The Block Cascade of Anchored Good-Set Clocks: Adapted Good Sets, Matched Escape Bounds, and the Energy Ledger

Adopt the setting, notation and conventions 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 $R$ and constant $K_{1}$, and $K_{2}=2\sqrt{l}\,(l-1)K_{1}$; the horizon $T>0$; the solution of the controlled $N$-agent dynamics with regular event $\Omega_{0}$, observation filtration $(\mathcal{G}_{t})_{t\in[0,T]}$ and system filtration $(\mathcal{F}^{\mathrm{sys}}_{t})_{t\in[0,T]}$; the realized control $\hat{\alpha}$ of the realized-control lemma; the point $x_{0}$ of the probability simplex $\Delta^{l}$; the realized mean-field flow $\Phi$; the map $S^{*}:[0,T]\to\mathbb{R}^{l}$ with continuous components and the deviation $Y_{t}=|\Phi_{t}-S^{*}_{t}|$, $|\cdot|$ being the Euclidean norm; the map $A:[0,T]\to\mathcal{A}$ with measurable components and the control energy $\mathcal{E}_{t}(\omega)=\int_{[0,t]}|\hat{\alpha}(s,\omega)-A_{s}|^{2}\,ds$; the real number $\delta>0$ and the clipped-out time $\mathcal{O}$ of that lemma. Stopping times are those of whichever of $(\mathcal{G}_{t})_{t\in[0,T]}$ and $(\mathcal{F}^{\mathrm{sys}}_{t})_{t\in[0,T]}$ is named; $\mathbb{E}$ is the expectation; $\mathbf{1}_{D}$ is the function equal to $1$ on a set $D$ and $0$ off it; $\int_{[0,t]}\cdot\,ds$ is the Lebesgue integral over a compact interval; $\exp$ is the exponential function and $\sqrt{\cdot}$ the nonnegative square root. Set $C_{a}=\sqrt{l}\,e^{\sqrt{l}\,\Lambda_{b}T}$, the constant of the anchored good-set clocks lemma.

Standing hypothesis on $S^{*}$. Assume, as in claim 4 of the extended good-set stopping-time lemma and of the anchored good-set clocks 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 $t\in[0,T]$ and $\gamma\in\{1,\dots,l\}$.

Block and level data. Let $T_{0}$ be a real number with $0<T_{0}\le T$, let $K$ be the least natural number with $K\,T_{0}\ge T$ (which exists by the Archimedean property), and put $t_{k}=\min(k\,T_{0},T)$ for $k\in\{0,1,\dots,K\}$ and $h_{k}=t_{k+1}-t_{k}$ for $k\in\{0,\dots,K-1\}$. Let $\varepsilon_{1}>0$, $\lambda_{c}>0$ and $\lambda_{o}>0$ be real numbers, and for $k\in\{0,\dots,K-1\}$ set

$$L_{k}=(2C_{a})^{k+1-K}\,\varepsilon_{1},\qquad c^{(k)}_{\mathcal{E}}=\frac{L_{k}^{2}}{\lambda_{c}},\qquad \theta^{(k)}_{\mathrm{out}}=\frac{L_{k}^{2}}{\delta^{2}\lambda_{o}},\qquad \Lambda_{\star}=\max\bigl(4\,C_{a}^{2}K_{2}^{2}\,T_{0},\ \lambda_{c},\ \lambda_{o}\bigr).$$

The cascade of clocks. For $k\in\{0,\dots,K-1\}$ let $\sigma^{(k)}$ be the anchored good-set clock of the anchored good-set clocks lemma with anchor $t_{k}$ and level $L_{k}$, formed for the instance of the setting above in which the energy threshold is $c^{(k)}_{\mathcal{E}}$ and the clipped-out threshold is $\theta^{(k)}_{\mathrm{out}}$, all other data — the rate family, the horizon, the solution, the realized control, the point $x_{0}$, the flow $\Phi$, the map $S^{*}$, the deviation $Y$, the map $A$, the energy $\mathcal{E}$, the real $\delta$ and the clipped-out time $\mathcal{O}$ — being those fixed above and independent of the two thresholds. Define the good sets and the leave events by

$$G_{0}=\Omega_{0},\qquad G_{k+1}=G_{k}\cap\{\sigma^{(k)}\ge t_{k+1}\},\qquad D_{k}=G_{k}\cap\{\sigma^{(k)}<t_{k+1}\}\qquad(k\in\{0,\dots,K-1\}),$$

and the block energy increments $\Delta_{k}\mathcal{E}=\mathcal{E}_{\min(\sigma^{(k)},\,t_{k+1})}-\mathcal{E}_{t_{k}}$, each a random variable with values in $[0,4R^{2}h_{k}]$ by claim 6 of the anchored clocks lemma.

