Ledger Decomposition of the Recentred N-Agent Cost over the Block Cascade and Its Near-Field Filtering Lower Bound

Adopt the setting, notation, parameters and standing hypotheses of the tracked energy bound lemma, formed for one and the same data as there. In particular: the affine-controlled transition-rate family $(\beta_{0},\beta_{1})$ on $l$ states with compact convex control set $\mathcal{A}\subseteq\mathbb{R}^{m}$ and control bound $R$, and its transition-rate family $\beta$ with rate bound $B$; the horizon $T>0$; the population cost data $(L,G)$ with its twice continuously differentiable extension, and the twice continuously differentiable extension $(U,V,\bar{\beta})$ of $\beta$ (whose derivative bound is not used below, so that the letter $K$ is free here for the number of blocks); the solution of the controlled $N$-agent dynamics with regular event $\Omega_{0}$, empirical state measure $\Sigma$, control $\alpha$, observation filtration $(\mathcal{G}_{t})_{t\in[0,T]}$ and system filtration $(\mathcal{F}^{\mathrm{sys}}_{t})_{t\in[0,T]}$; the stationary mean-field triple $(S,A,P)$ with the bound $C_{P}$; the fluctuation processes $\mathfrak{s}$, $\mathfrak{a}$ and $\mathfrak{z}=(\mathfrak{s},\mathfrak{a})$; the realized mean-field flow $\Phi$, the deviation $Y$, the energy $\mathcal{E}$, the real $q_{0}>0$, and the clipping threshold of the extended good-set stopping-time lemma, written $\delta$ there and $\delta_{\mathrm{cl}}$ here because the letter $\delta$ is reserved below for a state index; the noise majorant $Q$ with $\mathbb{E}[Q^{4}]\le c_{Q}\kappa_{0}N^{-2}$ and the noise event $\mathcal{N}=\{Q>q_{0}\}$; the block data $T_{0}$, $K$, $t_{k}$, $h_{k}$, the levels $L_{k}$ and the reals $\varepsilon_{1}$, $\lambda_{c}$, $\lambda_{o}$, the constants $C_{a}$, $\Lambda_{\star}$ and $C_{S}$, the anchored clocks $\sigma^{(k)}$, the good sets $G_{k}$, the leave events $D_{k}$, the block energy increments $\Delta_{k}\mathcal{E}$ and the tracked energy $\mathcal{Z}=N\sum_{k=0}^{K-1}\mathbb{E}[\mathbf{1}_{G_{k}}\Delta_{k}\mathcal{E}]$; the quantities $\mathcal{D}_{t}$, $\mathcal{D}_{G}$ and $\mathcal{J}_{N}$ with the bounds $C_{\mathcal{D}}$, $C_{\mathcal{D}G}$ and the constant $C_{3}$ of the first-order expansion lemma for the recentred $N$-agent cost; the constants $c_{J}$, $\rho^{*}$, $r_{0}$, $c_{\star}$, $C^{\vee}_{tg}$, $C_{ns}$, $\varepsilon_{tg}$, the real $\epsilon>0$ and the radius $\rho^{*}_{G}$; and the moduli of continuity $\omega_{L}$, $\omega_{b}$, $\omega_{G}$ and the radius $\rho_{t}=d((\Sigma_{t},\alpha_{t}),(S_{t},A_{t}))$ of the second-order expansion theorem, $d$ being the Euclidean distance and $|\cdot|$ the Euclidean norm. Adopt likewise the standing hypotheses (A), (U), (JC) with constant $c_{J}>0$, (LipC), the optimality $[A]\in\mathcal{M}^{*}_{S_{0}}$, (TG), the standing hypothesis on $S^{*}=S$ of the block cascade lemma, the smallness hypothesis (SM), and the additional requirement $\varepsilon_{1}+q_{0}\le\rho^{*}_{G}$ imposed in claim 4 of the tracked energy bound lemma.

