The Pointwise Supremum of a Locally Uniformly Bounded Family of Viscosity Subsolutions on the Wasserstein Space is a Viscosity Subsolution

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-coercive penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ whose penalty domain $\mathcal{D}$ has the map property, and let $F$ be a second-order equation operator over $\mathcal{D}_{\Sigma}$, with $\delta$-shifts relative to that pair. Viscosity subsolutions of $F$ relative to the pair, the $\delta$-envelopes $w^{-}_{\delta}$, functions on $\mathcal{D}$, and that a function is bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$ are those of the definitions cited. Let $\mathcal{S}$ be a nonempty set whose elements are functions from $\mathcal{P}_{2}(\mathbb{R}^{d})$ to $\mathbb{R}$, each a viscosity subsolution of $F$ relative to the penalty pair.

Assume that $\mathcal{S}$ is locally uniformly bounded above:¶ for every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ there are $c\in\mathbb{R}$ and a positive $r\in\mathbb{R}$ such that

For $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and such $c,r$, the set $\{v(\mu):v\in\mathcal{S}\}$ is nonempty, because $\mathcal{S}$ is, and bounded above by $c$, because $W_{2}(\mu,\mu)=0\le r$, $W_{2}$ being a metric by The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §metric; it therefore has a least upper bound, as recorded in The Real Numbers: Standing Notation and Background §bounds. Let $u:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be given by

Then the following hold.

1. (The supremum is locally bounded and dominates the family)¶ The function $u$ is bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, and $v(\mu)\le u(\mu)$ for every $v\in\mathcal{S}$ and every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. Consequently, for every positive $\delta\in\mathbb{R}$ and every $v\in\mathcal{S}$,

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