Aggregate Fluctuation Covariance

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 fluctuation covariance of $\beta$ is the function $\Theta$ assigning to each $(\Sigma,\alpha)$ in the probability simplex $\Delta^l$ times Euclidean space $\mathbb{R}^m$ the symmetric real matrix $\Theta(\Sigma,\alpha)$ of size $l\times l$ with entries

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

The matrix $\Theta(\Sigma,\alpha)$ is symmetric because the displayed off-diagonal formula is unchanged when $\gamma$ and $\delta$ are interchanged.