Stability of Comparison on the Wasserstein Space under Locally Vanishing Cost Defects

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 $F$, the reference operator, 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. Here $|s|$ is the absolute value of $s\in\mathbb{R}$, and $\lVert\cdot\rVert_{\nu}$ the norm of $L^{2}(\nu;\mathbb{R}^{d})$, which contains $\Sigma(\nu)$ for $\nu\in\mathcal{D}_{\Sigma}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair.

Let $H\in\mathbb{R}$ be nonnegative. For every $n\in\mathbb{N}$ let $F_{n}$ and $F'_{n}$ be second-order equation operators over $\mathcal{D}_{\Sigma}$ and let $h_{n},h'_{n}:\mathcal{D}_{\Sigma}\to\mathbb{R}$ satisfy $0\le h_{n}(\nu)\le H$ and $0\le h'_{n}(\nu)\le H$ for every $\nu\in\mathcal{D}_{\Sigma}$ and

for every $(\nu,q)$ in the bundle $\mathcal{V}(\mathcal{D}_{\Sigma})$, every $r\in\mathbb{R}$ and every $Y\in\mathcal{S}(d)$. Assume that the defects vanish locally uniformly: for all positive $R,C\in\mathbb{R}$ and every positive $\varepsilon\in\mathbb{R}$ there is $n_{0}\in\mathbb{N}$ such that

No semicontinuity of the defects is assumed.

Let $b,b'\in\mathbb{R}$, and for every $n\in\mathbb{N}$ let $u_{n},v_{n}:\mathcal{D}\to\mathbb{R}$ satisfy $u_{n}(\mu)\le b$ and $b'\le v_{n}(\mu)$ for every $\mu\in\mathcal{D}$; by The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §growth, $u_{n}$ has penalty-subordinate growth from above and $v_{n}$ from below. Assume that, for every $n$, $u_{n}$ is a viscosity subsolution of $F_{n}$ and $v_{n}$ is a viscosity supersolution of $F'_{n}$, both relative to the penalty pair.

(Asymptotic comparison)¶ For all positive $R,\theta\in\mathbb{R}$ there is $n_{1}\in\mathbb{N}$ such that