Adopt in addition the setting and notation of the stopped completion-of-squares lemma and of the cascade filtering lemma, formed for the same data, and with them the setting of the completion-of-squares theorem: the fluctuation Hessian coefficients $H_{ij}(t)$ ($i,j\in\{1,\dots,l+m\}$) and terminal coefficients $F_{\gamma\delta}$; the matrices $E_{t}$, $\mathsf{B}_{t}$, $Q_{t}$, $V_{t}$, $R_{t}$ and $\hat{F}$ and the entry pairing $x\cdot My=\sum_{p,q}M^{pq}x^{p}y^{q}$ defined there; hypotheses (H1) and (H2) with the Riccati family $Z=(Z_{t})_{t\in[0,T]}$, $W_{t}=Z_{t}\mathsf{B}_{t}+\tfrac{1}{2}V_{t}$ and $Z_{T}=\hat{F}$; the processes $u_{t}=\mathfrak{a}_{t}+R_{t}^{-1}W_{t}^{T}\mathfrak{s}_{t}$ and $e_{s}=g_{s}-E_{s}\mathfrak{s}_{s}-\mathsf{B}_{s}\mathfrak{a}_{s}$; the constants $C_{Z}$ and $c_{e}$; and, from the cascade filtering lemma, the matrix family $\Xi_{t}=W_{t}R_{t}^{-1}W_{t}^{T}$ and the tracked events of its claim 1. Adopt also the aggregate fluctuation covariance $\Theta$ of $\beta$, write $\Theta^{\star}_{s}=\Theta(S_{s},A_{s})$, and adopt the constant $c_{\Theta}$ of the covariance deviation lemma. Hypothesis (H1) holds, with $r=c_{J}$, by claim 5 of the localized joint coercivity lemma under (JC); hypothesis (H2) is assumed here.

Two notational collisions are resolved as follows. The matrices $Q_{t}$ of the completion-of-squares theorem always carry their time subscript and are distinct from the noise majorant $Q$, which never carries one; likewise $V_{t}$ is distinct from the control-side open set $V$ of the extension $(U,V,\bar{\beta})$, and $R_{t}$ from the control bound $R$. The quantity written $\Xi_{K}$ in claim 6 of the block cascade lemma is here written $\Upsilon_{\mathrm{esc}}$, the letter $\Xi$ being reserved for the matrix family $\Xi_{t}$ above.

Near-field parameters. Let $\eta>0$ and $\varrho>0$ be real numbers, and let $\rho_{\eta}>0$ be a real number such that

$$\tfrac{1}{2}(l+m)\bigl(\omega_{L}(v)+C_{P}\,\omega_{b}(v)\bigr)\ \le\ \eta\qquad\text{for every }v\in[0,\rho_{\eta}],$$

which exists by part (a) of the second-order expansion theorem, exactly as the radius of claim 2 of the localized joint coercivity lemma is obtained there. Assume the near-field smallness condition

$$\textbf{(SN)}\qquad (\varepsilon_{1}+q_{0})^{2}+\varrho^{2}\ \le\ \rho_{\eta}^{2}.$$

The radius $\rho^{*}_{G}$ is fixed here more precisely than in the tracked energy bound lemma: let $\rho^{*}_{G}>0$ be a real number with $\tfrac{1}{2}\,l\,\omega_{G}(v)\le\epsilon$ for every $v\in[0,\rho^{*}_{G}]$, which exists by part (a) of the second-order expansion theorem. By the terminal half of claim 1 of the localized joint coercivity lemma such a number has the property demanded of the radius in claim 4 of that lemma, so this is an admissible choice of the $\rho^{*}_{G}$ appearing in claim 4 of the tracked energy bound lemma.

Constants. Each of the families $t\mapsto H_{ij}(t)$, $t\mapsto W^{\gamma j}_{t}$, $t\mapsto V^{\gamma j}_{t}$, $t\mapsto Q^{\gamma\delta}_{t}$, $t\mapsto(W_{t}R_{t}^{-1}W_{t}^{T})^{\gamma\delta}$ and $s\mapsto\Theta^{\star\gamma\delta}_{s}$ is continuous on $[0,T]$ — the first by the extension definitions together with the continuity of $t\mapsto(S_{t},A_{t})$ and of $t\mapsto P_{t}$, the next four by conclusion (a) of the completion-of-squares theorem and (H2), and the last by clause (c) of the covariance deviation lemma — and hence bounded there by the extreme value theorem. Fix reals $C_{H}\ge0$, $C_{M}\ge0$ and $\bar{\Theta}\ge0$ bounding, in absolute value, the first family, the next four families, and the last family respectively, for all indices and all points of $[0,T]$, and set

$$C_{\mathfrak{h}}=\tfrac{1}{2}(l+m)\,C_{H},\qquad C_{\Psi}=\bigl(3\sqrt{lm}+2l\bigr)C_{M},\qquad C_{\mathcal{N}}=2TC_{\mathcal{D}}+2C_{\mathcal{D}G}+4lC_{Z}+4T(1+R^{2})\bigl(C_{\mathfrak{h}}+4l\,C_{Z}\,c_{e}\bigr).$$

