Proof of Fresh-Start Property of the Controlled N-Agent Dynamics

Write $\mathcal{T}^a_r$ for the consumed clock time of a generic clock label $a$ (a triple $(i,\sigma\gamma)$ or a pair $(i,\upsilon)$), $Y^a$ for the corresponding clock, and $\hat{Y}^a$ for the residual clock of the statement. Recall from Solution of the Controlled N-Agent Dynamics that the consumed clock times are defined on all of $\Omega$, vanish off the regular event, and satisfy $\mathcal{T}^a_r\le\bar{\beta}r$ with $\bar{\beta}=\max(B,\tilde{B})$; by part (iv) of Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics each $\mathcal{T}^a_r$ is $\mathcal{F}^{\mathrm{sys}}_r$-measurable.

Step 1: pathwise part of (a). For every $\omega$ and every clock, $u\mapsto\hat{Y}^a_u=Y^a_{\mathcal{T}^a_r+u}-Y^a_{\mathcal{T}^a_r}$ starts at $0$, is nondecreasing, integer-valued, right-continuous, and has unit jumps, all inherited from the counting path of $Y^a$; so every residual path is a counting path.

Step 2: grid approximation. Fix $\delta>0$. For each clock let $J_a$ be the smallest nonnegative integer with $\mathcal{T}^a_r\le J_a\delta$; then $J_a\le\lceil\bar{\beta}r/\delta\rceil$, and the events $C_{\mathbf{j}}=\{J_a=j_a\ \text{for all}\ a\}$, over the finitely many integer vectors $\mathbf{j}$, partition $\Omega$ and lie in $\mathcal{F}^{\mathrm{sys}}_r$, since each $\mathcal{T}^a_r$ is $\mathcal{F}^{\mathrm{sys}}_r$-measurable. For an integer vector $\mathbf{k}$ write $C'_{\mathbf{k}}=\{\mathcal{T}^a_r\le k_a\delta\ \text{for all}\ a\}$ and

so that $\mathcal{H}_{\mathbf{k}}\subseteq\mathcal{H}_{\mathbf{j}}$ whenever $\mathbf{k}\le\mathbf{j}$ componentwise. By the clock-reading bound (vi) with caps $k_a\delta$, each $C'_{\mathbf{k}}$ agrees up to a null event with an event of $\mathcal{H}_{\mathbf{k}}$. Since

(where $\mathbf{j}-e_a$ decrements the $a$-th cap; the set subtracted for $j_a=0$ is empty), the cell $C_{\mathbf{j}}$ agrees up to a null event with an event $G_{\mathbf{j}}\in\mathcal{H}_{\mathbf{j}}$. Moreover, for $F\in\mathcal{F}^{\mathrm{sys}}_r$, the clock-reading bound with caps $j_a\delta$ gives $H'\in\mathcal{H}_{\mathbf{j}}$ with $F\cap C'_{\mathbf{j}}$ equal to $H'\cap C'_{\mathbf{j}}$ up to a null event; intersecting with $C_{\mathbf{j}}\subseteq C'_{\mathbf{j}}$ and replacing $C_{\mathbf{j}}$ by $G_{\mathbf{j}}$,

Define the grid residuals $\hat{Y}^{a,\delta}_u=Y^a_{J_a\delta+u}-Y^a_{J_a\delta}$, which on $C_{\mathbf{j}}$ equal $Y^a_{j_a\delta+u}-Y^a_{j_a\delta}$. For fixed $\mathbf{j}$, the family of $\sigma$-algebras consisting of $\mathcal{H}_{\mathbf{j}}$ together with $\sigma(Y^a_{j_a\delta+u}-Y^a_{j_a\delta}:u\ge0)$, one per clock, is independent: the family consisting of $\sigma(\varsigma^1_0,\dots,\varsigma^N_0)$ and the full clock $\sigma$-algebras is independent by N-Agent Driving System, within each clock the pre-$j_a\delta$ $\sigma$-algebra and the shifted-increment $\sigma$-algebra are independent, which we check as follows. The lemma Increments Are Independent of the Natural Filtration Past gives, for any $s<t$, that the single increment $Y^a_t-Y^a_s$ is independent of the natural filtration of $Y^a$ up to time $s$; it does not by itself cover the whole shifted $\sigma$-algebra, so we build the latter from finitely many increments. Fix $0=u_0<u_1<\dots<u_p$, nonnegative integers $m_1,\dots,m_p$, an event $E$ of the $\sigma$-algebra generated by the variables $Y^a_u$ with $u\le j_a\delta$, and set $A_q=\{Y^a_{j_a\delta+u_q}-Y^a_{j_a\delta+u_{q-1}}=m_q\}$. The event $E\cap A_1\cap\dots\cap A_{p-1}$ lies in the natural filtration of $Y^a$ up to time $j_a\delta+u_{p-1}$, so the cited lemma applied at that time gives $P(E\cap A_1\cap\dots\cap A_p)=P(E\cap A_1\cap\dots\cap A_{p-1})\,P(A_p)$; descending induction on $p$ yields $P\big(E\cap\bigcap_qA_q\big)=P(E)\prod_qP(A_q)$. The sets $\bigcap_qA_q$, together with $\Omega$ and $\emptyset$, form a $\pi$-system generating $\sigma(Y^a_{j_a\delta+u}-Y^a_{j_a\delta}:u\ge0)$, so Dynkin's Pi-Lambda Theorem upgrades the displayed factorization to independence of the two $\sigma$-algebras. The two levels then combine by factorizing $P\big(D_0\cap\bigcap_a(P_a\cap R_a)\big)=P(D_0)\prod_aP(P_a\cap R_a)=P(D_0)\prod_aP(P_a)P(R_a)$ over the generating intersections; these intersections form a $\pi$-system generating the joint $\sigma$-algebra of the family, so Dynkin's Pi-Lambda Theorem again upgrades the factorization to independence of the whole family; each shifted process is a homogeneous Poisson process with rate $1$ (independent increments and Poisson increments are preserved by a deterministic shift).

