $\Sigma_t$, 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 $K$, and assume

this integral being well defined in $[0,\infty]$ by clause (a) of the \reftext{lem:fluctuation-state-moment-bound-2026a}{a priori second-moment bound}. Let $b$ be the \reftext{def:aggregate-state-drift-2026a}{aggregate state drift} of $\beta$, let $\bar{b}$ be the \reftext{def:extended-aggregate-state-drift-2026a}{extended aggregate state drift} of $(U,\bar{\beta})$, adopt the partial-derivative notation $\partial_i$, $\partial_j\partial_i$ of the extension definition, and 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), $\mathbf{1}_{D}$ for the function equal to $1$ on a set $D$ and $0$ off $D$, and set, for $s\in[0,T]$,

Define, as in the statement of the \reftext{thm:fluctuation-control-coercivity-2026a}{completion-of-squares and coercivity theorem} and with the same symbols, the \textbf{linearization coefficient matrices}: for $s\in[0,T]$, the real $l\times l$ matrix $E_s$ and the real $l\times m$ matrix $\mathsf{B}_s$ (the sans-serif $\mathsf{B}_s$ is distinct from the rate bound $B$, and the matrix $E_s$ is distinct from the expectation $\mathbb{E}$) with entries

which are defined because $S_s$ lies in the \reftext{def:probability-simplex-2026a}{probability simplex} $\Delta^l$, so that $(S_s,A_s)\in\Delta^l\times\mathbb{R}^m\subseteq U\times\mathbb{R}^m$ (the inclusion $\Delta^l\subseteq U$ is part of the \reftext{def:c2-transition-rate-extension-2026a}{extension definition}), and the partial derivatives of $\bar{b}$ exist there by the \reftext{lem:extended-drift-regularity-2026a}{regularity of the extended aggregate state drift}. Define the \textbf{linearization residual}, with the \reftext{def:matrix-vector-product-2026a}{matrix-vector products} $E_s\mathfrak{s}_s$ and $\mathsf{B}_s\mathfrak{a}_s$,

Then:

\textbf{(a) (Pointwise bounds.)} At every point of $[0,T]\times\Omega$,

The first bound coincides with part (b) of the \reftext{thm:fluctuation-control-coercivity-2026a}{completion-of-squares and coercivity theorem} (whose constant $c_e$ equals $\tfrac{3}{2}\,l\,\sqrt{l}\,(l+m)\,K$); neither bound uses the hypothesis $\mathcal{A}<\infty$.

\textbf{(b) (Well-definedness.)} Each map $(s,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)\,e^\gamma_s(\omega)$ is \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}. The set $\Omega_{\mathfrak{a}}$ of those $\omega\in\Omega_0$ at which the \reftext{lem:interval-lebesgue-toolkit-2026a}{Lebesgue integral} $\int_{[0,T]}|\mathfrak{a}_s(\omega)|^2\,ds$ is finite is an event of probability $1$ (by the \reftext{thm:tonelli-fubini-2026a}{Tonelli theorem} and $\mathcal{A}<\infty$), and at each $\omega\in\Omega_{\mathfrak{a}}$ the section $s\mapsto e^\gamma_s(\omega)$ is Lebesgue integrable over $[0,t]$ for every $t\in[0,T]$. Define the \textbf{residual process}

Each $\mathcal{R}^\gamma_t$ is a \reftext{def:probability-space-random-variable-2026a}{random variable}: the integrals over $[0,t]$ of the positive and negative parts of $\mathbf{1}_{\Omega_0}e^\gamma$ (product-measurable, being \reftext{lem:continuous-composition-measurable-2026a}{continuous functions} of it) define measurable $[0,\infty]$-valued functions of $\omega$ by the Tonelli theorem, both finite on $\Omega_{\mathfrak{a}}$, and $\mathcal{R}^\gamma_t$ is the difference of the functions equal to them on $\Omega_{\mathfrak{a}}$ and to $0$ off $\Omega_{\mathfrak{a}}$.

\textbf{(c) (Integral form of the fluctuation dynamics.)} At every $\omega\in\Omega_{\mathfrak{a}}$ that lies in the almost-sure event of clause (a) of the \reftext{thm:n-agent-martingale-decomposition-2026a}{martingale decomposition}, and hence \reftext{def:almost-surely-2026a}{almost surely}, for every $t\in[0,T]$,

all integrals existing componentwise at such $\omega$.

\textbf{(d) (Mean-square residual bound.)} For every $t\in[0,T]$, with both sides valued in $[0,\infty]$,

Here the maps $(s,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)\min(\cdots)^2$ and $(s,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)|z_s(\omega)|^4$ are product-measurable, being \reftext{lem:continuous-composition-measurable-2026a}{continuous functions} of the product-measurable maps of the \reftext{lem:n-agent-joint-measurability-2026a}{joint measurability lemma}, so the integrands $s\mapsto\mathbb{E}[\min(\cdots)^2]$ and $s\mapsto\mathbb{E}[|z_s|^4]$ (unchanged by the $\mathbf{1}_{\Omega_0}$ modification, $\Omega_0$ having probability $1$) are measurable $[0,\infty]$-valued functions of $s$ by the \reftext{thm:tonelli-fubini-2026a}{Tonelli theorem}. In particular