Derived quantities. Write $\mathbf{1}_{D}$ for the indicator of a set $D$, $\mathbb{E}$ for the expectation, $P$ for the probability of the driving system (the stationary co-state, also written $P$, always carries a time subscript), $\int_{[a,b]}\cdot\,ds$ for the Lebesgue integral over a compact interval, and, for a nonnegative real $x$, $x^{1/2}$ for the nonnegative square root and $x^{3/4}=(x^{1/4})^{3}$ as in the restricted moments lemma. For $k\in\{0,\dots,K-1\}$ and $s\in[0,T]$ set

$$\mathcal{T}_{k}(s)=G_{k}\cap\{s<\sigma^{(k)}\},\qquad \mathcal{T}^{\mathrm{nr}}_{k}(s)=\mathcal{T}_{k}(s)\cap\bigl\{|\mathfrak{a}_{s}|\le\sqrt{N}\varrho\bigr\},\qquad \mathcal{T}^{\mathrm{fr}}_{k}(s)=\mathcal{T}_{k}(s)\cap\bigl\{|\mathfrak{a}_{s}|>\sqrt{N}\varrho\bigr\},$$

so that $\mathcal{T}_{k}(s)$ is the union of the disjoint sets $\mathcal{T}^{\mathrm{nr}}_{k}(s)$ and $\mathcal{T}^{\mathrm{fr}}_{k}(s)$; and set

$$\mathfrak{h}_{s}=\tfrac{1}{2}\sum_{i=1}^{l+m}\sum_{j=1}^{l+m}H_{ij}(s)\,\mathfrak{z}^{i}_{s}\mathfrak{z}^{j}_{s},\qquad \Psi_{s}=u_{s}\cdot R_{s}u_{s}-\mathfrak{h}_{s},\qquad \mathfrak{s}^{\sharp}_{k}=\mathfrak{s}_{\min(t_{k+1},\sigma^{(k)})},\qquad Z^{\sharp}_{k}=Z_{\min(t_{k+1},\sigma^{(k)})},$$

the sampled functions being those of the sampled-process definition, taken componentwise, and

$$\mathcal{S}^{\mathrm{tr}}=\sum_{k=0}^{K-1}\int_{[t_{k},t_{k+1}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}_{k}(s)}|\mathfrak{s}_{s}|^{2}\bigr]ds,\qquad \mathcal{P}=\sum_{k=0}^{K-1}P(D_{k}),\qquad \Upsilon_{\mathrm{lev}}=\sum_{k=0}^{K-1}L_{k}^{-2}.$$

Finally set

$$\mathcal{B}_{N}=\bigl(\eta+2l\,C_{Z}\,c_{e}\,(\varepsilon_{1}+q_{0})\bigr)\bigl(\mathcal{S}^{\mathrm{tr}}+\mathcal{Z}\bigr)+C_{\Psi}\,\mathcal{Z}\Bigl(\frac{\varepsilon_{1}+q_{0}}{\varrho}+\frac{(\varepsilon_{1}+q_{0})^{2}}{\varrho^{2}}\Bigr)+2\epsilon\bigl(C_{S}^{2}\mathcal{Z}+c_{Q}^{1/2}\kappa_{0}^{1/2}\bigr) +\ \bigl(C^{\vee}_{tg}+l\,C_{Z}\bigr)\bigl(2\Lambda_{\star}\mathcal{Z}+2c_{Q}^{1/2}\kappa_{0}^{1/2}\mathcal{P}^{1/2}\bigr)+C_{ns}\bigl(\Lambda_{\star}\mathcal{Z}\bigr)^{3/4}N^{-1/4}\,\Upsilon_{\mathrm{esc}} +\ l^{2}C_{Z}\Bigl(\tfrac{1}{2}\,c_{\Theta}\,N^{-1/2}\bigl(T+\mathcal{S}^{\mathrm{tr}}+\mathcal{Z}\bigr)+\bar{\Theta}\,T\,\mathcal{P}\Bigr)\ +\ C_{\mathcal{N}}\,N\,P(\mathcal{N}).$$

Then the following hold.

