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

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $F$ be a second-order equation operator over $\mathcal{D}_{\Sigma}$. The $\delta$-envelopes $u^{-}_{\delta}$ and $u^{+}_{\delta}$, functions on $\mathcal{D}$, that a real-valued function is bounded above, or below, near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, and the viscosity subsolutions, supersolutions and solutions of $F$ relative to the penalty pair are those of the definitions cited. Let $u,\tilde{u}:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ satisfy

$$u(\nu)=\tilde{u}(\nu)\qquad\text{for every }\nu\in\mathcal{D}.$$

Then the following hold.

1. (The $\delta$-envelopes agree)(i) If $u$ and $\tilde{u}$ are bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, then $u^{-}_{\delta}(\nu)=\tilde{u}^{-}_{\delta}(\nu)$ for every positive $\delta\in\mathbb{R}$ and every $\nu\in\mathcal{D}$. If $u$ and $\tilde{u}$ are bounded below near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, then $u^{+}_{\delta}(\nu)=\tilde{u}^{+}_{\delta}(\nu)$ for every positive $\delta\in\mathbb{R}$ and every $\nu\in\mathcal{D}$.

2. (Subsolutions)(ii) If $u$ and $\tilde{u}$ are bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$ and $u$ is a viscosity subsolution of $F$ relative to the penalty pair, then $\tilde{u}$ is a viscosity subsolution of $F$ relative to the penalty pair.

3. (Supersolutions)(iii) If $u$ and $\tilde{u}$ are bounded below near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$ and $u$ is a viscosity supersolution of $F$ relative to the penalty pair, then $\tilde{u}$ is a viscosity supersolution of $F$ relative to the penalty pair.

4. (Solutions)(iv) If $u$ and $\tilde{u}$ are bounded above near each point and bounded below near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$ and $u$ is a viscosity solution of $F$ relative to the penalty pair, then $\tilde{u}$ is a viscosity solution of $F$ relative to the penalty pair.