Ties in the Aggregate Recursion: Consumed Levels and First Hitting Times, Conflict-Freeness in the Absence of Ties, and the Point-Deletion Identity

Adopt the setting and notation of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution (and hence of Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks): natural numbers $N\ge1$, $l\ge2$, $m\ge1$, the control set $\mathcal{A}$, real numbers $B\ge0$ and $T>0$, the transition-rate family $\beta$ with rate bound $B$, the aggregate lattice $\mathbb{G}_N\subseteq\Delta^l$ (with $\Delta^l$ the probability simplex), the transition labels $c=(\sigma,\gamma)$ with vectors $v_c=\delta_\gamma-\delta_\sigma$, clock families $p=(p^{c})$ of counting paths with jump times $\tau_j(\cdot)$, control paths $a$, and, for data $(p,a,x_0)$ with $x_0\in\mathbb{G}_N$, the aggregate recursion with its times $\theta_k$, points $x^{(k)}$, stopping index $K$ and the quantities $\kappa^{c}_k$, $\mathsf{C}^{c,(k)}$, $\lambda^{c}_k$, $h^{c}_k$, $\mathcal{J}_k$ defined there (which we call the consumed levels, step consumed-time functions, next jump levels, hitting times and firing sets respectively), the notion of conflict-free data, and the recursion consumed times $\mathsf{C}^{\mathrm{rec},c}_t$ ($t\in[0,T]$). Greatest lower bounds are as there ($+\infty$ for the empty set), $\mathbf{1}\{\cdot\}$ is the indicator of a condition, and a real number $u>0$ is a jump time of a counting path $q$ if $q(u)>q(u-)$, where $q(u-)$ is the least upper bound of $\{q(s):0\le s<u\}$, as in Counting Path and Its Jump Times. Fix data $(p,a,x_0)$ and run the recursion. A step $k<K$ is called a tie if the firing set $\mathcal{J}_k$ has at least two elements.

1. (Consumed levels and first hitting times) For every label $c=(\sigma,\gamma)$: the map $t\mapsto\mathsf{C}^{\mathrm{rec},c}_t$ is nondecreasing on $[0,T]$ with $|\mathsf{C}^{\mathrm{rec},c}_t-\mathsf{C}^{\mathrm{rec},c}_s|\le NB(t-s)$ for $0\le s\le t\le T$, and $\mathsf{C}^{\mathrm{rec},c}_0=0$; for every $k<K$, $\kappa^{c}_k\le\kappa^{c}_{k+1}$, $\mathsf{C}^{\mathrm{rec},c}_t\le\kappa^{c}_{k+1}$ for $t\in[0,\theta_{k+1}]$, and $\mathsf{C}^{\mathrm{rec},c}_t<\lambda^{c}_k$ for $t\in[0,\theta_{k+1})$; and if $k<K$ and $c\in\mathcal{J}_k$, then $x^{(k),\sigma}>0$, $\kappa^{c}_{k+1}=\lambda^{c}_k$, and $\theta_{k+1}=h^{c}_k=\inf\{t\in[0,T]:\ \mathsf{C}^{\mathrm{rec},c}_t\ge\lambda^{c}_k\},$ the first time at which the recursion consumed time of $c$ reaches the next jump level $\lambda^{c}_k$.

2. (No ties implies conflict-free) If no step $k<K$ is a tie, then $x^{(k)}\in\mathbb{G}_N$ for every $k\le K$, and the data $(p,a,x_0)$ are conflict-free.

3. (Point deletion) Let $c_0$ be a label and $u>0$ a jump time of $p^{c_0}$, and let $p^{-}=(p^{-,c})$ be the family with $p^{-,c}=p^{c}$ for $c\ne c_0$ and $p^{-,c_0}(t)=p^{c_0}(t)-\mathbf{1}\{t\ge u\}$ for $t\ge0$. Then $p^{-,c_0}$ is a counting path, so $p^{-}$ is a clock family; run the recursion for the data $(p^{-},a,x_0)$ and mark its quantities by a minus sign. Suppose that for some step $k<K$ one has $c_0\in\mathcal{J}_k$, $\lambda^{c_0}_k=u$, and $\mathcal{J}_k$ contains a label $c'\ne c_0$ (so that $k$ is a tie); put $v=\lambda^{c'}_k$, a jump time of $p^{c'}=p^{-,c'}$. Then the recursion for $p^{-}$ performs the steps $0,\dots,k$ with $\theta^{-}_{i}=\theta_i$, $x^{-(i)}=x^{(i)}$ and $\kappa^{-,c}_{i}=\kappa^{c}_i$ for all $i\le k$ and all $c$, and moreover $\theta^{-}_{k+1}=\theta_{k+1}$, $\mathsf{C}^{-,\mathrm{rec},c}_t=\mathsf{C}^{\mathrm{rec},c}_t$ for all $c$ and $t\in[0,\theta_{k+1}]$, $\theta_{k+1}=\inf\{t\in[0,T]:\ \mathsf{C}^{-,\mathrm{rec},c'}_t\ge v\},\qquad\text{and}\qquad u=\mathsf{C}^{-,\mathrm{rec},c_0}_{\theta_{k+1}} .$