Pathwise Estimand Linearisation of an Open-Loop Aggregate Solution Along a Comparison Pair: Cell-Count Coefficients from the Fundamental Solution and the Three Error Terms

Adopt the setting and notation of Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound: natural numbers $l\ge2$, $m\ge1$, the control set $\mathcal{A}\subseteq\mathbb{R}^m$, real numbers $B\ge0$ and $T>0$, the transition-rate family $\beta$ with its twice continuously differentiable extension of derivative bound $K$, the labels $c=(\sigma,\gamma)\in\mathcal{L}$ with vectors $v_c$, the label rates $\psi_c$, state gradients $g^{c}$, drift Jacobian $\mathcal{E}$, drift $b$, the constants $\Lambda_2$, $\Lambda_3$, $\Lambda_{\mathcal{E}}$, the notions measurable and bounded measurable for maps on $[0,T]$, the comparison pair $(S,\mathsf{A})$ with $\mathcal{E}^{\star}_u=\mathcal{E}(S_u,\mathsf{A}_u)$ continuous in $u$, the two-parameter fundamental solution $\Phi^{\mathcal{E}}(t,u)$ and its bound $\bar{\Phi}$, and, for a control path $a$ and paths $x,y$, the objects $e$, $\rho$, $\mathsf{d}$ and $L_t(\cdot)$ defined there. Adopt also from Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks a natural number $N\ge1$, the aggregate lattice $\mathbb{G}_N\subseteq\Delta^l$, clock families, control paths, and open-loop aggregate solutions with their consumed clock times $\mathsf{C}^{c}$; and from Cell-Count Form of a Compensated Counting-Path Functional Along a Time Change: Window-Discrepancy Bounds for the Time-Change and Partial-Cell Errors, for a counting path $p$ and a real number $R>0$, the compensated path $\bar{M}(x)=p(x)-x$, the window discrepancy $\mathrm{Disc}_w(p)$ (formed with this $R$), the compensated functional $\Lambda(\cdot)$, and, for a bounded measurable $H:[0,T]\to\mathbb{R}^l$, the norm $\lVert H\rVert_1=\int_{[0,T]}|H_u|\,du$. Write $|\cdot|$ for the Euclidean norm, $x\cdot y$ for the dot product, and $\sqrt{N}$ for the nonnegative square root.

Cells and clocks. Let $R>0$ be a real number with $R\ge NBT$; for every label $c$ let $J_c\ge1$ be a natural number and $0=\mathsf{b}^{c}_0<\mathsf{b}^{c}_1<\dots<\mathsf{b}^{c}_{J_c}=R$ real numbers, with cell lengths $\mu_{c,j}=\mathsf{b}^{c}_j-\mathsf{b}^{c}_{j-1}$; let $\mathsf{L}$ be the set of pairs $q=(c,j)$ with $1\le j\le J_c$, write $\mu_q=\mu_{c,j}$ and $\mu_{\max}=\max_{q\in\mathsf{L}}\mu_q$ (the cells of Injection Weights from a Bounded Profile Along Mean-Field Clocks: Label-Rate Form of the Fluctuation Covariance, Weight Bounds, the Prior Quadratic Form, and the Step-Function Injection). Let $p=(p^{c})_{c\in\mathcal{L}}$ be a clock family, $a$ a control path, $x_0\in\mathbb{G}_N$, and let $\Sigma:[0,T]\to\mathbb{R}^l$ be an open-loop aggregate solution for the data $(p,a,x_0)$, with consumed clock times $\mathsf{C}^{c}$; define the cell counts $\mathsf{K}_{c,j}=p^{c}(\mathsf{b}^{c}_j)-p^{c}(\mathsf{b}^{c}_{j-1})$ and write $\mathsf{K}_q=\mathsf{K}_{c,j}$. Define the mean-field label rates and mean-field clocks

$$\phi_c(t)=\psi_c(S_t,\mathsf{A}_t),\qquad \bar{\mathsf{C}}^{c}_t=N\int_{[0,t]}\phi_c(u)\,du\qquad(c\in\mathcal{L},\ t\in[0,T]),$$

as in Injection Weights from a Bounded Profile Along Mean-Field Clocks: Label-Rate Form of the Fluctuation Covariance, Weight Bounds, the Prior Quadratic Form, and the Step-Function Injection, whose cell boundaries $b^{c}_j$ are written $\mathsf{b}^{c}_j$ here (each $\phi_c$ is measurable with values in $[0,B]$ by claim 1). Let $y:[0,T]\to\Delta^l$ be a measurable map with $y_t=y_0+\int_{[0,t]}b(y_u,a_u)\,du$ for every $t$ (the flow), and let $\mathbf{c}\in\mathbb{R}^l$ (the estimand direction; the bold letter is unrelated to the labels $c$). Put $e_t=\Sigma_t-y_t$, and let $\rho_u$ and $\mathsf{d}_u$ be the residual and the control-gradient discrepancy of Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound for $x=\Sigma$ and this $y$. Assume the clock-discrepancy hypothesis: a real number $w_1\ge0$ with

$$\textbf{(CD)}\qquad|\mathsf{C}^{c}_u-\bar{\mathsf{C}}^{c}_u|\le w_1\qquad\text{for every }c\in\mathcal{L}\text{ and }u\in[0,T].$$

