Uniqueness and Continuity on Energy Sublevel Sets of Bounded Viscosity Solutions on the Wasserstein Space

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 $\mathcal{P}_{2}(\mathbb{R}^{d})$ with closed score along couplings, whose penalty domain $\mathcal{D}$ has the map property. Let $F$ be a second-order equation operator over $\mathcal{D}_{\Sigma}$, with $\delta$-shifts relative to that pair, that is locally strictly proper and satisfies the shift-coercivity condition, the shift-semicontinuity condition and the second-order structure condition at uniquely mapped pairs. If $u:\mathcal{D}\to\mathbb{R}$ is bounded, with bound $b$, then $-b\le u(\mu)\le b$ for every $\mu\in\mathcal{D}$ by claim 6 of Properties of the Absolute Value in an Ordered Field, so $u$ has penalty-subordinate growth from above and from below by The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §growth, so that being a viscosity solution of $F$ relative to the penalty pair is defined for it.

1. (Uniqueness)¶ If $u,v:\mathcal{D}\to\mathbb{R}$ are bounded viscosity solutions of $F$ relative to the penalty pair, then $u(\mu)=v(\mu)$ for every $\mu\in\mathcal{D}$.

2. (Continuity on sublevel sets)¶ If $u:\mathcal{D}\to\mathbb{R}$ is a bounded viscosity solution of $F$ relative to the penalty pair and $c\in\mathbb{R}$, then the restriction of $u$ to $K_{c}=\{\mu\in\mathcal{D}:\mathcal{E}(\mu)\le c\}$ is uniformly continuous on $K_{c}$, for the metric spaces $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ and $\mathbb{R}$ with the metric of The Absolute Value Metric on the Real Line.