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 $p\in\mathbb{N}$, let $\Gamma\in\mathcal{M}_{p\times d}(\mathbb{R})$ be a real $p\times d$ matrix, with transpose $\Gamma^{\top}$, 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 matrix $\Gamma$, control cost $\theta$ and running cost $g$, a second-order equation operator over $\mathcal{D}_{\Sigma}$. The letter $p$ denotes the number of rows of $\Gamma$ here; the dimensions written $p$ in Probability Measures on Euclidean Space and Random Vectors: Standing Notation (i) and in Euclidean Space and Real Matrices: Standing Notation (ii) are not used. Then $F$ is degenerate elliptic.