Law Identity between the N-Agent Aggregate Path with Its Observation Record and the Synthetic Copy's Regularised Path with Its Record

Adopt the setting and notation of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, with the probability space of that lemma renamed $(\Omega^\flat,\mathcal{F}^\flat,P^\flat)$: natural numbers $N\ge1$, $l\ge2$, $m\ge1$, $\tilde{l}\ge1$; a nonempty control set $\mathcal{A}\subseteq\mathbb{R}^m$ in Euclidean space; real numbers $B\ge0$, $\tilde{B}\ge0$ and $T>0$; a transition-rate family $\beta$ with control set $\mathcal{A}$ and rate bound $B$; an observation-rate family $\tilde{\beta}$ with rate bound $\tilde{B}$, its aggregate observation drift $\tilde{b}=(\tilde{b}^\upsilon)$ and the total observation rate $\tilde{b}^{\mathrm{tot}}=\sum_{\upsilon=1}^{\tilde{l}}\tilde{b}^\upsilon$ on the probability simplex $\Delta^l$; the aggregate lattice $\mathbb{G}_N\subseteq\Delta^l$ and the point $x_0\in\mathbb{G}_N$; the observation record space $(\mathbf{R},\mathcal{R},\rho)$ with horizon $T$ and $\tilde{l}$ channels, records written $r=(k,\mathbf{t},v)$ with $\mathbf{t}=(t_1,\dots,t_k)$ and $v=(v_1,\dots,v_k)$; an $\mathcal{A}$-valued observation-driven control policy $h$ with record-frozen control paths $a^r$; the real number $R$, for which we assume the strict inequality $R>NBT$; the cells, the index set $\mathsf{L}$, the dimension $d$, the smoothing parameter $\eta$ (a real number occurring only as the subscript of the Gaussian smoothing weight $\varphi_\eta$; the occupation indicators $\eta^{r,i,\gamma}$ below always carry superscripts), the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^d)$ and Lebesgue measure $\lambda_d$ on $\mathbb{R}^d$; the driving variables and the cell-count vector $\mathsf{K}$ on $(\Omega^\flat,\mathcal{F}^\flat,P^\flat)$; the copy clocks $\mathsf{P}^\sharp$; the regularised recursion paths $\bar{\Sigma}^{\sharp,r}_t(\omega^\flat)$, the likelihoods $\ell^{\sharp,\omega^\flat}(r)$ and the conflict-free set $\mathsf{G}^\sharp$ of claim 3 of that lemma; and the synthetic copy $(\Omega^\sharp,\mathcal{F}^\sharp,\mu^\sharp)$ with $\Omega^\sharp=\Omega^\flat\times\mathbb{R}^d\times\mathbf{R}$, $\mathcal{F}^\sharp=(\mathcal{F}^\flat\otimes\mathcal{B}(\mathbb{R}^d))\otimes\mathcal{R}$, and the record $\mathsf{D}(\omega^\flat,\theta,r)=r$. Integrals of $[0,\infty]$-valued measurable maps are those of Lebesgue Integral of a Nonnegative Measurable Function, products being formed with the convention $0\cdot\infty=0$ of Image Measures, Measures with Densities, and Change of Variables, and $\mathbf{1}_S$ denotes the indicator of a set $S$.

Let moreover $(\Omega,\mathcal{F},P)$ be an $N$-agent driving system with $l$ states and $\tilde{l}$ observation channels, with expectation $\mathbb{E}$ (extended to $[0,\infty]$-valued measurable maps as their integrals with respect to $P$); let a solution of the controlled $N$-agent dynamics on $[0,T]$ for the policy $h$ and for the same data $m$, $\mathcal{A}$, $\beta$, $B$, $\tilde{\beta}$, $\tilde{B}$ and $T$ be given, with regular event $\Omega_0$, empirical state measure $\Sigma_t$ and observation record $W$; and fix reconstruction data $(\eta^{r,i,\gamma},\sigma^{r,i},\tilde{A}^{r,i,\upsilon},G)$ for this driving system and policy, with reconstructed empirical state measures $\Sigma^r$ and, for each $r\in\mathbf{R}$, the event $\Omega^r$ of clause (d) of that lemma. Put $D_0=\{\Sigma_0=x_0\}$.

