The Hamilton-Jacobi Operator with Common Noise and Penalty Drift is Degenerate Elliptic

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$, let $\lambda_{0}\in\mathbb{R}$ and $\theta\in\mathbb{R}$ be positive, let $\kappa\in\mathbb{R}$ be nonnegative, let $g:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$, and let $F$ be the Hamilton-Jacobi operator with common noise and penalty drift with discount $\lambda_{0}$, common-noise intensity $\kappa$, control cost $\theta$ and running cost $g$, a second-order equation operator over $\mathcal{D}_{\Sigma}$. Then $F$ is degenerate elliptic.