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

In the setting of The Real Numbers: Standing Notation and Background, let the fine level, the levels $N\in\mathbb{N}$, the mode-restriction links $(\kappa^{(N)},p^{(N)},j^{(N)})$, the noise penalty pairs $\mathcal{P}=(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ and $\mathcal{P}^{(N)}=(\mathcal{D}^{(N)},\mathcal{D}^{(N)}_{\Sigma},\mathcal{E}^{(N)},\Sigma^{(N)})$ and the first-order equation operators $F$ and $F^{(N)}$ be as in 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, with that notation, and suppose that they satisfy its hypotheses 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, 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, 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 §uniform-penalty, 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 §uniform-proper, 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 §uniform-coercive, 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 §uniform-structure, 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 §uniform-momentum, 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 §link and 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 §defect. For $\mu\in\mathcal{D}$ one has $p^{(N)}_{\#}\mu\in\mathcal{D}^{(N)}$ by 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 §link and Compatible Noise Penalty Pairs at a Fine and a Coarse Level §compatible. By Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §growth, at the fine level and at level $N$, the pairs being noise-closed by 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 and 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, functions on $\mathcal{D}$ or on $\mathcal{D}^{(N)}$ that are bounded above, respectively below, have penalty-subordinate growth from above, respectively from below, so that viscosity subsolutions, supersolutions and solutions of $F$ relative to $\mathcal{P}$, and of $F^{(N)}$ relative to $\mathcal{P}^{(N)}$ at level $N$, are defined for them. Suppose moreover that there is $b_{0}\in\mathbb{R}$ and, for every $N\in\mathbb{N}$, a viscosity solution $w_{N}:\mathcal{D}^{(N)}\to\mathbb{R}$ of $F^{(N)}$ relative to $\mathcal{P}^{(N)}$ with $|w_{N}(\nu)|\le b_{0}$ for every $\nu\in\mathcal{D}^{(N)}$, where $|s|$ is the absolute value of $s\in\mathbb{R}$. Then the following hold.

1. (Comparison)¶ Let $u:\mathcal{D}\to\mathbb{R}$ be a viscosity subsolution of $F$ relative to $\mathcal{P}$ that is bounded above, and let $v:\mathcal{D}\to\mathbb{R}$ be a viscosity supersolution of $F$ relative to $\mathcal{P}$ that is bounded below. Then $u(\mu)\le v(\mu)$ for every $\mu\in\mathcal{D}$.

2. (Uniqueness)¶ Let $w,w':\mathcal{D}\to\mathbb{R}$ be viscosity solutions of $F$ relative to $\mathcal{P}$, each bounded above and below. Then $w(\mu)=w'(\mu)$ for every $\mu\in\mathcal{D}$.

3. (Convergence of the coarse solutions)¶ Let $w:\mathcal{D}\to\mathbb{R}$ be a viscosity solution of $F$ relative to $\mathcal{P}$ that is bounded above and below. Then for all $\theta,c\in\mathbb{R}$ with $0<\theta$ there is $N_{1}\in\mathbb{N}$ such that

$$\bigl|w_{N}(p^{(N)}_{\#}\mu)-w(\mu)\bigr|\le\theta\qquad\text{for every }N\in\mathbb{N}\text{ with }N_{1}\le N\text{ and every }\mu\in\mathcal{D}\text{ with }\mathcal{E}(\mu)\le c.$$

¶In particular, for every $\mu\in\mathcal{D}$ the sequence $(w_{N}(p^{(N)}_{\#}\mu))_{N\in\mathbb{N}}$ converges to $w(\mu)$.