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 $X\preceq Y$. By The Trace as a Sum of Quadratic Forms, its Monotonicity and a Norm Bound (iv), $\mathrm{tr}\,X\le\mathrm{tr}\,Y$.

The multiplier $\tfrac{\kappa}{2}$, the product of $\kappa$ with the multiplicative inverse of $2$, is nonnegative: that inverse is positive by claims 8 and 7 of Elementary Order Arithmetic in an Ordered Field, and claim 5 of Elementary Arithmetic in an Ordered Field, applied to $0\le\kappa$ with it as the nonnegative multiplier, gives $0\le\tfrac{\kappa}{2}$, since the product of it with $0$ is $0$ by claim 1 of Zero Products and Elementary Identities in a Field and multiplication is commutative in the ordered field $\mathbb{R}$. Hence claim 5 of Elementary Arithmetic in an Ordered Field gives

$$\frac{\kappa}{2}\,\mathrm{tr}\,X\le\frac{\kappa}{2}\,\mathrm{tr}\,Y ,$$

and claim 3 of that lemma, used in both directions, gives

$$-\frac{\kappa}{2}\,\mathrm{tr}\,Y\le-\frac{\kappa}{2}\,\mathrm{tr}\,X .$$

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 (i) gives

$$F(\nu,r,q,Y)\le F(\nu,r,q,X).$$

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