Mean-Field Trajectory Pair

Let $l$ and $m$ be natural numbers with $l\ge2$ and $m\ge1$, let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$, let $B$ be a nonnegative real number, let $\beta$ be a transition-rate family on $l$ states with control set $\mathcal{A}$ and rate bound $B$, let $b:\Delta^l\times\mathcal{A}\to\mathbb{R}^l$ be its aggregate state drift, defined on the probability simplex $\Delta^l\subset\mathbb{R}^l$ times the control set, and let $T>0$ be a real number, called the horizon.

A mean-field trajectory pair for $\beta$ with horizon $T$ is a pair $(S,A)$ of functions $S:[0,T]\to\Delta^l$ and $A:[0,T]\to\mathcal{A}$, with values written $S_t=(S^1_t,\dots,S^l_t)$ and $A_t=(A^1_t,\dots,A^m_t)$, such that:

1. (Continuity.) Every component $t\mapsto S^\gamma_t$ ($\gamma\in\{1,\dots,l\}$) and every component $t\mapsto A^j_t$ ($j\in\{1,\dots,m\}$) is continuous on $[0,T]$.

2. (Dynamics.) For every $\gamma\in\{1,\dots,l\}$ the map $s\mapsto b^\gamma(S_s,A_s)$ is continuous on $[0,T]$, and for every $t\in[0,T]$

where for $t>0$ the integral is the Riemann integral of the restriction of the integrand to $[0,t]$, which exists by claim 3 of the integral toolkit on a compact interval, and the integral is $0$ for $t=0$.