Aggregate State Drift

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

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