Proof of The Viscosity Property on the Wasserstein Space Depends Only on the Values on the Penalty Domain

Each result cited is universally quantified over the data in its own statement. Throughout, $\nu\in\mathcal{D}_{\Sigma}$ implies $\nu\in\mathcal{D}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair.

Claim 1. Let $\delta\in\mathbb{R}$ be positive and suppose that $u$ and $\tilde{u}$ are bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$. The functions $u-\delta\mathcal{E}$ and $\tilde{u}-\delta\mathcal{E}$ of The Delta-Envelopes of a Function on the Wasserstein Space Relative to a Penalty Pair take the values $u(\nu)-\delta\,\mathcal{E}(\nu)$ and $\tilde{u}(\nu)-\delta\,\mathcal{E}(\nu)$ at $\nu\in\mathcal{D}$, and these agree because $u$ and $\tilde{u}$ agree on $\mathcal{D}$; the two functions on $\mathcal{D}$ are therefore equal. Their upper semicontinuous envelopes are determined by their values on $\mathcal{D}$, so they too are equal, and $u^{-}_{\delta}=\tilde{u}^{-}_{\delta}$ on $\mathcal{D}$ by The Delta-Envelopes of a Function on the Wasserstein Space Relative to a Penalty Pair §minus. If instead $u$ and $\tilde{u}$ are bounded below near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, the same argument with the lower semicontinuous envelopes and The Delta-Envelopes of a Function on the Wasserstein Space Relative to a Penalty Pair §plus gives $u^{+}_{\delta}=\tilde{u}^{+}_{\delta}$ on $\mathcal{D}$.

Claim 2. Suppose that $u$ and $\tilde{u}$ are bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$ and that $u$ is a viscosity subsolution of $F$ relative to the penalty pair. Let $\delta\in\mathbb{R}$ be positive, let $\varphi$ be a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$, let $\hat{\mu}\in\mathcal{D}$ be a point at which the function on $\mathcal{D}$ with value $\tilde{u}^{-}_{\delta}(\mu)-\varphi(\mu)$ at $\mu$ has a local maximum relative to $\mathcal{D}$, and let $\varepsilon\in\mathbb{R}$ be positive. By claim 1 this function is the function on $\mathcal{D}$ with value $u^{-}_{\delta}(\mu)-\varphi(\mu)$ at $\mu$, so $\hat{\mu}$ is a point at which the latter has a local maximum relative to $\mathcal{D}$. Applying Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution to $u$, with these $\delta$, $\varphi$, $\hat{\mu}$ and $\varepsilon$, gives $\nu\in\mathcal{D}_{\Sigma}$, $\pi\in\Pi(\nu,\hat{\mu})$, $s\in\mathbb{R}$, $q\in L^{2}(\nu;\mathbb{R}^{d})$ and $Y\in\mathcal{S}(d)$ satisfying the six conditions listed there for $u$. Of those conditions, the two that mention $u$ do so only through the values $u^{-}_{\delta}(\nu)$ and $u^{-}_{\delta}(\hat{\mu})$, which equal $\tilde{u}^{-}_{\delta}(\nu)$ and $\tilde{u}^{-}_{\delta}(\hat{\mu})$ by claim 1, the points $\nu$ and $\hat{\mu}$ lying in $\mathcal{D}$. The same data therefore satisfy the six conditions for $\tilde{u}$, and since $\delta$, $\varphi$, $\hat{\mu}$ and $\varepsilon$ were arbitrary, $\tilde{u}$ is a viscosity subsolution of $F$ relative to the penalty pair.

Claim 3. The same argument, with Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §supersolution in place of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution, local minima in place of local maxima and the envelopes $u^{+}_{\delta}$, $\tilde{u}^{+}_{\delta}$ of claim 1 in place of $u^{-}_{\delta}$, $\tilde{u}^{-}_{\delta}$, shows that $\tilde{u}$ is a viscosity supersolution of $F$ relative to the penalty pair whenever $u$ is.

Claim 4. A viscosity solution is both a viscosity subsolution and a viscosity supersolution, by Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §solution; claims 2 and 3 transfer each property to $\tilde{u}$, which is therefore a viscosity solution of $F$ relative to the penalty pair.