The Langevin Hamilton-Jacobi Operator with a Density Cost Satisfies the Hypotheses of the Comparison Principle and of Perron's Method

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation. Let $V$ be a confining potential on $\mathbb{R}^{d}$, let $\lambda_{0},\sigma\in\mathbb{R}$ be positive, let $\theta\in\mathbb{R}$ satisfy $0<\theta\le1$, let $\kappa,L\in\mathbb{R}$ be nonnegative, let $g:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$, and let $\Phi$ be a convex Lipschitz integrand with constant $L$. Let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Langevin free-energy pair with potential $V$ and noise intensity $\sigma$, and let $F$ be the Langevin Hamilton-Jacobi operator with common noise and density cost with potential $V$, noise intensity $\sigma$, discount $\lambda_{0}$, common-noise intensity $\kappa$, control cost $\theta$, running cost $g$ and integrand $\Phi$, with $\delta$-shifts relative to the pair. Degenerate ellipticity, 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. The letter $\sigma$ denotes the noise intensity; the swap map written $\sigma$ in Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §swap is not used. Assume the following.

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

1. (Ellipticity)¶ $F$ is degenerate elliptic.

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