Let $\mathsf{Path}=\mathsf{Path}(\mathbb{G}_N,T)$ be the space of piecewise constant paths in $\mathbb{G}_N$ with horizon $T$, with its $\sigma$-algebra $\mathcal{C}=\mathcal{C}_T$ generated by the sets $\{p:p(u)=y\}$ ($u\in[0,T]$, $y\in\mathbb{G}_N$), and write $\mathcal{C}\otimes\mathcal{R}$ for the product $\sigma$-algebra on $\mathsf{Path}\times\mathbf{R}$. Define $\Pi$ on $\Omega$, with values in the set of maps $[0,T]\to\mathbb{G}_N$ (the empirical state measure $\Sigma_u(\omega)$ lies in $\mathbb{G}_N$ for every $\omega$ and $u$, its coordinates being agent counts divided by $N$), by $\Pi(\omega)(u)=\Sigma_u(\omega)$ for $\omega\in\Omega_0$ and $\Pi(\omega)(u)=x_0$ for $\omega\notin\Omega_0$; and define $\Pi^\sharp$ on $\Omega^\sharp$ by $\Pi^\sharp(\omega^\flat,\theta,r)(u)=\bar{\Sigma}^{\sharp,r}_u(\omega^\flat)$ ($u\in[0,T]$). Then:

1. (Measurability.) $D_0\in\mathcal{F}$; $\Pi$ and $\Pi^\sharp$ take their values in $\mathsf{Path}$ and are measurable with respect to $\mathcal{F}$, respectively $\mathcal{F}^\sharp$, and $\mathcal{C}$; $W$ is measurable with respect to $\mathcal{F}$ and $\mathcal{R}$, and $\mathsf{D}$ with respect to $\mathcal{F}^\sharp$ and $\mathcal{R}$.

2. (Law identity for path and record.) For every $F:\mathsf{Path}\times\mathbf{R}\to[0,\infty]$ measurable with respect to $\mathcal{C}\otimes\mathcal{R}$,

$$\mathbb{E}\bigl[\mathbf{1}_{D_0}\,F(\Pi,W)\bigr]=P(D_0)\int_{\Omega^\sharp}F(\Pi^\sharp,\mathsf{D})\,d\mu^\sharp\qquad\text{in }[0,\infty].$$

3. (Law identity for the aggregate at a fixed time and the record.) For every $s\in[0,T]$ and every $F':\mathbb{G}_N\times\mathbf{R}\to[0,\infty]$ measurable with respect to the product of the $\sigma$-algebra of all subsets of $\mathbb{G}_N$ and $\mathcal{R}$,

$$\mathbb{E}\bigl[\mathbf{1}_{D_0}\,F'(\Sigma_s,W)\bigr]=P(D_0)\int_{\Omega^\sharp}F'\bigl(\bar{\Sigma}^{\sharp,\mathsf{D}}_s,\mathsf{D}\bigr)\,d\mu^\sharp\qquad\text{in }[0,\infty],$$

where $\bar{\Sigma}^{\sharp,\mathsf{D}}_s(\omega^\flat,\theta,r)=\bar{\Sigma}^{\sharp,r}_s(\omega^\flat)$. In particular, if $P(D_0)=1$, the pair $(\Sigma_s,W)$ under $P$ and the pair $(\bar{\Sigma}^{\sharp,\mathsf{D}}_s,\mathsf{D})$ under $\mu^\sharp$ have the same image measure on the product of the $\sigma$-algebra of all subsets of $\mathbb{G}_N$ and $\mathcal{R}$.