The Empirical State Measure Deviates from the Realized Mean-Field Flow by at Most the Noise Majorant

Adopt the setting, hypotheses and notation of the pre-stopping envelope lemma: in particular the number of agents $N$ and the number of states $l$ (both natural numbers), the horizon $T>0$, the solution of the controlled $N$-agent dynamics with regular event $\Omega_0$ and empirical state measure $\Sigma$, the realized control $\hat{\alpha}$, the point $x_0$ of the probability simplex $\Delta^l$, the realized mean-field flow $\Phi$ of the extended good-set stopping-time lemma, $\Phi_t(\omega)=S_t(x_0,\hat{\alpha}(\omega))$ (the two-argument $S_t(\cdot,\cdot)$ is the mean-field flow of claim 2 of the flow stability lemma), the martingale part $M$, the integrals $\mathcal{M}_t$, the random variables $\overline{M}$ and $I$, the trajectory $S$ of the pair $(S,A)$ with $S_0=x_0$ (the one-argument $S_t$) and the initial state fluctuation $\mathfrak{s}_0=\sqrt{N}(\Sigma_0-S_0)$, the constant $\Lambda_b$, the barrier $\varepsilon_Y$, and the envelope majorant $\overline{\mathfrak{b}}$ with

$$Q=\overline{\mathfrak{b}}-\varepsilon_Y=\overline{M}+\Lambda_be^{\Lambda_bT}I+e^{\Lambda_bT}N^{-1/2}|\mathfrak{s}_0| .$$

Only the objects just listed and the relation $S_0=x_0$ are used; the map $S^{*}$ of that lemma enters only through its stipulation $S^{*}=S$ and the relation $S_0=x_0$, while the barriers $c_{\mathcal{E}}$, $\delta$, $\theta_{\mathrm{out}}$, the stopping times, the deviation $Y$ and the energy $\mathcal{E}$ play no role in the conclusion. Write $|\cdot|$ for the Euclidean norm on $\mathbb{R}^l$.

Then for every $\omega\in\Omega_0$ and every $t\in[0,T]$,

$$\bigl|\Sigma_t(\omega)-\Phi_t(\omega)\bigr|\ \le\ Q(\omega).$$