Mean-Square Linearization Residual of the Observation 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} $\beta$ on $l$ states with control dimension $m$, an \reftext{def:observation-rate-family-2026a}{observation-rate family} $\tilde{\beta}$ with $\tilde{l}$ observation channels and rate bound $\tilde{B}$, 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 $\Sigma_t$, observation processes $\Upsilon^\upsilon_t$, 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 state fluctuation process $\mathfrak{s}_t=\sqrt{N}(\Sigma_t-S_t)$. Assume that $\tilde{\beta}$ admits a \reftext{def:c2-observation-rate-extension-2026a}{twice continuously differentiable extension} $(\tilde{U},\bar{\tilde{\beta}})$ with derivative bound $\tilde{K}$. Let $\tilde{b}$ be the \reftext{def:aggregate-observation-drift-2026a}{aggregate observation drift} of $\tilde{\beta}$, with $\tilde{b}(\Sigma)$ the vector with components $\tilde{b}^1(\Sigma),\dots,\tilde{b}^{\tilde{l}}(\Sigma)$, let $\bar{\tilde{b}}$ be the \reftext{def:extended-aggregate-observation-drift-2026a}{extended aggregate observation drift} of $(\tilde{U},\bar{\tilde{\beta}})$, adopt the partial-derivative notation $\partial_\gamma$, $\partial_\delta\partial_\gamma$ of the extension definition, and let $\tilde{M}_t=(\tilde{M}^\upsilon_t)_{\upsilon\in\{1,\dots,\tilde{l}\}}$ 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), $\Delta^l$ for the \reftext{def:probability-simplex-2026a}{probability simplex}, and $\mathbf{1}_{D}$ for the function equal to $1$ on a set $D$ and $0$ off $D$, and set, for $s\in[0,T]$,

Define the \textbf{observation linearization matrix}: for $s\in[0,T]$, the real matrix $\tilde{\mathcal{E}}_s$ with $\tilde{l}$ rows and $l$ columns and entries

which is defined because $S_s$ lies in $\Delta^l\subseteq\tilde{U}$ (the inclusion being part of the \reftext{def:c2-observation-rate-extension-2026a}{extension definition}) and the partial derivatives of $\bar{\tilde{b}}$ exist there by the \reftext{lem:extended-observation-drift-regularity-2026a}{regularity of the extended aggregate observation drift}; when $(S,A)$ belongs to a \reftext{def:stationary-mean-field-triple-2026a}{stationary mean-field triple} and the observation-rate extension is the one chosen there, $\tilde{\mathcal{E}}_s$ is the observation matrix of the \reftext{def:fluctuation-lqg-data-2026a}{fluctuation LQG data}, with the same symbol, by the identical defining formula of clause 3 of that definition. Define the \textbf{observation linearization residual}, with the \reftext{def:matrix-vector-product-2026a}{matrix-vector product} $\tilde{\mathcal{E}}_s\mathfrak{s}_s$,

and the \textbf{observation fluctuation process}, with $\Upsilon_t$ the vector with components $\Upsilon^1_t,\dots,\Upsilon^{\tilde{l}}_t$,

the integral being the componentwise \reftext{lem:interval-lebesgue-toolkit-2026a}{Lebesgue integral} over the compact interval $[0,t]$ of a continuous integrand (each $r\mapsto\tilde{b}^\upsilon(S_r)$ is continuous, $S$ being continuous with values in $\Delta^l$ and $\tilde{b}^\upsilon$ agreeing there with the extension $\bar{\tilde{b}}^\upsilon$, which is continuous by part (i) of the \reftext{lem:extended-observation-drift-regularity-2026a}{regularity lemma}), equal to $0$ for $t=0$. Then:

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

moreover $|\mathfrak{s}_s|\le2\sqrt{N}$ at every point of $[0,T]\times\Omega$, so that $|\tilde{e}_s|\le4\,\tilde{\Lambda}\,\sqrt{N}$ everywhere.

\textbf{(b) (Well-definedness.)} Each map $(s,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)\,\tilde{e}^\upsilon_s(\omega)$ 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} and the continuity of the \reftext{def:mean-field-trajectory-pair-2026a}{mean-field trajectory pair}; and at each $\omega\in\Omega_0$ every section $s\mapsto\tilde{e}^\upsilon_s(\omega)$ is bounded and Lebesgue integrable over $[0,t]$ for every $t\in[0,T]$. Define the \textbf{observation residual process}

Each $\tilde{\mathcal{R}}^\upsilon_t$ is a \reftext{def:probability-space-random-variable-2026a}{random variable}, by the \reftext{thm:tonelli-fubini-2026a}{Tonelli theorem} applied to the positive and negative parts of $\mathbf{1}_{\Omega_0}\tilde{e}^\upsilon$.

\textbf{(c) (Integral form of the observation fluctuation process.)} At every $\omega\in\Omega_0$ 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 $t\in[0,T]$, all quantities below are finite and

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