Fluctuation Processes of the Controlled N-Agent Dynamics

Adopt the setting of the controlled $N$-agent dynamics with $N$ agents, $l$ states, $\tilde{l}$ observation channels, and control dimension $m$, and let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$: a transition-rate family $\beta$ on $l$ states with control set $\mathcal{A}$ and rate bound $B$, an observation-rate family $\tilde{\beta}$ on $l$ states with $\tilde{l}$ channels and rate bound $\tilde{B}$, a horizon $T>0$, an $N$-agent driving system $(\Omega,\mathcal{F},P)$, an observation-driven control policy $h$ which is $\mathcal{A}$-valued, and a solution $((\sigma^i)_{i=1}^N,(\Upsilon^\upsilon)_{\upsilon=1}^{\tilde{l}},\alpha)$ on $[0,T]$ for these data, with empirical state measure $\Sigma_t$ and control process $\alpha_t$. Let $(S,A)$ be a mean-field trajectory pair for $\beta$ with horizon $T$ (so that $A_t\in\mathcal{A}$ for every $t\in[0,T]$), and write $\sqrt{N}$ for the nonnegative square root of $N$.

The fluctuation processes of the solution about $(S,A)$ are the pair of families $(\mathfrak{s}_t)_{t\in[0,T]}$ and $(\mathfrak{a}_t)_{t\in[0,T]}$ defined componentwise by

$$\mathfrak{s}^\gamma_t=\sqrt{N}\,\big(\Sigma^\gamma_t-S^\gamma_t\big)\quad(\gamma\in\{1,\dots,l\}),\qquad\qquad \mathfrak{a}^j_t=\sqrt{N}\,\big(\alpha^j_t-A^j_t\big)\quad(j\in\{1,\dots,m\}),$$

written $\mathfrak{s}_t=\sqrt{N}(\Sigma_t-S_t)$ and $\mathfrak{a}_t=\sqrt{N}(\alpha_t-A_t)$ as $\mathbb{R}^l$-valued and $\mathbb{R}^m$-valued maps on the underlying probability space; $(\mathfrak{s}_t)_{t\in[0,T]}$ is called the state fluctuation process and $(\mathfrak{a}_t)_{t\in[0,T]}$ the control fluctuation process.