1. (Tracked moments and escape sums.) All the quantities below are finite, and:

(a) $\displaystyle\sum_{k=0}^{K-1}\int_{[t_{k},t_{k+1}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}_{k}(s)}|\mathfrak{a}_{s}|^{2}\bigr]ds=\mathcal{Z}$.

(b) $\mathcal{S}^{\mathrm{tr}}\le 2\,C_{S}^{2}\,T\,\mathcal{Z}+2\,T\,c_{Q}^{1/2}\kappa_{0}^{1/2}$.

(c) $\displaystyle\sum_{k=0}^{K-1}\int_{[t_{k},t_{k+1}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}^{\mathrm{fr}}_{k}(s)}\bigr]ds\le\frac{\mathcal{Z}}{N\varrho^{2}}$ and $\displaystyle\sum_{k=0}^{K-1}\int_{[t_{k},t_{k+1}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}^{\mathrm{fr}}_{k}(s)}|\mathfrak{a}_{s}|\bigr]ds\le\frac{\mathcal{Z}}{\sqrt{N}\,\varrho}$.

(d) $\mathcal{P}\le\Lambda_{\star}\mathcal{Z}\,\Upsilon_{\mathrm{lev}}\,N^{-1}$, and $\displaystyle\sum_{k=0}^{K-1}\sqrt{N}\,P(D_{k})^{3/4}\le(\Lambda_{\star}\mathcal{Z})^{3/4}N^{-1/4}\,\Upsilon_{\mathrm{esc}}$, and $\displaystyle\sum_{k=0}^{K-1}\int_{[t_{k},t_{k+1}]}\mathbb{E}\bigl[1-\mathbf{1}_{\mathcal{T}_{k}(s)}\bigr]ds\le T\,\mathcal{P}$.

(e) $\displaystyle\sum_{k=0}^{K-1}\mathbb{E}\bigl[\mathbf{1}_{D_{k}}|\mathfrak{s}_{\sigma^{(k)}}|^{2}\bigr]\le 2\Lambda_{\star}\mathcal{Z}+2\,c_{Q}^{1/2}\kappa_{0}^{1/2}\,\mathcal{P}^{1/2}$.

2. (The near-field tracked events are observable.) For every $k\in\{0,\dots,K-1\}$ and every $s\in[t_{k},T]$ the set $\mathcal{T}^{\mathrm{nr}}_{k}(s)$ is an event belonging to $\mathcal{G}_{s}$; consequently, by claim 3 of the cascade filtering lemma, for any choice of conditional expectations defining the filtering error $\varepsilon_{s}$ there,

$$\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}^{\mathrm{nr}}_{k}(s)}\,u_{s}\cdot R_{s}u_{s}\bigr]\ \ge\ \sum_{\gamma=1}^{l}\sum_{\delta=1}^{l}\Xi^{\gamma\delta}_{s}\,\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}^{\mathrm{nr}}_{k}(s)}\,\varepsilon^{\gamma}_{s}\varepsilon^{\delta}_{s}\bigr]\ \ge\ 0 .$$

3. (Master ledger identity.) With $\int_{[\varsigma,T]}X_{t}\,dt$ as in the post-exit comparison lemma,

$$\mathcal{J}_{N}\ =\ \sum_{k=0}^{K-1}\int_{[t_{k},t_{k+1}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}_{k}(s)}\,N\mathcal{D}_{s}\bigr]ds\ +\ \sum_{k=0}^{K-1}\mathbb{E}\Bigl[\mathbf{1}_{D_{k}}\Bigl(N\int_{[\sigma^{(k)},T]}\mathcal{D}_{t}\,dt+N\mathcal{D}_{G}\Bigr)\Bigr]\ +\ \mathbb{E}\bigl[\mathbf{1}_{G_{K}}\,N\mathcal{D}_{G}\bigr],$$

all terms being finite.

