The Hamilton-Jacobi Operator with Common Noise and Penalty Drift Satisfies the Hypotheses of the Comparison Principle for a Displacement Convex Pair

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$, with translation Hessian $H_{\mathcal{E}}$ and $\mathrm{tr}$ the trace, let $\lambda_{0},\theta\in\mathbb{R}$ satisfy $0<\lambda_{0}$ and $0<\theta\le1$, 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 of the pair with discount $\lambda_{0}$, common-noise intensity $\kappa$, control cost $\theta$ and running cost $g$, with $\delta$-shifts relative to the pair. Displacement convexity, being locally strictly proper, the shift-coercivity condition, the shift-semicontinuity condition and the second-order structure condition at uniquely mapped pairs are those of the definitions cited; boundedness and uniform continuity of $g$ refer to $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ and $\mathbb{R}$ with the metric of The Absolute Value Metric on the Real Line, and lower semicontinuity and continuity on subsets of $\mathcal{D}$ are taken relative to those subsets in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$. Assume the following.

(Convexity)¶ The pair is displacement convex.

(Semicontinuity)¶ $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$.

(Growth)¶ There is $C\in\mathbb{R}$ with $M_{2}(\mu)\le C(1+|\mathcal{E}(\mu)|)$ and $|\mathrm{tr}\,H_{\mathcal{E}}(\mu)|\le C(1+|\mathcal{E}(\mu)|)$ for every $\mu\in\mathcal{D}$.

(Hessian continuity)¶ For every positive $R\in\mathbb{R}$ the restriction of $\mu\mapsto\mathrm{tr}\,H_{\mathcal{E}}(\mu)$ to $\{\mu\in\mathcal{D}:|\mathcal{E}(\mu)|\le R\}$ is continuous.

(Running cost)¶ $g$ is bounded and uniformly continuous.

Then $F$ is locally strictly proper and satisfies the shift-coercivity condition, the shift-semicontinuity condition and the second-order structure condition at uniquely mapped pairs.¶