Well-Posedness of the Hamilton-Jacobi Equation with Gibbs Score Drift for a Semiconvex Cylindrical Potential: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution

In the settings of The Real Numbers: Standing Notation and Background and 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 $K\in\mathbb{R}$ be nonnegative and let $V$ be an admissible cylindrical potential with semiconvexity constant $K$; no relation between $K$ and $\beta/\kappa$ is assumed; 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 Gibbs entropy pair with potential $V$ and temperature $\beta$, whose hypothesis holds with this $\kappa$; it is a noise penalty pair on $\mathcal{P}^{a}_{\rho}$ by The Gibbs Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain §pair, and its penalty domain $\mathcal{D}$ is the set of the measures of finite relative entropy with respect to the Gibbs measure $\gamma^{V}_{\beta}$ of $V$ at temperature $\beta$ by The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §penalty-domain. $W_{a}$ is the noise Wasserstein distance and $|s|$ the absolute value of $s\in\mathbb{R}$.

(The data)¶ $\lambda_{0},\theta\in\mathbb{R}$ satisfy $0<\lambda_{0}$ and $0<\theta\le1$, and $g:\mathcal{D}\to\mathbb{R}$ is 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 Gibbs score drift relative to $\gamma^{V}_{\beta}$, with discount $\lambda_{0}$, control cost $\theta$ and running cost $g$.

1. (Comparison)¶ Let $u,w:\mathcal{D}\to\mathbb{R}$ and $b,b'\in\mathbb{R}$ satisfy $u(\mu)\le b$ and $b'\le w(\mu)$ for every $\mu\in\mathcal{D}$; the pair being noise-closed by The Gibbs Entropy Pair is Noise-Closed, and Its Penalty Bounds the Squared Noise Wasserstein Distance to the Reference Measure §noise-closed, Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §growth shows that $u$ has penalty-subordinate growth from above and $w$ from below. Let $u$ be a viscosity subsolution and $w$ a viscosity supersolution. Then $u(\mu)\le w(\mu)$ for every $\mu\in\mathcal{D}$.

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

3. (Uniqueness)¶ Let $u,u':\mathcal{D}\to\mathbb{R}$ be viscosity solutions, each bounded. Then $u(\mu)=u'(\mu)$ for every $\mu\in\mathcal{D}$; together with claim 2, the equation has exactly one bounded viscosity solution.