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

Each result cited is universally quantified over the data in its own statement.

Let $(\nu,q)\in\mathcal{V}(\mathcal{D}_{\Sigma})$, let $r\in\mathbb{R}$ and let $X,Y\in\mathcal{S}(d)$ satisfy $X\preceq Y$. Apply The Trace as a Sum of Quadratic Forms, its Monotonicity and a Norm Bound §monotone with its matrix $A$ taken to be $\Gamma$; the letters $m$ and $p$ of that lemma are its own dimensions, and here its $m$ is read as our $p$ (the number of rows of $\Gamma$) and its $p$ as our $d$, so that its $\mathcal{S}(p)$ is our $\mathcal{S}(d)$ and its $A^{\top}AX$ is our $\Gamma^{\top}\Gamma X$ (its hypotheses $1\le m$ and $1\le p$ hold because the matrix sets of Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §matrices, to which $\Gamma$ belongs, are formed only for dimensions at least $1$). It gives

The multiplier $\tfrac{1}{2}$, the product of $1$ with the multiplicative inverse of $2$, equals that inverse and is positive: $0<2$ and the inverse exists by claim 8 of Elementary Order Arithmetic in an Ordered Field, and the inverse is positive by claim 7 of that lemma. In particular $0\le\tfrac{1}{2}$, the strict order $<$ entailing $\le$ by its definition there. Hence claim 5 of Elementary Arithmetic in an Ordered Field, applied to the displayed inequality with $\tfrac{1}{2}$ as the nonnegative multiplier, gives

and the sign reversal of claim 4 of Elementary Order Arithmetic in an Ordered Field gives

Adding to both sides the real number $\lambda_{0}\,r+\tfrac{\theta}{2}\lVert q\rVert_{\nu}^{2}+\langle\Sigma(\nu),q\rangle_{\nu}-g(\nu)$, which does not depend on the matrix argument — the compatibility of the order with addition, an axiom of Ordered Field — and reading off the value of the operator from The Discounted Hamilton-Jacobi Equation with Common Noise and a Penalty Drift on the Wasserstein Space §operator gives

Since $(\nu,q)$, $r$, $X$ and $Y$ were arbitrary, $F$ is degenerate elliptic by Degenerate Elliptic Second-Order Equation Operators on the Wasserstein Space §elliptic.