Extended 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 $\mathbb{R}^m$, let $\beta$ be a transition-rate family on $l$ states with control set $\mathcal{A}$ and rate bound $B$, and let $(U,V,\bar{\beta})$ be a twice continuously differentiable extension of $\beta$ with derivative bound $K$, points of $U\times V$ being written $x=(\Sigma,\alpha)$ as in that definition.

The extended aggregate state drift of $(U,V,\bar{\beta})$ is the function $\bar{b}:U\times V\to\mathbb{R}^l$ whose components are

$$\bar{b}^\gamma(\Sigma,\alpha)=\sum_{\sigma:\sigma\neq\gamma}\Big(\Sigma^\sigma\,\bar{\beta}(\sigma,\gamma,\Sigma,\alpha)-\Sigma^\gamma\,\bar{\beta}(\gamma,\sigma,\Sigma,\alpha)\Big)\qquad(\gamma\in\{1,\dots,l\}),$$

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