Viscosity Subsolutions, Supersolutions and Solutions on the Noise Wasserstein Space Relative to a Noise Penalty Pair and a Profile

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 penalty pair on $\mathcal{P}^{a}_{\rho}$, let $F$ be a first-order equation operator over $\mathcal{D}_{\Sigma}$, let $\Phi$ be a noise intrinsic test function on $\mathcal{D}$, with gradients along noise couplings $\nabla\Phi(\nu)$, 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 noise field and the letter $r$ a real number.

1. (The shifted operator)¶ Let $(\nu,q)\in\mathcal{V}^{a}(\mathcal{D}_{\Sigma})$, the bundle of noise fields over $\mathcal{D}_{\Sigma}$, and $r\in\mathbb{R}$. Then $\nu\in\mathcal{D}$ by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, so $\nabla\Phi(\nu)\in T^{a}_{\nu}\subseteq L^{2}(\nu;X^{a})$ by property (b) of Noise Intrinsic Test Functions on the Noise Wasserstein Space §differentiability, and $q+\nabla\Phi(\nu)\in L^{2}(\nu;X^{a})$. The operator $F$ shifted by $\Phi$ is the first-order equation operator over $\mathcal{D}_{\Sigma}$ given by

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

with $\delta$-shifts $(F^{\Phi})^{-}_{\delta}$ and $(F^{\Phi})^{+}_{\delta}$ relative to the noise 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 noise penalty pair and the profile $\Phi$ if $u-\Phi$ is a viscosity subsolution of $F^{\Phi}$ relative to the noise 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 noise penalty pair and the profile $\Phi$ if $u-\Phi$ is a viscosity supersolution of $F^{\Phi}$ relative to the noise 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 noise penalty pair and the profile $\Phi$ if it is both a viscosity subsolution and a viscosity supersolution of $F$ relative to the noise penalty pair and the profile $\Phi$.