Transition-Rate Family

Let $l$ and $m$ be natural numbers with $l\ge 2$ and $m\ge 1$, let $\Delta^l\subset\mathbb{R}^l$ be the probability simplex, let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$, called the control set, and let $B$ be a nonnegative real number.

A transition-rate family on $l$ states with control set $\mathcal{A}$ and rate bound $B$ is a family of functions

indexed by the ordered pairs $(\sigma,\gamma)$ with $\sigma,\gamma\in\{1,\dots,l\}$ and $\sigma\neq\gamma$, such that for every such pair:

1. (Bounds.) $0\le\beta(\sigma,\gamma,\Sigma,\alpha)\le B$ for all $\Sigma\in\Delta^l$ and $\alpha\in\mathcal{A}$.

2. (Joint continuity.) Whenever $(\Sigma_n,\alpha_n)_{n\in\mathbb{N}}$ is a sequence in $\Delta^l\times\mathcal{A}$ such that the Euclidean distances $d(\Sigma_n,\Sigma)$ and $d(\alpha_n,\alpha)$ converge to $0$ for some $\Sigma\in\Delta^l$ and $\alpha\in\mathcal{A}$, then $\beta(\sigma,\gamma,\Sigma_n,\alpha_n)\to\beta(\sigma,\gamma,\Sigma,\alpha)$.