A Comparison Principle for Viscosity Solutions on the Lift of the Wasserstein Space

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, assume that $(\Omega,\mathcal{F},P)$ is rich, which Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (v) does not assume, and 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. Let $F$ be a second-order equation operator on the lift over $\mathcal{D}_{\Sigma}$ that is locally strictly proper and satisfies the shift-coercivity condition, the shift-semicontinuity condition and the second-order structure condition at optimally coupled pairs.

The Wasserstein space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ and the law $\mathcal{L}(X)$ of a class are those of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (i). Upper and lower semicontinuity of a real-valued function on $\mathcal{P}_{2}(\mathbb{R}^{d})$, relative to $\mathcal{P}_{2}(\mathbb{R}^{d})$, are understood in that metric space.

Let $u,v:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ and $b,b'\in\mathbb{R}$ be such that $u$ is upper semicontinuous, $v$ is lower semicontinuous, and $u(\sigma)\le b$ and $b'\le v(\sigma)$ for every $\sigma\in\mathcal{P}_{2}(\mathbb{R}^{d})$. Then $u$ is bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$ and $v$ is bounded below near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, since for $\sigma\in\mathcal{P}_{2}(\mathbb{R}^{d})$ the radius $1$ witnesses that $b$ lies in the set written $A_{u}(\sigma)$ there and that $b'$ lies in the set written $B_{v}(\sigma)$. Assume that $u$ is a viscosity subsolution of $F$ on the lift and that $v$ is a viscosity supersolution of $F$ on the lift, both relative to the penalty pair.

(Comparison on the penalty domain)(i) Then

$$u(\mu)\ \le\ v(\mu)\qquad\text{for every }\mu\in\mathcal{D}.$$