The Hamilton-Jacobi Operator with Penalty Drift on the Noise Wasserstein Space Satisfies the Hypotheses of the First-Order Comparison Principle for a Displacement Convex Noise Penalty Pair

In the setting of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a noise penalty pair on $\mathcal{P}^{a}_{\rho}$, let $\lambda_{0},\theta\in\mathbb{R}$ satisfy $0<\lambda_{0}$ and $0<\theta\le1$, let $g:\mathcal{D}\to\mathbb{R}$, and let $F$ be the Hamilton-Jacobi operator with penalty drift of the pair with discount $\lambda_{0}$, control cost $\theta$ and running cost $g$, a first-order equation operator over $\mathcal{D}_{\Sigma}$ by that clause, with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$ relative to the pair. Displacement convexity of the pair, being locally strictly proper with properness constants, the shift-coercivity condition, the shift-semicontinuity condition, the first-order structure condition at uniquely noise-mapped pairs and momentum-continuous shifts are those of the definitions cited. Lower semicontinuity of $\mathcal{E}$ and uniform continuity of $g$ are taken relative to $\mathcal{D}$ in the metric space $(\mathcal{P}^{a}_{\rho},W_{a})$ of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §space, with $\mathbb{R}$ carrying the metric of The Absolute Value Metric on the Real Line, and boundedness of $g$ is that of that definition; $|s|$ is the absolute value of $s\in\mathbb{R}$. For $\mu\in\mathcal{D}$ the number $W_{a}(\mu,\rho)$ is a nonnegative real number, because $\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair and $\rho\in\mathcal{P}^{a}_{\rho}$ by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §reference. Assume the following.

(Convexity)¶ The pair is displacement convex.

(Lower semicontinuity of the penalty)¶ $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$.

(Growth)¶ There is $C\in\mathbb{R}$ with

$$W_{a}(\mu,\rho)^{2}\le C\,\bigl(1+|\mathcal{E}(\mu)|\bigr)\qquad\text{for every }\mu\in\mathcal{D}.$$

(Running cost)¶ $g$ is bounded and uniformly continuous on $\mathcal{D}$; that is, there is $M\in\mathbb{R}$ with $0\le M$ and $|g(\mu)|\le M$ for every $\mu\in\mathcal{D}$, and for every positive $\varepsilon\in\mathbb{R}$ there is a positive $\gamma\in\mathbb{R}$ such that $|g(\mu)-g(\nu)|<\varepsilon$ for all $\mu,\nu\in\mathcal{D}$ with $W_{a}(\mu,\nu)<\gamma$.

Then the following hold.

1. (Local strict properness)¶ For every positive $R\in\mathbb{R}$, $\lambda_{0}$ is a properness constant for $F$ at $R$; hence $F$ is locally strictly proper.

2. (Shift-coercivity)¶ $F$ satisfies the shift-coercivity condition.

3. (Shift-semicontinuity)¶ $F$ satisfies the shift-semicontinuity condition.

4. (Structure)¶ $F$ satisfies the first-order structure condition at uniquely noise-mapped pairs.

5. (Momentum)¶ $F$ has momentum-continuous shifts relative to the pair.

6. (Conclusion)¶ Consequently $F$ is locally strictly proper, satisfies the shift-coercivity condition, the shift-semicontinuity condition and the first-order structure condition at uniquely noise-mapped pairs, and has momentum-continuous shifts relative to the pair.