Well-Posedness of the Lifted Hamilton-Jacobi Equation with a Wick-Square Cost Relative to a Diagonal Gaussian Measure, by Gaussian Dressing

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 $0<\theta\le1$, let $\beta\in\mathbb{R}^{d}$ satisfy $(a/c_{i})^{2}+\lambda_{0}a/c_{i}+2\theta\beta_{i}>0$ for every $i\in[d]$, 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. Let $c'$ be the dressed variances, $\Phi_{0}$ the quadratic profile and $e=\sum_{i=1}^{d}(a\,b_{i}-\beta_{i}c_{i})$ the real number of Gaussian Dressing of the Wick-Square Cost: the Riccati Coefficients, the Dressed Variances, the Quadratic Profile, the Shifted Pair and the Shifted Operator §operator, with $b_{1},\dots,b_{d}$ the Riccati coefficients of Gaussian Dressing of the Wick-Square Cost: the Riccati Coefficients, the Dressed Variances, the Quadratic Profile, the Shifted Pair and the Shifted Operator §riccati, all for these $c$, $a$, $\lambda_{0}$, $\theta$ and $\beta$. The function $\Phi_{0}$ is an intrinsic test function on $\mathcal{D}$ by Gaussian Dressing of the Wick-Square Cost: the Riccati Coefficients, the Dressed Variances, the Quadratic Profile, the Shifted Pair and the Shifted Operator §profile and Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §restriction, and the penalty domain of the Gaussian free-energy pair with variances $c'$ and temperature $a$ is $\mathcal{D}$ by Gaussian Dressing of the Wick-Square Cost: the Riccati Coefficients, the Dressed Variances, the Quadratic Profile, the Shifted Pair and the Shifted Operator §pairs. Viscosity subsolutions, supersolutions and solutions relative to the profile $\Phi_{0}$ are those of the lifted Wick-square Hamilton-Jacobi equation with discount $\lambda_{0}$, control cost $\theta$, couplings $\beta$ and running cost $g$; for $u:\mathcal{D}\to\mathbb{R}$ the function $u-\Phi_{0}$ on $\mathcal{D}$ takes the value $u(\mu)-\Phi_{0}(\mu)$ at $\mu$, and $\lambda_{0}^{-1}$ is the multiplicative inverse of $\lambda_{0}$.

1. (Comparison)¶ Let $u:\mathcal{D}\to\mathbb{R}$ be a viscosity subsolution and $v:\mathcal{D}\to\mathbb{R}$ a viscosity supersolution relative to the profile $\Phi_{0}$, and let $k,k'\in\mathbb{R}$ satisfy $u(\mu)-\Phi_{0}(\mu)\le k$ and $k'\le v(\mu)-\Phi_{0}(\mu)$ 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 $U:\mathcal{D}\to\mathbb{R}$ relative to the profile $\Phi_{0}$ with

$$-\lambda_{0}^{-1}\bigl(C+|e|\bigr)\le U(\mu)-\Phi_{0}(\mu)\le\lambda_{0}^{-1}\bigl(C+|e|\bigr)\qquad\text{for every }\mu\in\mathcal{D}.$$

3. (Uniqueness)¶ Let $U,U':\mathcal{D}\to\mathbb{R}$ be viscosity solutions relative to the profile $\Phi_{0}$ such that $U-\Phi_{0}$ and $U'-\Phi_{0}$ are bounded. Then $U(\mu)=U'(\mu)$ for every $\mu\in\mathcal{D}$.

4. (Representation)¶ Let $U:\mathcal{D}\to\mathbb{R}$ be such that $U-\Phi_{0}$ is bounded. Then $U$ is a viscosity solution relative to the profile $\Phi_{0}$ if and only if $U-\Phi_{0}$ is a viscosity solution of the lifted Ornstein-Uhlenbeck Hamilton-Jacobi equation with variances $c'$, temperature $a$, discount $\lambda_{0}$, control cost $\theta$ and running cost $\nu\mapsto g(\nu)+e$.

5. (Constant cost)¶ Suppose that $g$ is constant, with value $g_{0}$. Then the function $U_{0}:\mathcal{D}\to\mathbb{R}$, $U_{0}(\mu)=\Phi_{0}(\mu)+\lambda_{0}^{-1}(g_{0}+e)$, is a viscosity solution relative to the profile $\Phi_{0}$, and every viscosity solution $U$ relative to the profile $\Phi_{0}$ with $U-\Phi_{0}$ bounded equals $U_{0}$.