4. (Leaver bill.)

$$\sum_{k=0}^{K-1}\mathbb{E}\Bigl[\mathbf{1}_{D_{k}}\Bigl(N\int_{[\sigma^{(k)},T]}\mathcal{D}_{t}\,dt+N\mathcal{D}_{G}\Bigr)\Bigr]\ \ge\ -\,C^{\vee}_{tg}\bigl(2\Lambda_{\star}\mathcal{Z}+2c_{Q}^{1/2}\kappa_{0}^{1/2}\mathcal{P}^{1/2}\bigr)-C_{ns}\bigl(\Lambda_{\star}\mathcal{Z}\bigr)^{3/4}N^{-1/4}\Upsilon_{\mathrm{esc}}-\bigl(TC_{\mathcal{D}}+C_{\mathcal{D}G}\bigr)N\,P(\mathcal{N}).$$

5. (Terminal bill.)

$$\mathbb{E}\bigl[\mathbf{1}_{G_{K}}N\mathcal{D}_{G}\bigr]\ \ge\ \mathbb{E}\bigl[\mathbf{1}_{G_{K}}\,\mathfrak{s}_{T}\cdot Z_{T}\mathfrak{s}_{T}\bigr]-2\epsilon\bigl(C_{S}^{2}\mathcal{Z}+c_{Q}^{1/2}\kappa_{0}^{1/2}\bigr)-\bigl(C_{\mathcal{D}G}+4lC_{Z}\bigr)N\,P(\mathcal{N}).$$

6. (Block bound with the far field cancelled.) For every $k\in\{0,\dots,K-1\}$,

$$\int_{[t_{k},t_{k+1}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}_{k}(s)}N\mathcal{D}_{s}\bigr]ds\ \ge\ \mathbb{E}\bigl[\mathbf{1}_{G_{k}}\,\mathfrak{s}_{t_{k}}\cdot Z_{t_{k}}\mathfrak{s}_{t_{k}}\bigr]-\mathbb{E}\bigl[\mathbf{1}_{G_{k}}\,\mathfrak{s}^{\sharp}_{k}\cdot Z^{\sharp}_{k}\mathfrak{s}^{\sharp}_{k}\bigr]+\int_{[t_{k},t_{k+1}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}^{\mathrm{nr}}_{k}(s)}\,u_{s}\cdot R_{s}u_{s}\bigr]ds +\ \int_{[t_{k},t_{k+1}]}\sum_{\gamma=1}^{l}\sum_{\delta=1}^{l}Z^{\gamma\delta}_{s}\,\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}_{k}(s)}\,\Theta^{\gamma\delta}(\Sigma_{s},\alpha_{s})\bigr]ds\ -\ \mathcal{X}_{k},$$

where

$$\mathcal{X}_{k}=\bigl(\eta+2lC_{Z}c_{e}(\varepsilon_{1}+q_{0})\bigr)\int_{[t_{k},t_{k+1}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}_{k}(s)}|\mathfrak{z}_{s}|^{2}\bigr]ds+C_{\Psi}\int_{[t_{k},t_{k+1}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}^{\mathrm{fr}}_{k}(s)\setminus\mathcal{N}}\bigl(|\mathfrak{s}_{s}||\mathfrak{a}_{s}|+|\mathfrak{s}_{s}|^{2}\bigr)\bigr]ds +\ \Bigl(C_{\mathcal{D}}+4(1+R^{2})\bigl(C_{\mathfrak{h}}+4lC_{Z}c_{e}\bigr)\Bigr)\,h_{k}\,N\,P(\mathcal{N}).$$

7. (Assembled lower bound.)

$$\mathcal{J}_{N}\ \ge\ \mathbb{E}\bigl[\mathbf{1}_{\Omega_{0}}\,\mathfrak{s}_{0}\cdot Z_{0}\mathfrak{s}_{0}\bigr]\ +\ \int_{[0,T]}\sum_{\gamma=1}^{l}\sum_{\delta=1}^{l}Z^{\gamma\delta}_{s}\,\Theta^{\star\gamma\delta}_{s}\,ds\ +\ \sum_{k=0}^{K-1}\int_{[t_{k},t_{k+1}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}^{\mathrm{nr}}_{k}(s)}\,u_{s}\cdot R_{s}u_{s}\bigr]ds\ -\ \mathcal{B}_{N},$$

and in particular, the last sum being nonnegative by claim 2,

$$\mathcal{J}_{N}\ \ge\ \mathbb{E}\bigl[\mathbf{1}_{\Omega_{0}}\,\mathfrak{s}_{0}\cdot Z_{0}\mathfrak{s}_{0}\bigr]+\int_{[0,T]}\sum_{\gamma=1}^{l}\sum_{\delta=1}^{l}Z^{\gamma\delta}_{s}\,\Theta^{\star\gamma\delta}_{s}\,ds\ -\ \mathcal{B}_{N}.$$

No hypothesis on the moments of $\mathfrak{s}$ or $\mathfrak{a}$, uniform in $N$ or otherwise, is used.