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

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-closed penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ with closed score along couplings, whose penalty domain $\mathcal{D}$ has the map property. Let $F$ be a second-order equation operator over $\mathcal{D}_{\Sigma}$, with $\delta$-shifts relative to that pair, that does not depend on its matrix argument, that is,

$$F(\nu,r,q,Y)=F(\nu,r,q,Y')\qquad\text{for all }(\nu,q)\in\mathcal{V}(\mathcal{D}_{\Sigma}),\ r\in\mathbb{R}\text{ and }Y,Y'\in\mathcal{S}(d),$$

and that is locally strictly proper, satisfies the shift-coercivity condition, the shift-semicontinuity condition and the second-order structure condition at uniquely 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 vector 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 The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-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 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 $\mu\in\mathcal{D}$.