Changing the Penalty of a Noise Penalty Pair by a Noise Intrinsic Test Function and a Constant Does Not Change the Viscosity Notion

In the setting of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation, let $P=(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ and $P'=(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E}',\Sigma')$ be noise penalty pairs on $\mathcal{P}^{a}_{\rho}$ with the same penalty domain and the same score domain, and let $\Psi$ be a noise intrinsic test function on $\mathcal{D}$, with gradients along noise couplings $\nabla\Psi(\nu)$, and $C_{0}\in\mathbb{R}$ such that

$$\mathcal{E}'(\mu)=\mathcal{E}(\mu)+\Psi(\mu)+C_{0}\quad(\mu\in\mathcal{D}),\qquad\Sigma'(\nu)=\Sigma(\nu)+\nabla\Psi(\nu)\quad(\nu\in\mathcal{D}_{\Sigma}).$$

Let $F$ be a first-order equation operator over $\mathcal{D}_{\Sigma}$ and let $v:\mathcal{D}\to\mathbb{R}$. Penalty-subordinate growth, the $\delta$-envelopes of $v$, the $\delta$-shifts of $F$ and viscosity sub- and supersolutions (Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §subsolution, Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §supersolution) are taken relative to $P$ or to $P'$ according to the pair named.

(Change of the penalty)¶ If $v$ has penalty-subordinate growth from above relative to $P$ and relative to $P'$, then $v$ is a viscosity subsolution of $F$ relative to $P$ if and only if it is one relative to $P'$. If $v$ has penalty-subordinate growth from below relative to $P$ and relative to $P'$, then $v$ is a viscosity supersolution of $F$ relative to $P$ if and only if it is one relative to $P'$.