Well-Posedness of the Hamilton-Jacobi Equation with Gaussian Score Drift and a Score-Paired Wick-Square Cost on a Hilbert Space, by Gaussian Dressing

In the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, so that the reference measure is $\rho=\gamma_{c}$, let $\beta,\kappa\in\mathbb{R}$ be positive with $c_{k}\le\kappa\,a_{k}$ for every $k\in\mathbb{N}$, and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gaussian entropy pair with temperature $\beta$, whose hypothesis holds with this $\kappa$; $\mathcal{D}$ is the set of measures of finite relative entropy with respect to $\gamma_{c}$, by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §penalty-domain. Let $w$ be a sequence of Wick couplings, with a bound $W$, let $\lambda_{0},\theta\in\mathbb{R}$ satisfy $0<\lambda_{0}$ and $0<\theta\le1$, and let $\varepsilon_{0}\in\mathbb{R}$ be positive with

$$\frac{\beta a_{k}}{c_{k}}+\lambda_{0}+\frac{2\theta\,w_{k}c_{k}}{\beta}\ge\varepsilon_{0}\qquad\text{for every }k\in\mathbb{N}.$$

Let $|s|$ denote the absolute value of $s\in\mathbb{R}$, and let $g:\mathcal{D}\to\mathbb{R}$ be bounded and uniformly continuous on $\mathcal{D}$, relative to $\mathcal{D}$ in the metric space $(\mathcal{P}^{a}_{\rho},W_{a})$ of 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 §metric and with $\mathbb{R}$ carrying the metric of The Absolute Value Metric on the Real Line; fix $C\in\mathbb{R}$ with $0\le C$ and $|g(\mu)|\le C$ for every $\mu\in\mathcal{D}$. Let $\Phi_{0}$ be the quadratic profile, $c'$ the dressed variances and $e$ the real number of Gaussian Dressing of the Score-Paired Wick-Square Cost on a Hilbert Space: Riccati Coefficients, Dressed Variances, the Quadratic Profile, the Constant, the Dressed Pair and the Shifted Operator §constant, all for these data; $\Phi_{0}$ is a noise intrinsic test function on $\mathcal{D}$ by that clause. Let $\kappa'$ be the constant of Gaussian Dressing of the Score-Paired Wick-Square Cost on a Hilbert Space: Riccati Coefficients, Dressed Variances, the Quadratic Profile, the Constant, the Dressed Pair and the Shifted Operator §variances and $(\mathcal{D}',\mathcal{D}'_{\Sigma},\mathcal{E}',\Sigma')$ the Gaussian entropy pair relative to $\gamma_{c'}$ with temperature $\beta$, whose hypothesis holds with $\kappa'$; then $\mathcal{D}'=\mathcal{D}$ and $\mathcal{D}'_{\Sigma}=\mathcal{D}_{\Sigma}$ by Gaussian Dressing of the Score-Paired Wick-Square Cost on a Hilbert Space: Riccati Coefficients, Dressed Variances, the Quadratic Profile, the Constant, the Dressed Pair and the Shifted Operator §pairs. Viscosity subsolutions, supersolutions and solutions relative to the profile $\Phi_{0}$ are those of the Hamilton-Jacobi equation with Gaussian score drift and the Wick-square cost relative to $\gamma_{c}$, with temperature $\beta$, discount $\lambda_{0}$, control cost $\theta$, Wick couplings $w$ and running cost $g$. For $u:\mathcal{D}\to\mathbb{R}$, $u-\Phi_{0}$ is the function on $\mathcal{D}$ with 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 $B_{1},B_{2}\in\mathbb{R}$ satisfy $u(\mu)-\Phi_{0}(\mu)\le B_{1}$ and $B_{2}\le v(\mu)-\Phi_{0}(\mu)$ 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 Hamilton-Jacobi equation with Gaussian score drift relative to $\gamma_{c'}$ and the pair $(\mathcal{D}',\mathcal{D}'_{\Sigma},\mathcal{E}',\Sigma')$, with temperature $\beta$, discount $\lambda_{0}$, control cost $\theta$ and running cost $\nu\mapsto g(\nu)+e$ on $\mathcal{D}$.