Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks

Let $N$, $l$ and $m$ be natural numbers with $N\ge1$, $l\ge2$ and $m\ge1$, let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$, let $B\ge0$ and $T>0$ be real numbers, and let $\beta$ be a transition-rate family on $l$ states with control set $\mathcal{A}$ and rate bound $B$. Write $\mathbb{N}_0$ for the set consisting of $0$ and the natural numbers, $\Delta^l$ for the probability simplex with its standard basis vectors $\delta_1,\dots,\delta_l$, and, for a point $x\in\mathbb{R}^l$, $x^{\gamma}$ for its $\gamma$-th coordinate. The aggregate lattice with $N$ agents is the set

$$\mathbb{G}_N=\{x\in\Delta^l:\ Nx^\gamma\in\mathbb{N}_0\ \text{for every }\gamma\in\{1,\dots,l\}\}.$$

Throughout, $c=(\sigma,\gamma)$ ranges over the transition labels on $l$ states, and $v_c=\delta_\gamma-\delta_\sigma$, with sums and scalar multiples in $\mathbb{R}^l$.

A control path is a map $a:[0,T]\to\mathcal{A}$, written $s\mapsto a_s$, each of whose components is measurable with respect to the trace Borel $\sigma$-algebra on $[0,T]$ and the Borel $\sigma$-algebra of the real line. A clock family on $l$ states is a family $p=(p^{c})$ of counting paths, one for each transition label $c$.

Let $p$ be a clock family, let $a$ be a control path, and let $x_0\in\mathbb{G}_N$. An open-loop aggregate solution on $[0,T]$ for the data $(p,a,x_0)$ is a map $\Sigma:[0,T]\to\mathbb{R}^l$, written $t\mapsto\Sigma_t=(\Sigma^1_t,\dots,\Sigma^l_t)$, satisfying condition 1 below and, the following quantities being then well defined, condition 2. The consumed clock times and transition counters of $\Sigma$ are

$$\mathsf{C}^{c}_t=\int_{[0,t]}N\,\Sigma^\sigma_s\,\beta(\sigma,\gamma,\Sigma_s,a_s)\,ds,\qquad \mathsf{N}^{c}_t=p^{c}\bigl(\mathsf{C}^{c}_t\bigr)\qquad(c=(\sigma,\gamma),\ t\in[0,T]),$$

the first being the Lebesgue integral over the compact interval $[0,t]$ of a map which, under condition 1, is measurable with respect to the trace Borel $\sigma$-algebra on $[0,T]$ and the Borel $\sigma$-algebra of the real line and takes values in $[0,NB]$: under condition 1 it is a finite sum of products of indicators of intervals with maps $s\mapsto Nx^{\sigma}\beta(\sigma,\gamma,x,a_s)$, $x\in\mathbb{G}_N$; each of these maps is measurable by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable (the map $\alpha\mapsto\beta(\sigma,\gamma,x,\alpha)$ being sequentially continuous on $\mathcal{A}$ by condition 2 of Transition-Rate Family with constant first argument) and claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, the indicators, products and finite sum are measurable by claims 1, 3 and 2 of the latter, and the values lie in $[0,NB]$ because $0\le x^{\sigma}\le1$ for $x\in\Delta^l$ and $0\le\beta\le B$ by condition 1 of Transition-Rate Family; the integral over $[0,0]$ is $0$.

1. (Regularity.) $\Sigma_t\in\mathbb{G}_N$ for every $t\in[0,T]$, and there are a count $n$, either zero or a natural number, and times $0<t_1<\dots<t_n\le T$ such that $\Sigma$ is constant on $[0,t_1)$, constant on $[t_i,t_{i+1})$ for each $i\in\{1,\dots,n-1\}$, and constant on $[t_n,T]$, with the convention that for $n=0$ the map $\Sigma$ is constant on all of $[0,T]$.

2. (State identity.) For every $t\in[0,T]$,

$$\Sigma_t=x_0+\frac{1}{N}\sum_{c}v_{c}\,\mathsf{N}^{c}_t ,$$

the sum running over all transition labels.

Let moreover $\mathsf{P}$ be a family of aggregate transition clocks on $l$ states over a probability space $(\Omega,\mathcal{F},P)$. A family $(\Sigma_t)_{t\in[0,T]}$ of maps $\Sigma_t:\Omega\to\mathbb{R}^l$ is an open-loop aggregate solution driven by $\mathsf{P}$ for the control path $a$ and the initial point $x_0$ if for every $\omega\in\Omega$ the map $t\mapsto\Sigma_t(\omega)$ is an open-loop aggregate solution on $[0,T]$ for the data $(\mathsf{P}(\omega),a,x_0)$.