Jump Representation and Positive Semidefiniteness of the Aggregate Fluctuation Covariance

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 rate bound $B$, and let $\Theta$ be its \reftext{def:aggregate-fluctuation-covariance-2026a}{aggregate fluctuation covariance}. Write $\Delta^l$ for the \reftext{def:probability-simplex-2026a}{probability simplex}, write $e_\gamma$ ($\gamma\in\{1,\dots,l\}$) for the $\gamma$-th standard basis vector of \reftext{def:euclidean-space-rn-2026a}{Euclidean space} $\mathbb{R}^l$, identify vectors of $\mathbb{R}^l$ with matrices having $l$ rows and one column, let products of matrices be \reftext{def:product-real-matrices-2026a}{matrix products}, let sums of matrices of equal size be taken entry by entry, and let $(\cdot)^{\top}$ be the \reftext{def:transpose-real-matrix-2026a}{transpose}. Then for every $\Sigma\in\Delta^l$ and every $\alpha\in\mathbb{R}^m$:

\textbf{1. (Jump representation.)}

the sum running over all ordered pairs $(\sigma,\gamma)\in\{1,\dots,l\}^2$ with $\sigma\neq\gamma$.

\textbf{2. (Quadratic form.)} For every $x\in\mathbb{R}^l$ with components $x^1,\dots,x^l$:

over the same ordered pairs.

\textbf{3. (Positive semidefiniteness.)} $\Theta(\Sigma,\alpha)$ is \reftext{def:positive-semidefinite-matrix-2026a}{positive semidefinite}.