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 $m$: a transition-rate family $\beta$, an observation-rate family $\tilde{\beta}$, a horizon $T>0$, an $N$-agent driving system, an observation-driven control policy $h$, and a solution on $[0,T]$ with empirical state measure $\Sigma_t$ and control $\alpha_t$. Let $(S,A)$ be a mean-field trajectory pair for $\beta$ with horizon $T$, 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

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.