Shifting the Penalised Viscosity Notion by an Intrinsic Test Function: Composition of Shifts, the Shift Identity, Invariance under a Change of Profile, and a Change of the Penalty

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $P=(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $F$ be a second-order equation operator over $\mathcal{D}_{\Sigma}$. Intrinsic test functions on $\mathcal{D}$, their gradients along couplings and their translation Hessians are those of that definition. For an intrinsic test function $\Phi$ on $\mathcal{D}$, $F^{\Phi}$ is the operator $F$ shifted by $\Phi$, and viscosity subsolutions and supersolutions of $F$ relative to a penalty pair and the profile $\Phi$ are those of that definition (and its supersolution clause). Viscosity subsolutions and supersolutions of an operator relative to a penalty pair are those of that definition, and penalty-subordinate growth from above and from below relative to a penalty pair is that of that definition. For functions $v,\Theta$ on $\mathcal{D}$, $v-\Theta$ takes the value $v(\mu)-\Theta(\mu)$ at $\mu$.

1. (Composition of shifts)¶ The zero function on $\mathcal{P}_{2}(\mathbb{R}^{d})$, written $0$, is an intrinsic test function on $\mathcal{D}$, and $F^{0}=F$. For intrinsic test functions $\Phi$ and $\Theta$ on $\mathcal{D}$, the function $\Phi+\Theta$ is an intrinsic test function on $\mathcal{D}$ by Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §linear, and $(F^{\Phi})^{\Theta}=F^{\Phi+\Theta}$.

2. (The shift identity)¶ Let $v:\mathcal{D}\to\mathbb{R}$ and let $\Theta$ be an intrinsic test function on $\mathcal{D}$. If $v$ and $v-\Theta$ have penalty-subordinate growth from above relative to $P$, then $v$ is a viscosity subsolution of $F$ relative to $P$ if and only if $v-\Theta$ is a viscosity subsolution of $F^{\Theta}$ relative to $P$. If $v$ and $v-\Theta$ have penalty-subordinate growth from below relative to $P$, then $v$ is a viscosity supersolution of $F$ relative to $P$ if and only if $v-\Theta$ is a viscosity supersolution of $F^{\Theta}$ relative to $P$.

3. (Invariance under a change of profile)¶ Let $\Phi$ and $\Phi'$ be intrinsic test functions on $\mathcal{D}$ and let $u:\mathcal{D}\to\mathbb{R}$. If $u-\Phi$ and $u-\Phi'$ have penalty-subordinate growth from above relative to $P$, then $u$ is a viscosity subsolution of $F$ relative to $P$ and the profile $\Phi$ if and only if it is one relative to $P$ and the profile $\Phi'$; if $u-\Phi$ and $u-\Phi'$ have penalty-subordinate growth from below relative to $P$, the same holds for supersolutions. In particular, if $u$ and $u-\Phi$ have penalty-subordinate growth from above relative to $P$, then $u$ is a viscosity subsolution of $F$ relative to $P$ and the profile $\Phi$ if and only if it is a viscosity subsolution of $F$ relative to $P$; and likewise for supersolutions, with growth from below.

4. (A change of the penalty)¶ Let $P'=(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E}',\Sigma')$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ with the same penalty domain and score domain as $P$, and let $\Psi$ be an intrinsic test function on $\mathcal{D}$ with

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

Let $v:\mathcal{D}\to\mathbb{R}$, and let the $\delta$-envelopes of $v$ and the $\delta$-shifts of $F$ be taken relative to $P$ or to $P'$ according to the pair named. 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 $P'$, the same holds for supersolutions.