The Fluctuation Linear-Quadratic Cost Functional

Let $l$, $m$, $\beta$, $(U,\bar{\beta})$ with derivative bound $K$, $(L,G)$, $(V,\bar{L},\bar{G})$ with second-derivative bound $K_c$, $T>0$, and $(S,A)$ be as in the definition of a \reftext{def:stationary-mean-field-triple-2026a}{stationary mean-field triple}, let $\bar{b}$ be the \reftext{def:extended-aggregate-state-drift-2026a}{extended aggregate state drift} of $(U,\bar{\beta})$, let $P$ be a stationary co-state for these data, and adopt the partial-derivative notation $\partial_j\partial_i$ of the \reftext{def:c2-transition-rate-extension-2026a}{extension} \reftext{def:c2-population-cost-extension-2026a}{definitions}. Define, for $t\in[0,T]$ and $i,j\in\{1,\dots,l+m\}$, the \textbf{fluctuation Hessian coefficients}

where the second-order partial derivatives of $\bar{b}$ exist and are continuous by the \reftext{lem:extended-drift-regularity-2026a}{regularity of the extended aggregate state drift}. Each $H_{ij}$ is continuous on $[0,T]$, being a composition of continuous maps along the \reftext{def:mean-field-trajectory-pair-2026a}{continuous trajectory pair} and the continuous co-state, and is therefore \reftext{lem:continuous-compact-interval-bounded-2026a}{bounded}.

Let $(x_t)_{t\in[0,T]}$ and $(a_t)_{t\in[0,T]}$ be families of $\mathbb{R}^l$-valued and $\mathbb{R}^m$-valued random vectors on a common \reftext{def:probability-space-random-variable-2026a}{probability space} (each component a random variable), and write $z_t=(x_t,a_t)$, identified with an $\mathbb{R}^{l+m}$-valued map with components $z^1_t,\dots,z^{l+m}_t$, such that:

\textbf{(i)} there is an event $\Omega_1$ of probability $1$ such that each map $(t,\omega)\mapsto\mathbf{1}_{\Omega_1}(\omega)\,z^i_t(\omega)$ on $[0,T]\times\Omega$ 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 the $\sigma$-algebra of the space, where $\mathbf{1}_{\Omega_1}$ is the function equal to $1$ on $\Omega_1$ and $0$ off $\Omega_1$;

\textbf{(ii)} $\displaystyle\int_0^T\mathbb{E}\big[|x_t|^2+|a_t|^2\big]\,dt<\infty$, where $|\cdot|$ is the Euclidean norm (\reftext{def:euclidean-distance-rn-2026a}{Euclidean distance} to the origin), the integrand is a measurable $[0,\infty]$-valued function of $t$ by the \reftext{thm:tonelli-fubini-2026a}{Tonelli theorem} applied to the maps of (i) (its value is unchanged when $z_t$ is replaced by $\mathbf{1}_{\Omega_1}z_t$, since $\Omega_1$ has probability $1$), and the integral is the \reftext{lem:interval-lebesgue-toolkit-2026a}{Lebesgue integral} with value in $[0,\infty]$;

\textbf{(iii)} $\mathbb{E}\big[|x_T|^2\big]<\infty$, with the \reftext{def:expectation-variance-2026a}{expectation}.

The \textbf{fluctuation linear-quadratic cost} of the pair $((x),(a))$ relative to the stationary mean-field triple $(S,A,P)$ and the chosen extensions is the real number

where the first expectation is taken of the random variable that vanishes off $\Omega_1$ and at each $\omega\in\Omega_1$ equals the Lebesgue integral over $[0,T]$ of the section $t\mapsto\tfrac{1}{2}\sum_{i,j}H_{ij}(t)\,z^i_t(\omega)\,z^j_t(\omega)$ (sections of product-measurable maps are measurable, as in the \reftext{thm:tonelli-fubini-2026a}{Tonelli theorem}). Both expectations are well defined and finite because the $\mathbf{1}_{\Omega_1}$-modified integrand is product-measurable and bounded in absolute value by a constant multiple of $|x_t|^2+|a_t|^2$ by the boundedness of the $H_{ij}$, the terminal term is bounded by a constant multiple of $|x_T|^2$, and (ii), (iii), and the \reftext{thm:tonelli-fubini-2026a}{Fubini theorem} apply; and the value of $LQG[(x),(a)]$ does not depend on the choice of $\Omega_1$, since two such events differ by an event of probability zero.