A Comparison Principle for First-Order Equations on the Noise Wasserstein Space Relative to a Noise-Closed Penalty Pair

In the setting of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a noise-closed noise penalty pair on $\mathcal{P}^{a}_{\rho}$ with closed score along noise couplings, whose penalty domain $\mathcal{D}$ has the noise map property. Let $F$ be a first-order equation operator over $\mathcal{D}_{\Sigma}$, with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$ relative to that pair, that is locally strictly proper, satisfies the shift-coercivity condition, the shift-semicontinuity condition and the first-order structure condition at uniquely noise-mapped pairs, and has momentum-continuous shifts relative to the pair. In this item the letter $r$ denotes a real number and the letter $q$ a noise field.

Let $u,v:\mathcal{D}\to\mathbb{R}$ and $b,b'\in\mathbb{R}$ satisfy $u(\mu)\le b$ and $b'\le v(\mu)$ for every $\mu\in\mathcal{D}$. By Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §growth, $u$ has penalty-subordinate growth from above and $v$ from below, so that for positive $\delta\in\mathbb{R}$ the $\delta$-envelopes $u^{-}_{\delta}$ and $v^{+}_{\delta}$, functions on $\mathcal{D}$, are defined. Assume that $u$ is a viscosity subsolution of $F$ and that $v$ is a viscosity supersolution of $F$, both relative to the noise penalty pair.

1. (Comparison of the envelopes)¶ For every positive $\theta\in\mathbb{R}$ there is a positive $\delta_{0}\in\mathbb{R}$ such that

$$u^{-}_{\delta}(\mu)-v^{+}_{\delta}(\mu)\le\theta\qquad\text{for every }\mu\in\mathcal{D}\text{ and every }\delta\in\mathbb{R}\text{ with }0<\delta<\delta_{0}.$$

2. (Comparison on the penalty domain)¶ $u(\mu)\le v(\mu)$ for every $\mu\in\mathcal{D}$.