Perron's Method on the Noise Wasserstein Space: Existence of a Viscosity Solution of a First-Order Equation Between a Subsolution and a Supersolution

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 $\mathcal{P}^{a}_{\rho}$ with regular penalised maxima, 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 $F^{-}_{\delta}$ and $F^{+}_{\delta}$ relative to that pair. Viscosity subsolutions, supersolutions and solutions of $F$ relative to the noise penalty pair, and penalty-subordinate growth from above and from below of a function on $\mathcal{D}$, are those of the items cited.

Let $f:\mathcal{D}\to\mathbb{R}$ be a viscosity subsolution of $F$ that has penalty-subordinate growth from below, let $g:\mathcal{D}\to\mathbb{R}$ be a viscosity supersolution of $F$ that has penalty-subordinate growth from above, both relative to the noise penalty pair, and assume that

$$f(\nu)\le g(\nu)\qquad\text{for every }\nu\in\mathcal{D}.$$

Let $\mathcal{G}$ be the set of all viscosity subsolutions $v:\mathcal{D}\to\mathbb{R}$ of $F$ relative to the noise penalty pair with $f(\nu)\le v(\nu)\le g(\nu)$ for every $\nu\in\mathcal{D}$. Then $f\in\mathcal{G}$, and for $\nu\in\mathcal{D}$ the set $\{v(\nu):v\in\mathcal{G}\}$ is nonempty and bounded above by $g(\nu)$, so that it 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

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

Then the following hold.

1. (The supremum lies between the data)¶ For every $\nu\in\mathcal{D}$, $f(\nu)\le u(\nu)$ and $u(\nu)\le g(\nu)$; consequently $u$ has penalty-subordinate growth from above and from below.

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