Anchored Pre-Stopping Envelope and Restricted Moment Bounds for the State Fluctuation Process

Adopt the setting, notation, hypotheses and definitions of the pre-stopping envelope lemma: the affine-controlled transition-rate family $(\beta_0,\beta_1)$ on $l$ states with compact convex control set $\mathcal{A}\subseteq\mathbb{R}^m$, its transition-rate family $\beta$ with rate bound $B$ and state-Lipschitz constant $\Lambda_b$, the horizon $T>0$, the solution of the controlled $N$-agent dynamics with $N$ agents, regular event $\Omega_0$, empirical state measure $\Sigma$, observation filtration $(\mathcal{G}_t)_{t\in[0,T]}$ and system filtration $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$, the realized control $\hat{\alpha}$, the point $x_0\in\Delta^l$ of the probability simplex, the realized mean-field flow $\Phi$, the mean-field trajectory pair $(S,A)$ with control $A$, initial state $S_0=x_0$ and $S^*=S$, the deviation $Y_t=|\Phi_t-S^*_t|$ — $|\cdot|$ being the Euclidean norm and $\sqrt{\cdot}$ the nonnegative square root — the energy $\mathcal{E}$, the reals $c_{\mathcal{E}}>0$, $\delta>0$, $\theta_{\mathrm{out}}>0$ and the clipped-out time $\mathcal{O}$ of the extended good-set stopping-time lemma adopted there, the martingale part $M=(M^1,\dots,M^l)$ with the nondecreasing integrals $\mathcal{M}_t$ of claim 1 of the pathwise tracking lemma adopted there (only claims 1 and 2 of that lemma are used here, so its population cost data may be taken arbitrary (for instance identically zero, which is convex in the control) and its hypothesis (LipC) is not assumed), the state fluctuation process $\mathfrak{s}_t=\sqrt{N}(\Sigma_t-S_t)$, the constants $c_M$, $\kappa_T$, $K_M$, $\kappa_0$ and $c_Q$, and the random variables $\overline{M}$, $I$ and $Q\ge0$, so that $\mathbb{E}[Q^4]\le c_Q\,\kappa_0\,N^{-2}$ by claim 2 of the pre-stopping envelope lemma, $\mathbb{E}$ being the expectation. The barrier $\varepsilon_Y>0$ of the pre-stopping envelope lemma and its stopping time $\tau^*$ play no role in the conclusions below.

Fix a real number $t_0\in[0,T]$ and a real number $\varepsilon_1$, and let $\sigma_Y$, $\sigma_{\mathcal{E}}$, $\sigma_{\mathrm{out}}$ and the anchored good-set clock $\sigma^*=\min(\sigma_Y,\sigma_{\mathcal{E}},\sigma_{\mathrm{out}})$ be as in the anchored good-set clocks lemma, applied to the same data — the same transition-rate family, horizon, solution, realized control, flow $\Phi$, map $S^*=S$, deviation $Y$, map $A$, energy $\mathcal{E}$, reals $\delta$, $\theta_{\mathrm{out}}$, $c_{\mathcal{E}}$, clipped-out time $\mathcal{O}$, and point $x_0$ — with the anchor $t_0$ and the level $\varepsilon_1$. Write $\mathbf{1}_{D}$ for the function equal to $1$ on a set $D$ and $0$ off it.

Then the following hold.

1. (Anchored envelope.) For every $\omega\in\Omega_0$ with $Y_{t_0}(\omega)<\varepsilon_1$ and every $t\in[t_0,T]$, writing $u=\min(t,\sigma^*(\omega))$,

$$\bigl|\mathfrak{s}_u(\omega)\bigr|\ \le\ \sqrt{N}\Bigl(\varepsilon_1+|M_u(\omega)|+\Lambda_be^{\Lambda_bT}\mathcal{M}_u(\omega)+e^{\Lambda_bT}N^{-1/2}|\mathfrak{s}_0(\omega)|\Bigr)\ \le\ \sqrt{N}\bigl(\varepsilon_1+Q(\omega)\bigr);$$

in particular $\mathbf{1}_{\{t<\sigma^*\}}(\omega)\,|\mathfrak{s}_t(\omega)|\le\sqrt{N}\bigl(\varepsilon_1+Q(\omega)\bigr)$ for every $t\in[t_0,T]$ and every $\omega\in\Omega_0$ with $Y_{t_0}(\omega)<\varepsilon_1$.

2. (Measurability.) For all $s,t\in[t_0,T]$ the functions $\mathbf{1}_{\Omega_0}|\mathfrak{s}_{\min(t,\sigma^*)}|^2$ (with $\mathfrak{s}_{\min(t,\sigma^*)}$ the componentwise sampled function, $\mathfrak{s}^\gamma_{\min(t,\sigma^*)}(\omega)=\mathfrak{s}^\gamma_{\min(t,\sigma^*(\omega))}(\omega)$) and $\mathbf{1}_{\Omega_0}\mathbf{1}_{\{s<\sigma^*\}}|\mathfrak{s}_s|^2$ are random variables; moreover $\{Y_{t_0}<\varepsilon_1\}\in\mathcal{G}_{t_0}$.

3. (Restricted second and fourth moments.) For all $s,t\in[t_0,T]$ and every event $D\in\mathcal{F}$ with $D\subseteq\{Y_{t_0}<\varepsilon_1\}$,

$$\mathbb{E}\bigl[\mathbf{1}_{D}\mathbf{1}_{\Omega_0}\,|\mathfrak{s}_{\min(t,\sigma^*)}|^2\bigr]\ \le\ N\,\mathbb{E}\bigl[\mathbf{1}_{D}(\varepsilon_1+Q)^2\bigr]\ \le\ 2N\varepsilon_1^2\,P(D)+2\,c_Q^{1/2}\kappa_0^{1/2}\,P(D)^{1/2},$$

and the same bounds hold with $\mathbf{1}_{\Omega_0}|\mathfrak{s}_{\min(t,\sigma^*)}|^2$ replaced by $\mathbf{1}_{\Omega_0}\mathbf{1}_{\{s<\sigma^*\}}|\mathfrak{s}_s|^2$; moreover

$$\mathbb{E}\bigl[\mathbf{1}_{D}\mathbf{1}_{\Omega_0}\,|\mathfrak{s}_{\min(t,\sigma^*)}|^4\bigr]\ \le\ N^2\,\mathbb{E}\bigl[\mathbf{1}_{D}(\varepsilon_1+Q)^4\bigr]\ \le\ 8N^2\varepsilon_1^4\,P(D)+8\,c_Q\,\kappa_0 .$$