Proof of Comparison, Uniqueness and Convergence of Coarse Solutions Inherited by a Fine First-Order Equation on the Noise Wasserstein Space from a Consistent Family of Coarse Levels

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

Preliminaries. Let $N\in\mathbb{N}$. By hypothesis $w_{N}$ is a viscosity solution of $F^{(N)}$ relative to $\mathcal{P}^{(N)}$, hence, by Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §solution taken at level $N$, both a viscosity subsolution and a viscosity supersolution of $F^{(N)}$ relative to $\mathcal{P}^{(N)}$. Since $|w_{N}(\nu)|\le b_{0}$ for every $\nu\in\mathcal{D}^{(N)}$, the two-sided bound Properties of the Absolute Value in an Ordered Field §two-sided gives $-b_{0}\le w_{N}(\nu)$ and $w_{N}(\nu)\le b_{0}$ for every $\nu\in\mathcal{D}^{(N)}$. Likewise, by Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §solution, a viscosity solution of $F$ relative to $\mathcal{P}$ is both a viscosity subsolution and a viscosity supersolution of $F$ relative to $\mathcal{P}$. Finally, $\mathcal{E}$ is a function from $\mathcal{D}$ to $\mathbb{R}$ by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, so $\mathcal{E}(\mu)$ is a real number for every $\mu\in\mathcal{D}$, and $p^{(N)}_{\#}\mu\in\mathcal{D}^{(N)}$ for $\mu\in\mathcal{D}$, as recorded in the statement, so that $w_{N}(p^{(N)}_{\#}\mu)$ is defined.

Claim (comparison). Choose $b,b'\in\mathbb{R}$ with $u(\mu)\le b$ and $b'\le v(\mu)$ for every $\mu\in\mathcal{D}$. Fix $\mu\in\mathcal{D}$, put $c=\mathcal{E}(\mu)$, and let $\theta\in\mathbb{R}$ with $0<\theta$. The choices are made in this order. First, A Cross-Level Comparison Principle for First-Order Equations on a Fine Noise Wasserstein Space and a Family of Coarse Levels Linked by Mode Restrictions §fine-below, applied with the numbers $b$, $-b_{0}$, $\theta$ and $c$ in place of its $b$, $b'$, $\theta$ and $c$, provides $N'\in\mathbb{N}$. Second, A Cross-Level Comparison Principle for First-Order Equations on a Fine Noise Wasserstein Space and a Family of Coarse Levels Linked by Mode Restrictions §coarse-below, applied with the numbers $b_{0}$, $b'$, $\theta$ and $c$ in place of its $b$, $b'$, $\theta$ and $c$, provides $N''\in\mathbb{N}$. Let $N$ be the larger of $N'$ and $N''$. Since $N'\le N$, the first of these results, applied to the viscosity subsolution $u$ (bounded above by $b$) and to the viscosity supersolution $w_{N}$ of $F^{(N)}$ (bounded below by $-b_{0}$ by the preliminaries), gives, because $\mathcal{E}(\mu)\le c$,

$$u(\mu)\le w_{N}(p^{(N)}_{\#}\mu)+\theta .$$

Since $N''\le N$, the second result, applied to the viscosity subsolution $w_{N}$ of $F^{(N)}$ (bounded above by $b_{0}$) and to the viscosity supersolution $z=v$ (bounded below by $b'$), gives

$$w_{N}(p^{(N)}_{\#}\mu)\le v(\mu)+\theta .$$

Hence $u(\mu)\le v(\mu)+2\theta$ for every positive $\theta$. If $u(\mu)\le v(\mu)$ failed, then $\theta=(u(\mu)-v(\mu))/4$ would be positive and the last inequality would give $u(\mu)-v(\mu)\le(u(\mu)-v(\mu))/2$, which is false for the positive number $u(\mu)-v(\mu)$. Therefore $u(\mu)\le v(\mu)$, and $\mu\in\mathcal{D}$ was arbitrary.

Claim (uniqueness). By the preliminaries, $w$ is a viscosity subsolution of $F$ relative to $\mathcal{P}$ that is bounded above and $w'$ is a viscosity supersolution of $F$ relative to $\mathcal{P}$ that is bounded below, so claim 1 gives $w(\mu)\le w'(\mu)$ for every $\mu\in\mathcal{D}$. Exchanging the roles of $w$ and $w'$, claim 1 gives $w'(\mu)\le w(\mu)$. Hence $w(\mu)=w'(\mu)$ for every $\mu\in\mathcal{D}$.

Claim (convergence). Choose $b_{w},b'_{w}\in\mathbb{R}$ with $b'_{w}\le w(\mu)\le b_{w}$ for every $\mu\in\mathcal{D}$, and let $\theta,c\in\mathbb{R}$ with $0<\theta$. First, A Cross-Level Comparison Principle for First-Order Equations on a Fine Noise Wasserstein Space and a Family of Coarse Levels Linked by Mode Restrictions §fine-below, applied with $b_{w}$, $-b_{0}$, $\theta$ and $c$ in place of its $b$, $b'$, $\theta$ and $c$, provides $N'\in\mathbb{N}$; second, A Cross-Level Comparison Principle for First-Order Equations on a Fine Noise Wasserstein Space and a Family of Coarse Levels Linked by Mode Restrictions §coarse-below, applied with $b_{0}$, $b'_{w}$, $\theta$ and $c$, provides $N''\in\mathbb{N}$. Let $N_{1}$ be the larger of $N'$ and $N''$, and let $N\in\mathbb{N}$ with $N_{1}\le N$ and $\mu\in\mathcal{D}$ with $\mathcal{E}(\mu)\le c$. The first result, applied to the viscosity subsolution $u=w$ of $F$ (bounded above by $b_{w}$) and the viscosity supersolution $v=w_{N}$ of $F^{(N)}$ (bounded below by $-b_{0}$), gives $w(\mu)\le w_{N}(p^{(N)}_{\#}\mu)+\theta$. The second result, applied to the viscosity subsolution $w_{N}$ of $F^{(N)}$ (bounded above by $b_{0}$) and the viscosity supersolution $z=w$ of $F$ (bounded below by $b'_{w}$), gives $w_{N}(p^{(N)}_{\#}\mu)\le w(\mu)+\theta$. Thus $-\theta\le w_{N}(p^{(N)}_{\#}\mu)-w(\mu)\le\theta$, and the two-sided bound Properties of the Absolute Value in an Ordered Field §two-sided gives

$$\bigl|w_{N}(p^{(N)}_{\#}\mu)-w(\mu)\bigr|\le\theta .$$

Claim (pointwise). Let $\mu\in\mathcal{D}$ and let $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$. Apply claim 3 with $\theta=\varepsilon/2$, which is positive, and $c=\mathcal{E}(\mu)$, to obtain $N_{1}\in\mathbb{N}$. For every $N\in\mathbb{N}$ with $N_{1}\le N$, $\mu$ satisfies $\mathcal{E}(\mu)\le c$, so $|w_{N}(p^{(N)}_{\#}\mu)-w(\mu)|\le\varepsilon/2<\varepsilon$. By Limit of a Sequence of Real Numbers, the sequence $(w_{N}(p^{(N)}_{\#}\mu))_{N\in\mathbb{N}}$ converges to $w(\mu)$.