The Tracked Energy Bound for the Block Cascade under Local Joint Coercivity

Adopt simultaneously the settings and notation of the block cascade lemma, of the anchored pre-stopping envelope lemma, of the localized joint coercivity lemma and of the post-exit comparison lemma, all formed for one and the same data: the affine-controlled transition-rate family $(\beta_{0},\beta_{1})$ on $l$ states with compact convex control set $\mathcal{A}$ and its transition-rate family $\beta$ with rate bound $B$, control bound $R$ and state-Lipschitz constant $\Lambda_{b}$; the horizon $T>0$; the population cost data $(L,G)$; the solution with regular event $\Omega_{0}$, empirical state measure $\Sigma$, control $\alpha$ and system filtration $(\mathcal{F}^{\mathrm{sys}}_{t})_{t\in[0,T]}$; the realized control $\hat{\alpha}$; the mean-field trajectory pair $(S,A)$ together with the extensions and the stationary co-state $P$, so that $(S,A,P)$ is a stationary mean-field triple; the point $x_{0}=S_{0}$ of the probability simplex, the realized mean-field flow $\Phi$, the choice $S^{*}=S$, the deviation $Y_{t}=|\Phi_{t}-S_{t}|$, the energy $\mathcal{E}$, the reals $\delta>0$ and the clipped-out time $\mathcal{O}$; the fluctuation processes $\mathfrak{s}$, $\mathfrak{a}$ and $\mathfrak{z}=(\mathfrak{s},\mathfrak{a})$; the noise majorant $Q\ge0$ with $\mathbb{E}[Q^{4}]\le c_{Q}\kappa_{0}N^{-2}$; and the quantities $\mathcal{D}_{t}$, $\mathcal{D}_{G}$, $\mathcal{J}_{N}$ with the bounds $C_{\mathcal{D}}$, $C_{\mathcal{D}G}$ and the constant $C_{3}=C_{2}+C_{1}^{2}/(2r_{0})$ of the first-order expansion lemma for the recentred $N$-agent cost. From the block cascade lemma adopt the block data $T_{0}$, $K$, $t_{k}$, the levels $L_{k}$, the matched thresholds $c^{(k)}_{\mathcal{E}}=L_{k}^{2}/\lambda_{c}$ and $\theta^{(k)}_{\mathrm{out}}=L_{k}^{2}/(\delta^{2}\lambda_{o})$, the constant $\Lambda_{\star}$, the clocks $\sigma^{(k)}$, the good sets $G_{k}$, the leave events $D_{k}$, the increments $\Delta_{k}\mathcal{E}$ and the quantity $\Xi_{K}$; from the localized coercivity lemma adopt the radius $\rho^{*}$ of its claim 2 and the constant $c_{J}$; and from the post-exit comparison lemma adopt $C^{\vee}_{tg}$, $C_{ns}$ and $\varepsilon_{tg}$. Write $\mathbf{1}_{D}$ for the indicator of a set $D$ and $\mathbb{E}$ for the expectation, and let $C_{S}=e^{\Lambda_{b}T}\sqrt{l}\,K_{2}\sqrt{T}$ be the constant of claim 4 of the extended good-set stopping-time lemma.

Standing hypotheses. Hypothesis (A) and hypothesis (U) of the quadratic growth lemma; hypothesis (JC) of the localized coercivity lemma, with constant $c_{J}>0$ and with $r_{0}>0$ the constant furnished by claim 5 of that lemma together with (U); hypothesis (LipC) of the pathwise tracking lemma; the optimality $[A]\in\mathcal{M}^{*}_{S_{0}}$ and hypothesis (TG) of the to-go comparison lemma; and the standing hypothesis on $S^{*}$ of the block cascade lemma.

Parameters. Let $q_{0}>0$ be a real number, put $\mathcal{N}=\{Q>q_{0}\}$, and set

$$c_{\star}=\min\Bigl(\frac{c_{J}}{2},\ \frac{r_{0}}{4}\Bigr),\qquad \mathcal{Z}=N\sum_{k=0}^{K-1}\mathbb{E}\bigl[\mathbf{1}_{G_{k}}\,\Delta_{k}\mathcal{E}\bigr].$$

(SM) (Smallness.) Assume

$$(\varepsilon_{1}+q_{0})^{2}+\delta^{2}\le(\rho^{*})^{2},\qquad C_{3}\,(\varepsilon_{1}+q_{0})^{2}\le\frac{r_{0}}{4}\,\delta^{2},\qquad \varepsilon_{1}+q_{0}\le\varepsilon_{tg}.$$

Then the following hold.

