Jump Representation and Positive Semidefiniteness of the Aggregate Fluctuation Covariance

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

1. (Jump representation.)

$$\Theta(\Sigma,\alpha)=\sum_{(\sigma,\gamma):\,\sigma\neq\gamma}\Sigma^\sigma\,\beta(\sigma,\gamma,\Sigma,\alpha)\,(e_\gamma-e_\sigma)(e_\gamma-e_\sigma)^{\top},$$

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

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

$$\sum_{p=1}^{l}\sum_{q=1}^{l}\Theta^{pq}(\Sigma,\alpha)\,x^p\,x^q=\sum_{(\sigma,\gamma):\,\sigma\neq\gamma}\Sigma^\sigma\,\beta(\sigma,\gamma,\Sigma,\alpha)\,\big(x^\gamma-x^\sigma\big)^2,$$

over the same ordered pairs.

3. (Positive semidefiniteness.) $\Theta(\Sigma,\alpha)$ is positive semidefinite.