The Pointwise Supremum of a Uniformly Subordinate Family of Viscosity Subsolutions on the Noise Wasserstein Space is a Viscosity Subsolution

In the setting of 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-closed noise penalty pair on $\mathcal{P}^{a}_{\rho}$ whose penalty domain $\mathcal{D}$ has the noise map property, and let $F$ be a first-order equation operator over $\mathcal{D}_{\Sigma}$, with $\delta$-shifts relative to that pair. Viscosity subsolutions of $F$ relative to the pair, penalty-subordinate growth from above of a function $w:\mathcal{D}\to\mathbb{R}$, and the $\delta$-envelopes $w^{-}_{\delta}$ of such a function, functions on $\mathcal{D}$, are those of the definitions cited. Let $\mathcal{S}$ be a nonempty set whose elements are functions from $\mathcal{D}$ to $\mathbb{R}$, each a viscosity subsolution of $F$ relative to the noise penalty pair.

Assume that $\mathcal{S}$ is uniformly subordinate from above:¶ for every positive $\delta\in\mathbb{R}$ there is $C\in\mathbb{R}$ such that

$$v(\nu)\le C+\delta\,\mathcal{E}(\nu)\qquad\text{for every }v\in\mathcal{S}\text{ and every }\nu\in\mathcal{D}.$$

For $\mu\in\mathcal{D}$ and such a $C$ for $\delta=1$, the set $\{v(\mu):v\in\mathcal{S}\}$ is nonempty, because $\mathcal{S}$ is, and bounded above by $C+\mathcal{E}(\mu)$; it therefore has a least upper bound, as recorded in The Real Numbers: Standing Notation and Background §bounds. Let $u:\mathcal{D}\to\mathbb{R}$ be given by

$$u(\mu)=\sup\{v(\mu):v\in\mathcal{S}\}\qquad\text{for }\mu\in\mathcal{D}.$$

Then the following hold.

1. (Growth of the supremum, and domination)¶ The function $u$ has penalty-subordinate growth from above, and $v(\mu)\le u(\mu)$ for every $v\in\mathcal{S}$ and every $\mu\in\mathcal{D}$. Consequently, for every positive $\delta\in\mathbb{R}$ and every $v\in\mathcal{S}$,

$$v^{-}_{\delta}(\nu)\le u^{-}_{\delta}(\nu)\qquad\text{for every }\nu\in\mathcal{D}.$$

2. (The supremum is a viscosity subsolution)¶ The function $u$ is a viscosity subsolution of $F$ relative to the noise penalty pair.