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

Each result cited is universally quantified over the data in its own statement.

Throughout, $F$ is the Hamilton-Jacobi operator with penalty drift of the pair with discount $\lambda_{0}$, control cost $\theta$ and running cost $g$; by The Discounted Hamilton-Jacobi Equation with a Penalty Drift on the Noise Wasserstein Space §operator it is a first-order equation operator over $\mathcal{D}_{\Sigma}$, and its $\delta$-shifts relative to the pair are those of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §operators. By The Discounted Hamilton-Jacobi Equation with a Penalty Drift on the Noise Wasserstein Space §equation, a viscosity subsolution, supersolution or solution of the equation is precisely a viscosity subsolution, supersolution or solution of $F$ relative to the pair in the sense of 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, which is the notion used in A Comparison Principle for First-Order Equations on the Noise Wasserstein Space Relative to a Noise-Closed Penalty Pair and in Perron's Method on the Noise Wasserstein Space: Existence of a Viscosity Solution of a First-Order Equation Between a Subsolution and a Supersolution. Elementary order and arithmetic of real numbers, including the properties of the absolute value, is carried by The Real Numbers: Standing Notation and Background §background.

Step 0 (the bound $C$). By (The data), $C$ is a real number with $0\le C$ and $|g(\mu)|\le C$ for every $\mu\in\mathcal{D}$, that is, a nonnegative bound for $g$ in the sense of Bounded Real-Valued Function on a Set.

Step 1 (properties of the operator). By (The pair) the pair is noise-closed, so $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$ by Basic Properties of a Noise-Closed Noise Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets and Bounded Distances §lsc. We verify the hypotheses of The Hamilton-Jacobi Operator with Penalty Drift on the Noise Wasserstein Space Satisfies the Hypotheses of the First-Order Comparison Principle for a Displacement Convex Noise Penalty Pair for the pair, the reals $\lambda_{0},\theta$ (with $0<\lambda_{0}$ and $0<\theta\le1$ by (The data)) and the function $g$. Convexity holds because the pair is displacement convex by (The pair); lower semicontinuity of the penalty was just shown; growth holds with the constant $K$ of (The pair) in the role of its $C$; and the running-cost hypothesis holds with the bound $C$ of Step 0 in the role of its $M$ and with the uniform continuity of (The data). Hence, by The Hamilton-Jacobi Operator with Penalty Drift on the Noise Wasserstein Space Satisfies the Hypotheses of the First-Order Comparison Principle for a Displacement Convex Noise Penalty Pair §conclusion, $F$ is locally strictly proper, satisfies the shift-coercivity condition, the shift-semicontinuity condition and the first-order structure condition at uniquely noise-mapped pairs, and has momentum-continuous shifts relative to the pair.

Step 2 (comparison, claim 1). Let $u$, $v$, $b$, $b'$ be as in claim 1. By (The pair) the pair is a noise-closed noise penalty pair with closed score along noise couplings whose penalty domain has the noise map property, and by Step 1 the operator $F$ is a first-order equation operator over $\mathcal{D}_{\Sigma}$, with $\delta$-shifts relative to the pair, that is locally strictly proper, satisfies the shift-coercivity, shift-semicontinuity and first-order structure conditions, and has momentum-continuous shifts. Moreover $u$ is a viscosity subsolution and $v$ a viscosity supersolution of $F$ relative to the pair, with $u\le b$ and $b'\le v$ on $\mathcal{D}$. Hence A Comparison Principle for First-Order Equations on the Noise Wasserstein Space Relative to a Noise-Closed Penalty Pair §comparison gives $u(\mu)\le v(\mu)$ for every $\mu\in\mathcal{D}$.

