Aggregate State Drift

Let $l$ and $m$ be natural numbers with $l\ge2$ and $m\ge1$, let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$, and let $\beta$ be a transition-rate family on $l$ states with control set $\mathcal{A}$ and rate bound $B$. The aggregate state drift of $\beta$ is the function $b:\Delta^l\times\mathcal{A}\to\mathbb{R}^l$, defined on the probability simplex times the control set, whose components are

where the sum runs over $\sigma\in\{1,\dots,l\}$ with $\sigma\neq\gamma$.