Stationary Mean-Field Triple

Let $l$ and $m$ be \reftext{def:natural-numbers-2026a}{natural numbers} with $l\ge2$ and $m\ge1$. Let $\beta$ be a \reftext{def:transition-rate-family-2026a}{transition-rate family} on $l$ states with control dimension $m$, let $(U,\bar{\beta})$ be a \reftext{def:c2-transition-rate-extension-2026a}{twice continuously differentiable extension} of $\beta$ with derivative bound $K$, let $\bar{b}$ be the \reftext{def:extended-aggregate-state-drift-2026a}{extended aggregate state drift} of $(U,\bar{\beta})$, let $(L,G)$ be \reftext{def:population-cost-data-2026a}{population cost data} on $l$ states with control dimension $m$, let $(V,\bar{L},\bar{G})$ be a \reftext{def:c2-population-cost-extension-2026a}{twice continuously differentiable extension} of $(L,G)$ with second-derivative bound $K_c$, let $T>0$ be a \reftext{def:real-numbers-c54-2026c}{real number}, and let $(S,A)$ be a \reftext{def:mean-field-trajectory-pair-2026a}{mean-field trajectory pair} for $\beta$ with horizon $T$. Adopt the coordinate and partial-derivative notation $\partial_i$ of the extension definitions, so that $\partial_\gamma$ with $\gamma\le l$ differentiates in the $\gamma$-th state coordinate and $\partial_{l+j}$ with $j\le m$ in the $j$-th control coordinate; the first-order partial derivatives of $\bar{b}$ used below exist and are continuous by the \reftext{lem:extended-drift-regularity-2026a}{regularity of the extended aggregate state drift}. Clauses 2 and 3 below are the co-state system for the generalized Hamiltonian $P\cdot\bar{b}-\bar{L}$.

A \textbf{stationary co-state} for these data is a function $P:[0,T]\to\mathbb{R}^l$, with values $P_t=(P^1_t,\dots,P^l_t)$, such that:

\textbf{1. (Continuity.)} Each component $t\mapsto P^\gamma_t$ ($\gamma\in\{1,\dots,l\}$) is \reftext{def:continuity-closed-interval-c54-2026b}{continuous} on $[0,T]$.

\textbf{2. (Co-state equation.)} For every $\gamma\in\{1,\dots,l\}$ the map $s\mapsto\sum_{\delta=1}^{l}\partial_\gamma\bar{b}^\delta(S_s,A_s)\,P^\delta_s-\partial_\gamma\bar{L}(S_s,A_s)$ is continuous on $[0,T]$, and for every $t\in[0,T]$

where for $t<T$ the integral is the \reftext{def:riemann-integrable-closed-interval-c54-2026b}{Riemann integral} of the restriction of the integrand to $[t,T]$, which exists by \reftext{lem:continuous-implies-riemann-integrable-c54-2026b}{continuity}, and the integral is $0$ for $t=T$.

\textbf{3. (Stationarity.)} For every $t\in[0,T]$ and every $j\in\{1,\dots,m\}$:

When $P$ is a stationary co-state for these data, the triple $(S,A,P)$ is called a \textbf{stationary mean-field triple} for $\beta$, $(L,G)$, and the chosen extensions.