Perron's Method on the Wasserstein Space: Existence of a Viscosity Solution Between a Subsolution and a Supersolution

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a Wasserstein-closed penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ with regular penalised maxima, whose penalty domain $\mathcal{D}$ has the map property, and let $F$ be a second-order equation operator over $\mathcal{D}_{\Sigma}$ that is degenerate elliptic. Viscosity subsolutions, supersolutions and solutions of $F$ relative to the penalty pair, and penalty-subordinate growth from above and from below of a function on $\mathcal{D}$, are those of the definitions 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 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 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 (v). 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)(i) 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)(ii) The function $u$ is a viscosity solution of $F$ relative to the penalty pair.