Closure of Approximate Test Data for Viscosity Sub- and Supersolutions 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, and let $F$ be a second-order equation operator over $\mathcal{D}_{\Sigma}$, with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$ relative to that pair, that satisfies the shift-coercivity condition and the shift-semicontinuity condition. Intrinsic test functions $\varphi$ on $\mathcal{D}$, their gradients along couplings $\nabla\varphi(\rho)\in L^{2}(\rho;\mathbb{R}^{d})$ and translation Hessians $H_{\varphi}(\rho)\in\mathcal{S}(d)$ at $\rho\in\mathcal{D}$, local maxima and local minima relative to $\mathcal{D}$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$, the cost $I(\pi)$ of a coupling, the discrepancy of two fields along it, and the bundle $\mathcal{V}(\mathcal{D}_{\Sigma})$ are those of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space and of the setting; $|s|$ is the absolute value of $s\in\mathbb{R}$. Let $\delta\in\mathbb{R}$ satisfy $0<\delta<1$.

1. (Subsolutions)¶ Let $u:\mathcal{D}\to\mathbb{R}$ be bounded above and a viscosity subsolution of $F$ relative to the penalty pair; its $\delta$-envelope $u^{-}_{\delta}$ is defined by The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §growth. Let $\rho^{*}\in\mathcal{D}$, $V_{*}\in L^{2}(\rho^{*};\mathbb{R}^{d})$ and $\mathbb{X}\in\mathcal{S}(d)$, and suppose that for every positive $\varepsilon\in\mathbb{R}$ there are $\rho\in\mathcal{D}$, an intrinsic test function $\varphi$ on $\mathcal{D}$ such that the function $\mathcal{D}\to\mathbb{R}$ with value $u^{-}_{\delta}(\rho')-\varphi(\rho')$ at $\rho'$ has a local maximum relative to $\mathcal{D}$ at $\rho$, and $\pi\in\Pi(\rho,\rho^{*})$ with

Then $\rho^{*}\in\mathcal{D}_{\Sigma}$, so that $(\rho^{*},V_{*})\in\mathcal{V}(\mathcal{D}_{\Sigma})$, and $F^{-}_{\delta}\bigl(\rho^{*},u^{-}_{\delta}(\rho^{*}),V_{*},\mathbb{X}\bigr)\le0$.

2. (Supersolutions)¶ Let $v:\mathcal{D}\to\mathbb{R}$ be bounded below and a viscosity supersolution of $F$ relative to the penalty pair; its $\delta$-envelope $v^{+}_{\delta}$ is defined by The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §growth. Let $\sigma^{*}\in\mathcal{D}$, $W_{*}\in L^{2}(\sigma^{*};\mathbb{R}^{d})$ and $\mathbb{Y}\in\mathcal{S}(d)$, and suppose that for every positive $\varepsilon\in\mathbb{R}$ there are $\sigma\in\mathcal{D}$, an intrinsic test function $\psi$ on $\mathcal{D}$ such that the function $\mathcal{D}\to\mathbb{R}$ with value $v^{+}_{\delta}(\sigma')-\psi(\sigma')$ at $\sigma'$ has a local minimum relative to $\mathcal{D}$ at $\sigma$, and $\gamma\in\Pi(\sigma,\sigma^{*})$ with

Then $\sigma^{*}\in\mathcal{D}_{\Sigma}$, so that $(\sigma^{*},W_{*})\in\mathcal{V}(\mathcal{D}_{\Sigma})$, and $0\le F^{+}_{\delta}\bigl(\sigma^{*},v^{+}_{\delta}(\sigma^{*}),W_{*},\mathbb{Y}\bigr)$.