Anchored Good-Set Clocks on a Subinterval and the Block Escape Bound

Adopt the setting, notation and definitions of the extended good-set stopping-time 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 $B$, aggregate state drift $b$, state-Lipschitz constant $\Lambda_b$, control bound $R$, the constant $K_1$ of that lemma, and $K_2=2\sqrt{l}\,(l-1)K_1$; the horizon $T>0$; the solution of the controlled $N$-agent dynamics with 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 set $\mathcal{U}_{\mathcal{A}}$, the flow notation $S(z_0,\xi)$, and the admissible representatives of the extended lemma, together with the element $\zeta_A$ of $\mathcal{U}_{\mathcal{A}}$ represented by $A$ (claim 4 of the extended lemma); the point $x_0$ of the probability simplex $\Delta^l$, the realized mean-field flow $\Phi$, the map $S^*$ with continuous components and the deviation $Y_t=|\Phi_t-S^*_t|$; the map $A:[0,T]\to\mathcal{A}$ with measurable components and the energy $\mathcal{E}$; the reals $\delta>0$, $\theta_{\mathrm{out}}>0$, $c_{\mathcal{E}}>0$ and the clipped-out time $\mathcal{O}$ of that lemma; and its conventions for stopping times, order-relation events, Lebesgue integrals over compact intervals, the expectation $\mathbb{E}$, and continuity relative to subintervals of the real numbers with the metric of the real line.

Fix a real number $t_0\in[0,T]$, called the anchor, and a real number $\varepsilon_1$, and set $C_a=\sqrt{l}\,e^{\sqrt{l}\,\Lambda_bT}$, with $\exp$ the exponential function and $\sqrt{\cdot}$ the nonnegative square root. For $\omega\in\Omega$ put

and define $\sigma_Y(\omega)$, $\sigma_{\mathcal{E}}(\omega)$, $\sigma_{\mathrm{out}}(\omega)$ to be the greatest lower bounds of these sets when they are nonempty (existing by the existence theorem for infima, the sets being bounded below by $t_0$) and $T$ otherwise, and the anchored good-set clock

Then the following hold.

1. (Stopping.) Each of $\sigma_Y$, $\sigma_{\mathcal{E}}$, $\sigma_{\mathrm{out}}$ and $\sigma^*$ takes values in $[t_0,T]$ and is a stopping time of $(\mathcal{G}_t)_{t\in[0,T]}$ and of $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$, and $\{t<\sigma^*\}\in\mathcal{G}_t$ for every $t\in[0,T]$. Moreover $\{\sigma_{\mathcal{E}}\le q\}=\{\mathcal{E}_q-\mathcal{E}_{t_0}\ge c_{\mathcal{E}}\}$ and $\{\sigma_{\mathrm{out}}\le q\}=\{\mathcal{O}_q-\mathcal{O}_{t_0}\ge\theta_{\mathrm{out}}\}$ for every $q\in[t_0,T)$.

2. (Pre- and stopped bounds.) For every $\omega\in\Omega$: if $t\in[t_0,T]$ and $t<\sigma^*(\omega)$ then

for every $t\in[t_0,T]$, writing $u=\min(t,\sigma^*(\omega))$, one has $\mathcal{E}_u(\omega)-\mathcal{E}_{t_0}(\omega)\le c_{\mathcal{E}}$ and $\mathcal{O}_u(\omega)-\mathcal{O}_{t_0}(\omega)\le\theta_{\mathrm{out}}$, and if $Y_{t_0}(\omega)<\varepsilon_1$ then also $Y_u(\omega)\le\varepsilon_1$. Moreover $\delta^2\bigl(\mathcal{O}_t(\omega)-\mathcal{O}_{t_0}(\omega)\bigr)\le\mathcal{E}_t(\omega)-\mathcal{E}_{t_0}(\omega)$ for all $t\in[t_0,T]$.

3. (Hitting values.) For every $\omega\in\Omega$: if $\sigma_Y(\omega)<T$ then $Y_{\sigma_Y(\omega)}(\omega)\ge\varepsilon_1$; if $\sigma_{\mathcal{E}}(\omega)<T$ then $\mathcal{E}_{\sigma_{\mathcal{E}}(\omega)}(\omega)-\mathcal{E}_{t_0}(\omega)\ge c_{\mathcal{E}}$; and if $\sigma_{\mathrm{out}}(\omega)<T$ then $\mathcal{O}_{\sigma_{\mathrm{out}}(\omega)}(\omega)-\mathcal{O}_{t_0}(\omega)\ge\theta_{\mathrm{out}}$.

4. (Anchored deviation-energy bound.) Suppose, as in claim 4 of the extended good-set stopping-time lemma, that $S^*$ takes values in $\Delta^l$, that $S^*_0=x_0$, and that $S^{*\gamma}_t=x^\gamma_0+\int_{[0,t]}b^\gamma(S^*_s,A_s)\,ds$ for all $t$ and $\gamma$. Then for every $\omega\in\Omega$ and all $t_0\le t\le T$,

5. (Block escape bound.) Assume the hypotheses of claim 4 and $\varepsilon_1>0$, and set

then $m^*_b>0$. The function $\omega\mapsto\mathcal{E}_{\sigma^*(\omega)}(\omega)-\mathcal{E}_{t_0}(\omega)$ is a random variable with values in $[0,4R^2T]$, and

consequently, for every event $D\in\mathcal{F}$ with $D\subseteq\{Y_{t_0}\le\varepsilon_1/(2C_a)\}$,

where $\mathbf{1}_{D}$ is the function equal to $1$ on $D$ and $0$ off it.

6. (Truncated block escape bound.) Assume the hypotheses of claim 4 and $\varepsilon_1>0$, and let $h$ be a real number with $0<h\le T-t_0$. Set

then $m^{*,h}_b>0$. The function $\omega\mapsto\mathcal{E}_{\min(\sigma^*(\omega),\,t_0+h)}(\omega)-\mathcal{E}_{t_0}(\omega)$ is a random variable with values in $[0,4R^2h]$, and

consequently, for every event $D\in\mathcal{F}$ with $D\subseteq\{Y_{t_0}\le\varepsilon_1/(2C_a)\}$,