Well-Posedness of the Hamilton-Jacobi Equation with Penalty Drift on the Noise Wasserstein Space: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution

In the settings of The Real Numbers: Standing Notation and Background and First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a noise penalty pair on $\mathcal{P}^{a}_{\rho}$, and let $|s|$ denote the absolute value of $s\in\mathbb{R}$. For $\mu\in\mathcal{D}$ the number $W_{a}(\mu,\rho)$ is a nonnegative real number, because $\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair and $\rho\in\mathcal{P}^{a}_{\rho}$ by 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 §reference. Assume the following.

(The pair)¶ The pair is noise-closed, has closed score along noise couplings, has regular penalised maxima and is displacement convex; its penalty domain $\mathcal{D}$ has the noise map property of The Noise Map Property of a Set of Probability Measures §map-property; and there is $K\in\mathbb{R}$ with

$$W_{a}(\mu,\rho)^{2}\le K\,\bigl(1+|\mathcal{E}(\mu)|\bigr)\qquad\text{for every }\mu\in\mathcal{D}.$$

(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 First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §space and with $\mathbb{R}$ carrying the metric of The Absolute Value Metric on the Real Line: for every positive $\varepsilon\in\mathbb{R}$ there is a positive $\gamma\in\mathbb{R}$ with $|g(\mu)-g(\nu)|<\varepsilon$ for all $\mu,\nu\in\mathcal{D}$ with $W_{a}(\mu,\nu)<\gamma$. 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 penalty drift on the noise Wasserstein space of the pair, with discount $\lambda_{0}$, control cost $\theta$ and running cost $g$, read as in that clause through Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §subsolution, Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §supersolution and Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §solution.

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 $\mu\in\mathcal{D}$; the pair being noise-closed by (The pair), 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 $v$ from below. 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 $u:\mathcal{D}\to\mathbb{R}$.