Shared-Clock Coupling of One-Agent-Moved Reconstructions

Adopt the setting of claim 1 of One-Agent-Move Ratio of the Record Density Kernel: the controlled $N$-agent dynamics with transition-rate family $\beta$ on $l$ states with control dimension $m$, nonempty control set $\mathcal{A}\subseteq\mathbb{R}^m$, and rate bound $B$, observation-rate family $\tilde{\beta}$, horizon $T>0$, driving system $(\Omega,\mathcal{F},P)$ with clocks $Y^{i,\sigma\gamma}$ and $\tilde{Y}^{i,\upsilon}$, policy $h$, moved agent $i_0$ and moved initial states, the observation record space $(\mathbf{R},\mathcal{R})$, the $\sigma$-algebras $\mathcal{T}'\subseteq\mathcal{T}$, and fixed reconstruction data $(\eta^{r,i,\gamma},\sigma^{r,i},\tilde{A}^{r,i,\upsilon},G)$ for the original system and $(\eta'^{r,i,\gamma},\sigma'^{r,i},\tilde{A}'^{r,i,\upsilon},G')$ for the moved system, both for the same policy $h$ and the same clocks, with reconstructed empirical state measures $\Sigma^r$ and $\Sigma'^r$. For $r\in\mathbf{R}$ let $a^r$ be the record-frozen control path, which is the same for both systems and takes values in $\mathcal{A}$, as noted in claim (d) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records, since $h$ is $\mathcal{A}$-valued, and define for both data sets the reconstructed transition consumed times $A^{r,i,\sigma\gamma}_u(\omega)=\int_{[0,u]}\eta^{r,i,\sigma}_x(\omega)\,\beta\bigl(\sigma,\gamma,\Sigma^r_x(\omega),a^r(x)\bigr)\,dx,\qquad A'^{r,i,\sigma\gamma}_u(\omega)=\int_{[0,u]}\eta'^{r,i,\sigma}_x(\omega)\,\beta\bigl(\sigma,\gamma,\Sigma'^r_x(\omega),a^r(x)\bigr)\,dx,$ as Lebesgue integrals over compact intervals; for $\omega$ in the probability-one events $\Omega^r$ and $\Omega'^r$ of claim (d) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records at $r$, these are the consumed transition clock times of the respective per-record solutions there.

Assume additionally that $\beta$ is Lipschitz in the state argument: there is a real $K_\beta\ge0$ such that $|\beta(\sigma,\gamma,\Sigma,a)-\beta(\sigma,\gamma,\bar{\Sigma},a)|\le K_\beta\sum_{\delta=1}^{l}|\Sigma^\delta-\bar{\Sigma}^\delta|$ for all $\sigma\neq\gamma$, all $a\in\mathcal{A}$, and all $\Sigma,\bar{\Sigma}$ in the probability simplex $\Delta^l$; and that $\tilde{\beta}$ is Lipschitz in the state argument with a real constant $K_{\tilde{\beta}}\ge0$: $|\tilde{\beta}(\sigma,\upsilon,\Sigma)-\tilde{\beta}(\sigma,\upsilon,\bar{\Sigma})|\le K_{\tilde{\beta}}\sum_{\delta=1}^{l}|\Sigma^\delta-\bar{\Sigma}^\delta|$ for all $\sigma$, all $\upsilon$, and all $\Sigma,\bar{\Sigma}\in\Delta^l$. Write $e_1,\dots,e_l$ for the standard basis vectors of Euclidean space $\mathbb{R}^l$.

For $(r,\omega)\in G\cap G'$ let $D^r(\omega)=\{u\in[0,T]:\ \sigma^{r,i}_u(\omega)\neq\sigma'^{r,i}_u(\omega)\ \text{for some}\ i\neq i_0\}$.

1. (Decoupling time) For every $(r,\omega)\in G\cap G'$ with $D^r(\omega)$ nonempty, the set $D^r(\omega)$ has a least element, and it is strictly positive. Define $\zeta^r(\omega)$ as this least element, $\zeta^r(\omega)=T+1$ for $(r,\omega)\in G\cap G'$ with $D^r(\omega)$ empty, and $\zeta^r(\omega)=T+1$ for $(r,\omega)\notin G\cap G'$; $\zeta^r(\omega)$ is called the decoupling time.

2. (Coupling before decoupling) For every $(r,\omega)\in G\cap G'$ and every $u\in[0,T]$ with $u<\zeta^r(\omega)$: $\sigma^{r,i}_u(\omega)=\sigma'^{r,i}_u(\omega)$ for all $i\neq i_0$; $N\bigl(\Sigma'^r_u(\omega)-\Sigma^r_u(\omega)\bigr)=e_{\sigma'^{r,i_0}_u(\omega)}-e_{\sigma^{r,i_0}_u(\omega)},$ the right side being $0$ when the moved agent occupies the same state in both reconstructions; in particular $\sum_{\delta=1}^{l}|\Sigma'^{r,\delta}_u(\omega)-\Sigma^{r,\delta}_u(\omega)|\le 2/N$. The same identities hold for the left limits at every $u\in(0,T]$ with $u\le\zeta^r(\omega)$.

3. (Consumed-time mismatch bounds) For every $(r,\omega)\in G\cap G'$, every $u\in[0,T]$ with $u\le\zeta^r(\omega)$, every $i\neq i_0$, every ordered pair $(\sigma,\gamma)$ with $\sigma\neq\gamma$, and every $\upsilon$: $\bigl|A'^{r,i,\sigma\gamma}_u(\omega)-A^{r,i,\sigma\gamma}_u(\omega)\bigr|\le\frac{2K_\beta u}{N},\qquad \bigl|\tilde{A}'^{r,i,\upsilon}_u(\omega)-\tilde{A}^{r,i,\upsilon}_u(\omega)\bigr|\le\frac{2K_{\tilde{\beta}}u}{N}.$

4. (Decoupling mechanism) For every $\omega\in\Omega^r\cap\Omega'^r$ with $(r,\omega)\in G\cap G'$ and $\zeta^r(\omega)\le T$, there exist $i\neq i_0$ and an ordered pair $(\sigma,\gamma)$ with $\sigma\neq\gamma$ such that, writing $\zeta=\zeta^r(\omega)$: $A^{r,i,\sigma\gamma}_\zeta(\omega)\neq A'^{r,i,\sigma\gamma}_\zeta(\omega)$; the closed interval with these two endpoints contains a jump time of the path $u\mapsto Y^{i,\sigma\gamma}_u(\omega)$; and $|A^{r,i,\sigma\gamma}_\zeta(\omega)-A'^{r,i,\sigma\gamma}_\zeta(\omega)|\le 2K_\beta\zeta/N$.

5. (Measurability) The map $(r,\omega)\mapsto\zeta^r(\omega)$ is measurable with respect to $\mathcal{R}\otimes\mathcal{T}$ (product $\sigma$-algebra).