Adding a Constant to a Viscosity Subsolution or Supersolution on the Wasserstein Space

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 $\mathcal{P}_{2}(\mathbb{R}^{d})$, and let $F$ and $F'$ be second-order equation operators over $\mathcal{D}_{\Sigma}$, with $\delta$-shifts $F^{-}_{\delta},F^{+}_{\delta}$ and $F'^{-}_{\delta},F'^{+}_{\delta}$ relative to that pair. Penalty-subordinate growth from above and from below, the $\delta$-envelopes $w^{-}_{\delta}$ and $w^{+}_{\delta}$ of a function $w:\mathcal{D}\to\mathbb{R}$, and viscosity subsolutions and supersolutions relative to the penalty pair are those of the definitions cited. Let $s\in\mathbb{R}$ and $u:\mathcal{D}\to\mathbb{R}$, and let $u+s:\mathcal{D}\to\mathbb{R}$ be the function with value $u(\mu)+s$ at $\mu$. In this lemma the letters $r$ and $s$ denote real numbers; the dimension written $r$ in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation is not used.

1. (Subsolutions)¶ Suppose that $u$ is a viscosity subsolution of $F$ relative to the penalty pair (so that $u$ has penalty-subordinate growth from above), and that

Then $u+s$ has penalty-subordinate growth from above, $(u+s)^{-}_{\delta}(\mu)=u^{-}_{\delta}(\mu)+s$ for every positive $\delta\in\mathbb{R}$ and every $\mu\in\mathcal{D}$, and $u+s$ is a viscosity subsolution of $F'$ relative to the penalty pair.

2. (Supersolutions)¶ Suppose that $u$ is a viscosity supersolution of $F$ relative to the penalty pair (so that $u$ has penalty-subordinate growth from below), and that

Then $u+s$ has penalty-subordinate growth from below, $(u+s)^{+}_{\delta}(\mu)=u^{+}_{\delta}(\mu)+s$ for every positive $\delta\in\mathbb{R}$ and every $\mu\in\mathcal{D}$, and $u+s$ is a viscosity supersolution of $F'$ relative to the penalty pair.