Step 3: the limit identity. Let $0=u_0<u_1<\dots<u_p$, let $m^a_q$ be nonnegative integers ($q\in\{1,\dots,p\}$, finitely many clocks), write $\pi(\mu;m)=e^{-\mu}\mu^m/m!$ for the Poisson probabilities, and fix $F\in\mathcal{F}^{\mathrm{sys}}_r$. Take $\delta=\delta_n=2^{-n}$. The caps $J_a\delta_n$ decrease to $\mathcal{T}^a_r$ pointwise as $n\to\infty$ (they lie in $[\mathcal{T}^a_r,\mathcal{T}^a_r+\delta_n)$), so by right-continuity of the clock paths, for every $\omega$ and every $u$,

and since all values are integers the corresponding indicator variables converge pointwise. For each fixed $n$, partitioning by the cells, replacing $F\cap C_{\mathbf{j}}$ by $H^F_{\mathbf{j}}$ (a null modification, changing no expectation), and using that on $C_{\mathbf{j}}$ the grid-residual indicator product is a function of the shifted processes of the cell while $H^F_{\mathbf{j}}\in\mathcal{H}_{\mathbf{j}}$ — but noting that the indicator product must first be rewritten cellwise, $\mathbf{1}_{C_{\mathbf{j}}}\prod\mathbf{1}_{\{\cdots\}}=\mathbf{1}_{C_{\mathbf{j}}}\prod\mathbf{1}_{\{(Y^a_{j_a\delta+u_q}-Y^a_{j_a\delta})-(Y^a_{j_a\delta+u_{q-1}}-Y^a_{j_a\delta})=m^a_q\}}$, and then $\mathbf{1}_{C_{\mathbf{j}}}$ replaced by $\mathbf{1}_{G_{\mathbf{j}}}$ up to a null event — the independence of Step 2 gives

Summing over $\mathbf{j}$ and using $\sum_{\mathbf{j}}P(H^F_{\mathbf{j}})=\sum_{\mathbf{j}}P(F\cap C_{\mathbf{j}})=P(F)$, then letting $n\to\infty$ with dominated convergence,

Step 4: conclusion. Taking $F=\Omega$ and one clock at a time, the increments of $\hat{Y}^a$ over any finite partition $0=u_0<u_1<\dots<u_p$ anchored at $0$ are independent with the Poisson distributions of parameters $u_q-u_{q-1}$ (the joint probability mass function factorizes into the required product for every such partition, and events involving the increments of integer-valued variables are unions of such atoms). For times $0<v_1<\dots<v_p$ not anchored at $0$, apply this to the partition $0<v_1<\dots<v_p$ and sum the resulting product formula over all values of the leading increment $\hat{Y}^a_{v_1}-\hat{Y}^a_0$, whose Poisson probabilities sum to $1$; this marginalization gives the same factorization and the same Poisson laws for the increments over $v_1<\dots<v_p$. Moreover $\hat{Y}^a_0=0$; with Step 1 this proves (a): each residual clock is a homogeneous Poisson process with rate $1$ all of whose paths are counting paths.

For (b): for each clock, the collection of events $\{\hat{Y}^a_{u_1}=w_1,\dots,\hat{Y}^a_{u_p}=w_p\}$ over finite time sets and integer values, together with $\Omega$ and $\emptyset$, is a $\pi$-system generating $\sigma(\hat{Y}^a_u:u\ge0)$ (the variables are integer-valued, so these cylinder atoms generate). Step 3 shows that for every $F\in\mathcal{F}^{\mathrm{sys}}_r$ and every choice of one such event $E_a$ per clock (finitely many clocks, the remaining ones taken to be $\Omega$),

since $P(E_a)$ is exactly the corresponding product of Poisson probabilities by (a). Fixing all but one entry and letting the remaining entry range over a $\pi$-system, Dynkin's $\pi$-$\lambda$ theorem upgrades each $\pi$-system in turn to the generated $\sigma$-algebra, one at a time; after finitely many applications this yields the factorization for arbitrary events from $\mathcal{F}^{\mathrm{sys}}_r$ and from the residual clock $\sigma$-algebras, which is the asserted independence of the family. The final assertion of the statement follows since the time-$r$ states are $\mathcal{F}^{\mathrm{sys}}_r$-measurable by part (iv) of Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics, so all requirements of N-Agent Driving System hold for the initial states $\sigma^1_r,\dots,\sigma^N_r$ and the residual clocks. $\blacksquare$