Then the following hold.

1. (Grid.) $K\ge1$, $t_{0}=0$, $t_{K}=T$, and $0<h_{k}\le T_{0}$ with $t_{k}<t_{k+1}$ for every $k\in\{0,\dots,K-1\}$. Moreover $0<L_{k}\le\varepsilon_{1}$, $L_{K-1}=\varepsilon_{1}$, and $L_{k}=2C_{a}L_{k-1}$ for $k\in\{1,\dots,K-1\}$; and $2C_{a}>1$.

2. (Adaptedness and partition.) Each $\sigma^{(k)}$ is a stopping time of $(\mathcal{G}_{t})_{t\in[0,T]}$ and of $(\mathcal{F}^{\mathrm{sys}}_{t})_{t\in[0,T]}$ with values in $[t_{k},T]$. For every $k\in\{0,\dots,K\}$ the set $G_{k}$ is an event belonging to $\mathcal{F}^{\mathrm{sys}}_{t_{k}}$, and $G_{0}\supseteq G_{1}\supseteq\dots\supseteq G_{K}$. For every $k\in\{0,\dots,K-1\}$ the leave event $D_{k}$ satisfies $D_{k}\subseteq\Omega_{0}$ and $D_{k}\in\mathcal{F}_{\sigma^{(k)}}$, the $\sigma$-algebra of events prior to $\sigma^{(k)}$ for $(\mathcal{F}^{\mathrm{sys}}_{t})_{t\in[0,T]}$. The sets $D_{0},\dots,D_{K-1},G_{K}$ are pairwise disjoint with union $\Omega_{0}$.

3. (Entry caps.) For every $k\in\{0,\dots,K-1\}$,

$$G_{k}\ \subseteq\ \Bigl\{Y_{t_{k}}\le\frac{L_{k}}{2C_{a}}\Bigr\}\ \subseteq\ \{Y_{t_{k}}<L_{k}\}.$$

4. (Matched escape bound.) For every $k\in\{0,\dots,K-1\}$,

$$L_{k}^{2}\,P(D_{k})\ \le\ \Lambda_{\star}\,\mathbb{E}\bigl[\mathbf{1}_{G_{k}}\,\Delta_{k}\mathcal{E}\bigr].$$

5. (Energy telescope.) $\displaystyle\sum_{k=0}^{K-1}\mathbb{E}\bigl[\mathbf{1}_{G_{k}}\,\Delta_{k}\mathcal{E}\bigr]\ \le\ \mathbb{E}\bigl[\mathbf{1}_{\Omega_{0}}\,\mathcal{E}_{T}\bigr].$

6. (The ledger.) Assume in addition:

(EB) (Tracked energy bound.) $C_{\dagger}$ is a real number with

$$N\sum_{k=0}^{K-1}\mathbb{E}\bigl[\mathbf{1}_{G_{k}}\,\Delta_{k}\mathcal{E}\bigr]\ \le\ C_{\dagger}.$$

By claim 5 it is sufficient — but not necessary — that $N\,\mathbb{E}[\mathbf{1}_{\Omega_{0}}\mathcal{E}_{T}]\le C_{\dagger}$; the left-hand side above counts only the energy accrued while the path is still tracked, and is the quantity the applications bound.

Then $C_{\dagger}\ge0$ and

$$\sum_{k=0}^{K-1}N\,L_{k}^{2}\,P(D_{k})\ \le\ \Lambda_{\star}\,C_{\dagger},\qquad\qquad \sum_{k=0}^{K-1}\sqrt{N}\;P(D_{k})^{3/4}\ \le\ \bigl(\Lambda_{\star}C_{\dagger}\bigr)^{3/4}\,N^{-1/4}\,\Xi_{K},$$

where $x^{3/4}=(x^{1/4})^{3}$ for $x\ge0$ as in the restricted moments lemma and

$$\Xi_{K}=\sum_{k=0}^{K-1}L_{k}^{-3/2}=\varepsilon_{1}^{-3/2}\,\frac{(2C_{a})^{3K/2}-1}{(2C_{a})^{3/2}-1},$$

where, $x^{1/2}$ denoting the nonnegative square root, $x^{3/2}=(x^{1/2})^{3}$ for $x\ge0$, and, for $x>0$, $x^{-3/2}$ and $x^{-1/4}$ are the reciprocals of $x^{3/2}$ and of $x^{1/4}$ respectively.