Extended Aggregate State Drift

Let $l$ and $m$ be \reftext{def:natural-numbers-2026a}{natural numbers} with $l\ge2$ and $m\ge1$, let $\beta$ be a \reftext{def:transition-rate-family-2026a}{transition-rate family} on $l$ states with control dimension $m$, and let $(U,\bar{\beta})$ be a \reftext{def:c2-transition-rate-extension-2026a}{twice continuously differentiable extension} of $\beta$ with derivative bound $K$, points of $U\times\mathbb{R}^m$ being written $x=(\Sigma,\alpha)$ as in that definition.

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

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