Viscosity Subsolutions, Supersolutions and Solutions Relative to a Penalty Pair and a Profile

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$, let $F$ be a second-order equation operator over $\mathcal{D}_{\Sigma}$, let $\Phi$ be an intrinsic test function on $\mathcal{D}$, with gradients along couplings $\nabla\Phi(\nu)$ and translation Hessians $H_{\Phi}(\nu)$, and let $u:\mathcal{D}\to\mathbb{R}$. The function $u-\Phi$ on $\mathcal{D}$ takes the value $u(\mu)-\Phi(\mu)$ at $\mu$. In this item the letter $q$ denotes a vector field and the letter $r$ a real number.

1. (The shifted operator)¶ Let $(\nu,q)\in\mathcal{V}(\mathcal{D}_{\Sigma})$, $r\in\mathbb{R}$ and $Y\in\mathcal{S}(d)$. Then $\nu\in\mathcal{D}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, so $\nabla\Phi(\nu)\in T_{\nu}\subseteq L^{2}(\nu;\mathbb{R}^{d})$ by property (b) of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §differentiability and $q+\nabla\Phi(\nu)\in L^{2}(\nu;\mathbb{R}^{d})$, and $Y+H_{\Phi}(\nu)\in\mathcal{S}(d)$ by claim 1 of The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure. The operator $F$ shifted by $\Phi$ is the second-order equation operator over $\mathcal{D}_{\Sigma}$ given by

$$F^{\Phi}(\nu,r,q,Y)=F\bigl(\nu,\ r+\Phi(\nu),\ q+\nabla\Phi(\nu),\ Y+H_{\Phi}(\nu)\bigr);$$

with $\delta$-shifts $(F^{\Phi})^{-}_{\delta}$ and $(F^{\Phi})^{+}_{\delta}$ relative to the penalty pair.

2. (Subsolution)¶ Suppose that $u-\Phi$ has penalty-subordinate growth from above. The function $u$ is a viscosity subsolution of $F$ relative to the penalty pair and the profile $\Phi$ if $u-\Phi$ is a viscosity subsolution of $F^{\Phi}$ relative to the penalty pair.

3. (Supersolution)¶ Suppose that $u-\Phi$ has penalty-subordinate growth from below. The function $u$ is a viscosity supersolution of $F$ relative to the penalty pair and the profile $\Phi$ if $u-\Phi$ is a viscosity supersolution of $F^{\Phi}$ relative to the penalty pair.

4. (Solution)¶ Suppose that $u-\Phi$ has penalty-subordinate growth from above and from below. The function $u$ is a viscosity solution of $F$ relative to the penalty pair and the profile $\Phi$ if it is both a viscosity subsolution and a viscosity supersolution of $F$ relative to the penalty pair and the profile $\Phi$.