Step 3 (existence, claim 2). Put $\kappa_{-}=-\lambda_{0}^{-1}C$ and $\kappa_{+}=\lambda_{0}^{-1}C$, and let $u_{\kappa_{-}},u_{\kappa_{+}}:\mathcal{D}\to\mathbb{R}$ be the constant functions with values $\kappa_{-}$ and $\kappa_{+}$. Then $\lambda_{0}\kappa_{-}=-C$ and $\lambda_{0}\kappa_{+}=C$. For $\nu\in\mathcal{D}_{\Sigma}\subseteq\mathcal{D}$ one has $|g(\nu)|\le C$, hence $-C\le g(\nu)\le C$ by Properties of the Absolute Value in an Ordered Field §two-sided; that is, $\lambda_{0}\kappa_{-}\le g(\nu)$ and $g(\nu)\le\lambda_{0}\kappa_{+}$. The pair is noise-closed with regular penalised maxima by (The pair), and $\lambda_{0},\theta$ are positive; so by Constant Viscosity Subsolutions and Supersolutions of the Hamilton-Jacobi Equation with Penalty Drift on the Noise Wasserstein Space §subsolution the function $u_{\kappa_{-}}$ is a viscosity subsolution of $F$ relative to the pair, by Constant Viscosity Subsolutions and Supersolutions of the Hamilton-Jacobi Equation with Penalty Drift on the Noise Wasserstein Space §supersolution the function $u_{\kappa_{+}}$ is a viscosity supersolution of $F$ relative to the pair, and by Constant Viscosity Subsolutions and Supersolutions of the Hamilton-Jacobi Equation with Penalty Drift on the Noise Wasserstein Space §growth both have penalty-subordinate growth from above and from below. As $0\le C$ (Step 0) and $\lambda_{0}^{-1}$ is positive, $0\le\lambda_{0}^{-1}C$, and hence $u_{\kappa_{-}}(\nu)=-\lambda_{0}^{-1}C\le\lambda_{0}^{-1}C=u_{\kappa_{+}}(\nu)$ for every $\nu\in\mathcal{D}$.

We now apply Perron's Method on the Noise Wasserstein Space: Existence of a Viscosity Solution of a First-Order Equation Between a Subsolution and a Supersolution with the pair, which is noise-closed with regular penalised maxima and has penalty domain with the noise map property by (The pair), with the first-order equation operator $F$ over $\mathcal{D}_{\Sigma}$, and with the subsolution $u_{\kappa_{-}}$ (having penalty-subordinate growth from below) and the supersolution $u_{\kappa_{+}}$ (having penalty-subordinate growth from above) in the roles of the functions written $f$ and $g$ there (the latter is not the running cost $g$), which satisfy $u_{\kappa_{-}}\le u_{\kappa_{+}}$ on $\mathcal{D}$. Let $u:\mathcal{D}\to\mathbb{R}$ be the pointwise supremum of the viscosity subsolutions lying between them, as defined there. By Perron's Method on the Noise Wasserstein Space: Existence of a Viscosity Solution of a First-Order Equation Between a Subsolution and a Supersolution §solution, $u$ is a viscosity solution of $F$ relative to the pair, hence a viscosity solution of the equation. By Perron's Method on the Noise Wasserstein Space: Existence of a Viscosity Solution of a First-Order Equation Between a Subsolution and a Supersolution §bounds, for every $\mu\in\mathcal{D}$,

$$-\lambda_{0}^{-1}C=u_{\kappa_{-}}(\mu)\le u(\mu)\le u_{\kappa_{+}}(\mu)=\lambda_{0}^{-1}C,$$

which is claim 2.

Step 4 (uniqueness, claim 3). Let $u,u'$ be bounded viscosity solutions. By Bounded Real-Valued Function on a Set there are real numbers $M,M'$ with $|u(\mu)|\le M$ and $|u'(\mu)|\le M'$ for every $\mu\in\mathcal{D}$, hence $-M\le u(\mu)\le M$ and $-M'\le u'(\mu)\le M'$ by Properties of the Absolute Value in an Ordered Field §two-sided. By Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §solution each of $u,u'$ is both a viscosity subsolution and a viscosity supersolution of $F$ relative to the pair, that is, of the equation. Applying claim 1 (Step 2) with the subsolution $u$, the supersolution $u'$, $b=M$ and $b'=-M'$ gives $u(\mu)\le u'(\mu)$ for every $\mu\in\mathcal{D}$; applying it with the subsolution $u'$, the supersolution $u$, $b=M'$ and $b'=-M$ gives $u'(\mu)\le u(\mu)$ for every $\mu\in\mathcal{D}$. By antisymmetry of the order of $\mathbb{R}$, $u(\mu)=u'(\mu)$ for every $\mu\in\mathcal{D}$.

Finally, the solution $u$ of claim 2 satisfies $|u(\mu)|\le\lambda_{0}^{-1}C$ for every $\mu\in\mathcal{D}$ by Properties of the Absolute Value in an Ordered Field §two-sided, with $0\le\lambda_{0}^{-1}C$, so it is bounded in the sense of Bounded Real-Valued Function on a Set; with the uniqueness just proved, it is the only bounded viscosity solution of the equation. $\blacksquare$