1. (Finiteness and the noise tail.) $0\le\mathcal{Z}\le4R^{2}TN<\infty$, the set $\mathcal{N}$ is an event with $P(\mathcal{N})\le c_{Q}\kappa_{0}\,q_{0}^{-4}N^{-2}$, and

$$N\sum_{k=0}^{K-1}\mathbb{E}\bigl[\mathbf{1}_{G_{k}\cap\mathcal{N}}\,\Delta_{k}\mathcal{E}\bigr]\ \le\ 4R^{2}T\,c_{Q}\kappa_{0}\,q_{0}^{-4}\,N^{-1}.$$

2. (Pointwise coercivity on the tracked region.) Let $k\in\{0,\dots,K-1\}$, let $\omega\in G_{k}\setminus\mathcal{N}$ and let $t\in[t_{k},T]$ with $t<\sigma^{(k)}(\omega)$. Then $N\mathcal{D}_{t}(\omega)\ \ge\ c_{\star}\,|\mathfrak{a}_{t}(\omega)|^{2}$.

3. (Terminal control on the surviving set.) $\mathbb{E}\bigl[\mathbf{1}_{G_{K}}|\mathfrak{s}_{T}|^{2}\bigr]\ \le\ 2\,C_{S}^{2}\,\mathcal{Z}+2\,c_{Q}^{1/2}\kappa_{0}^{1/2}$.

4. (Skeleton inequality.) Let $\epsilon>0$ be a real number and let $\rho^{*}_{G}>0$ be the radius furnished for it by claim 4 of the localized coercivity lemma, and assume in addition $\varepsilon_{1}+q_{0}\le\rho^{*}_{G}$. Then

$$c_{\star}\,\mathcal{Z}\ \le\ \mathcal{J}_{N}\ +\ 2\,C^{\vee}_{tg}\,\Lambda_{\star}\,\mathcal{Z}\ +\ 2\,\epsilon\,C_{S}^{2}\,\mathcal{Z}\ +\ C_{ns}\,\bigl(\Lambda_{\star}\mathcal{Z}\bigr)^{3/4}N^{-1/4}\,\Xi_{K}\ +\ \mathcal{R}_{N},$$

where the residual

$$\mathcal{R}_{N}=2\,C^{\vee}_{tg}\,c_{Q}^{1/2}\kappa_{0}^{1/2}\,\sqrt{K}\ +\ 2\,\epsilon\,c_{Q}^{1/2}\kappa_{0}^{1/2}\ +\ 4R^{2}T\,c_{\star}\,c_{Q}\kappa_{0}\,q_{0}^{-4}N^{-1}\ +\ 2\,N\bigl(C_{\mathcal{D}}T+C_{\mathcal{D}G}\bigr)\,c_{Q}\kappa_{0}\,q_{0}^{-4}N^{-2}$$

is a finite real number depending on $N$ only through the last two terms, which are $O(N^{-1})$.

5. (The tracked energy bound.) Assume in addition the absorption condition

$$2\,C^{\vee}_{tg}\,\Lambda_{\star}+2\,\epsilon\,C_{S}^{2}\ \le\ \frac{c_{\star}}{4},$$

and assume $\mathcal{J}_{N}\le\mathcal{J}^{\sharp}$ for a real number $\mathcal{J}^{\sharp}$. Then

$$\mathcal{Z}\ \le\ C_{\dagger}:=\frac{2}{c_{\star}}\bigl(\mathcal{J}^{\sharp}+\mathcal{R}_{N}\bigr)\ +\ \frac{128\,\bigl(C_{ns}\,\Lambda_{\star}^{3/4}\,\Xi_{K}\bigr)^{4}}{c_{\star}^{4}},$$

and consequently hypothesis (EB) of the block cascade lemma holds with this constant $C_{\dagger}$.