Extended Aggregate Observation Drift

Let $l$ and $\tilde{l}$ be natural numbers with $l\ge2$ and $\tilde{l}\ge1$, let $\tilde{\beta}$ be an observation-rate family on $l$ states with $\tilde{l}$ observation channels and rate bound $\tilde{B}$, and let $(\tilde{U},\bar{\tilde{\beta}})$ be a twice continuously differentiable extension of $\tilde{\beta}$ with derivative bound $\tilde{K}$, points of $\tilde{U}$ being written $\Sigma$ as in that definition.

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

$$\bar{\tilde{b}}^\upsilon(\Sigma)=\sum_{\sigma=1}^{l}\Sigma^\sigma\,\bar{\tilde{\beta}}(\sigma,\upsilon,\Sigma)\qquad(\upsilon\in\{1,\dots,\tilde{l}\}).$$