For $c\in\mathcal{L}$ and $u\in[0,T]$ put $H^{c}_u=\Phi^{\mathcal{E}}(T,u)\,\mathcal{E}^{\star}_u\,v_c\in\mathbb{R}^l$. Finally define the entry times and the cell coefficients: for $q=(c,j)\in\mathsf{L}$ with $\bar{\mathsf{C}}^{c}_T\ge \mathsf{b}^{c}_j$ let $\bar{\tau}_q$ be the least $u\in[0,T]$ with $\bar{\mathsf{C}}^{c}_u\ge \mathsf{b}^{c}_j$ (it exists by claim 1) and put $\alpha_q=\mathbf{c}\cdot\bigl(\Phi^{\mathcal{E}}(T,\bar{\tau}_q)\,v_c\bigr)$; for $q=(c,j)$ with $\bar{\mathsf{C}}^{c}_T<\mathsf{b}^{c}_j$ put $\alpha_q=0$.

1. (Perturbation form of the solution.) Each $\phi_c$ is measurable with values in $[0,B]$; $\Sigma$ is bounded measurable with values in $\mathbb{G}_N\subseteq\Delta^l$, and $e$ is bounded measurable with $|e_t|\le2$; each $\mathsf{C}^{c}$ and each $\bar{\mathsf{C}}^{c}$ is continuous and nondecreasing on $[0,T]$ with values in $[0,NBT]\subseteq[0,R]$; for every $q=(c,j)$ with $\bar{\mathsf{C}}^{c}_T\ge \mathsf{b}^{c}_j$ the set $\{u\in[0,T]:\bar{\mathsf{C}}^{c}_u\ge \mathsf{b}^{c}_j\}$ has a least element $\bar{\tau}_q$ and equals $[\bar{\tau}_q,T]$; and, with the perturbation

$$\mathfrak{m}_t=\frac{1}{N}\sum_{c\in\mathcal{L}}v_c\,\bar{M}^{c}(\mathsf{C}^{c}_t),\qquad \bar{M}^{c}(x)=p^{c}(x)-x\quad(x\in[0,R]),$$

which is bounded measurable with $\mathfrak{m}_0=0$, one has $\Sigma_t=x_0+\int_{[0,t]}b(\Sigma_u,a_u)\,du+\mathfrak{m}_t$ for every $t\in[0,T]$. Consequently Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound applies to $x=\Sigma$, $y$, $a$ and $\mathfrak{m}$.

2. (Linear response in cell-count form.) $\sqrt{N}\,\mathbf{c}\cdot L_T(\mathfrak{m})=\frac{1}{\sqrt{N}}\sum_{c\in\mathcal{L}}\mathbf{c}\cdot\Lambda^{c}(\mathsf{C}^{c})$, where $\Lambda^{c}$ is the compensated functional of Cell-Count Form of a Compensated Counting-Path Functional Along a Time Change: Window-Discrepancy Bounds for the Time-Change and Partial-Cell Errors formed from $p^{c}$, $R$, $s=T$, $v=v_c$ and $H=H^{c}$ (bounded measurable with $\lVert H^{c}\rVert_1\le\sqrt{2}\,\bar{\Phi}^{2}\Lambda_{\mathcal{E}}T$); and

$$\Bigl|\sqrt{N}\,\mathbf{c}\cdot L_T(\mathfrak{m})-\frac{1}{\sqrt{N}}\sum_{q\in\mathsf{L}}\alpha_q\,(\mathsf{K}_q-\mu_q)\Bigr|\ \le\ \frac{|\mathbf{c}|}{\sqrt{N}}\sum_{c\in\mathcal{L}}\bigl(\sqrt{2}+\lVert H^{c}\rVert_1\bigr)\bigl(\mathrm{Disc}_{w_1}(p^{c})+\mathrm{Disc}_{\mu_{\max}}(p^{c})\bigr)=:\mathsf{E}_{\mathrm{cell}} .$$

3. (Estimand linearisation.) Writing $\bar{e}=\sup_{u\in[0,T]}\sqrt{N}\,|e_u|$,

$$\Bigl|\sqrt{N}\,\mathbf{c}\cdot(\Sigma_T-y_T)-\frac{1}{\sqrt{N}}\sum_{q\in\mathsf{L}}\alpha_q\,(\mathsf{K}_q-\mu_q)\Bigr|\ \le\ |\mathbf{c}|\,\bar{\Phi}^{2}\,\sqrt{N}\,|x_0-y_0|+|\mathbf{c}|\,\bar{\Phi}^{2}\,\mathsf{E}_{\mathrm{res}}+\mathsf{E}_{\mathrm{cell}},$$

where

$$\mathsf{E}_{\mathrm{res}}=\sqrt{N}\int_{[0,T]}|\rho_u|\,du\ \le\ \sqrt{2}\,l(l-1)\Lambda_2T\,\frac{\bar{e}^{2}}{\sqrt{N}}+\sqrt{2}\,l(l-1)\Lambda_3\,\bar{e}\int_{[0,T]}|y_u-S_u|\,du+\sqrt{2}\,\bar{e}\int_{[0,T]}\mathsf{d}_u\,du ,$$

the two last integrands being bounded measurable.