Well-Posedness of the Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation Relative to a Diagonal Gaussian Measure: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution

In the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $c$ be a variance vector, let $a\in\mathbb{R}$ be positive, and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gaussian free-energy pair with variances $c$ and temperature $a$. Let $\lambda_{0},\theta\in\mathbb{R}$ satisfy $0<\lambda_{0}$ and $0<\theta\le1$, and let $g:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be bounded and uniformly continuous from $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ to $\mathbb{R}$ with the metric of The Absolute Value Metric on the Real Line; fix $C\in\mathbb{R}$ with $|g(\mu)|\le C$ for every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, where $|s|$ is the absolute value. Viscosity subsolutions, supersolutions and solutions are those of the lifted Ornstein-Uhlenbeck Hamilton-Jacobi equation with discount $\lambda_{0}$, control cost $\theta$ and running cost $g$.

1. (Comparison)¶ Let $u:\mathcal{D}\to\mathbb{R}$ be a viscosity subsolution and $v:\mathcal{D}\to\mathbb{R}$ a viscosity supersolution, and let $b,b'\in\mathbb{R}$ satisfy $u(\mu)\le b$ and $b'\le v(\mu)$ for every $\mu\in\mathcal{D}$. Then $u(\mu)\le v(\mu)$ for every $\mu\in\mathcal{D}$.

2. (Existence)¶ There is a viscosity solution $u:\mathcal{D}\to\mathbb{R}$ with, $\lambda_{0}^{-1}$ denoting the multiplicative inverse of $\lambda_{0}$,

$$-\lambda_{0}^{-1}C\le u(\mu)\le\lambda_{0}^{-1}C\qquad\text{for every }\mu\in\mathcal{D}.$$

3. (Uniqueness)¶ Let $u,u':\mathcal{D}\to\mathbb{R}$ be viscosity solutions, each bounded. Then $u(\mu)=u'(\mu)$ for every $\mu\in\mathcal{D}$.