Well-Posedness of the Hamilton-Jacobi Equation with Gaussian Score Drift on a Hilbert Space: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution

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 $\beta$ and $\kappa$ be positive real numbers 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 $\kappa$; it is a noise penalty pair on $\mathcal{P}^{a}_{\rho}$ by The Gaussian Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain §pair, and $\mathcal{D}$ 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 $|s|$ denote the absolute value of $s\in\mathbb{R}$. Let $\lambda_{0},\theta\in\mathbb{R}$ satisfy $0<\lambda_{0}$ and $0<\theta\le1$, 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 $\mu\in\mathcal{D}$; such a $C$ exists, since if $C'$ bounds $|g|$ on $\mathcal{D}$ as in Bounded Real-Valued Function on a Set, the larger of $C'$ and $0$ serves. Viscosity subsolutions, supersolutions and solutions are those of the Hamilton-Jacobi equation with Gaussian score drift relative to $\gamma_{c}$ with temperature $\beta$, discount $\lambda_{0}$, control cost $\theta$ and running cost $g$.

1. (Comparison)¶ Let $u,v:\mathcal{D}\to\mathbb{R}$ and $b,b'\in\mathbb{R}$ satisfy $u(\mu)\le b$ and $b'\le v(\mu)$ for every $\mu\in\mathcal{D}$, let $u$ be a viscosity subsolution and $v$ a viscosity supersolution. 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 $\mu\in\mathcal{D}$. Consequently, together with claim 2, the equation has exactly one bounded viscosity solution $u:\mathcal{D}\to\mathbb{R}$.