Proof of 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

Each result cited is universally quantified over the data in its own statement. Elementary ordered-field arithmetic and order in $\mathbb{R}$ are used without further citation (The Real Numbers: Standing Notation and Background §background). Sums and differences of elements of $L^{2}(\nu;\mathbb{R}^{d})$ are formed in that real Hilbert space (Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields); sums and differences of elements of $\mathcal{S}(d)$ lie in $\mathcal{S}(d)$ by claim 1 of The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure, and matrix arithmetic is entrywise. For every penalty pair the score domain is contained in the penalty domain (Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair); hence for an intrinsic test function $\Theta$ on $\mathcal{D}$ and $\nu\in\mathcal{D}_{\Sigma}$ the gradient $\nabla\Theta(\nu)$ is defined and lies 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 $H_{\Theta}(\nu)\in\mathcal{S}(d)$ by Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §hessian. The distance $W_{2}$ is a metric (The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §metric), in particular symmetric, and $W_{2}(\nu,\mu)^{2}\le I(\pi)$ for every $\pi\in\Pi(\nu,\mu)$ by The Quadratic Wasserstein Distance on Euclidean Space §distance; costs $I(\pi)$ and discrepancies are nonnegative real numbers (The Intrinsic Calculus on the Wasserstein Space: Standing Notation §couplings).

Step 0. Three auxiliary claims.

Claim A (envelopes and continuous functions). Let $g:\mathcal{D}\to\mathbb{R}$, let $\Theta':\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be continuous in the sense of Continuous Map Between Metric Spaces (for $W_{2}$ and the absolute-value metric), and let $g-\Theta'$ be the function on $\mathcal{D}$ with value $g(\mu)-\Theta'(\mu)$. (i) If $g$ is bounded above near each point of $\mathcal{D}$ (Upper and Lower Semicontinuous Envelopes of a Real-Valued Function §near-bounds), then so is $g-\Theta'$, and the upper semicontinuous envelopes (Upper and Lower Semicontinuous Envelopes of a Real-Valued Function §upper) satisfy $(g-\Theta')^{*}(x)=g^{*}(x)-\Theta'(x)$ for every $x\in\mathcal{D}$. (ii) If $g$ is bounded below near each point of $\mathcal{D}$, then so is $g-\Theta'$, and the lower semicontinuous envelopes (Upper and Lower Semicontinuous Envelopes of a Real-Valued Function §lower) satisfy $(g-\Theta')_{*}(x)=g_{*}(x)-\Theta'(x)$ for every $x\in\mathcal{D}$.

Proof of (i). Fix $x\in\mathcal{D}$ and a positive $\kappa\in\mathbb{R}$. By continuity of $\Theta'$ at $x$ there is a positive $r_{0}$ with $|\Theta'(y)-\Theta'(x)|<\kappa$ whenever $W_{2}(x,y)<r_{0}$. Let $c\in A_{g}(x)$, with radius $r$ as in Upper and Lower Semicontinuous Envelopes of a Real-Valued Function, and put $r'=\min\{r,r_{0}/2\}$, a positive number. For $y\in\mathcal{D}$ with $W_{2}(y,x)\le r'$ we have $W_{2}(x,y)<r_{0}$, hence $g(y)-\Theta'(y)\le c-\Theta'(x)+\kappa$. Thus $c-\Theta'(x)+\kappa\in A_{g-\Theta'}(x)$. Since $A_{g}(x)$ is nonempty, so is $A_{g-\Theta'}(x)$, and $g-\Theta'$ is bounded above near each point. Moreover, $(g-\Theta')^{*}(x)=\inf A_{g-\Theta'}(x)\le c-\Theta'(x)+\kappa$ for every $c\in A_{g}(x)$. So $(g-\Theta')^{*}(x)+\Theta'(x)-\kappa$ is a lower bound of $A_{g}(x)$, and hence it is at most the greatest lower bound $g^{*}(x)$. As $\kappa$ was arbitrary, $(g-\Theta')^{*}(x)\le g^{*}(x)-\Theta'(x)$. The function $-\Theta'$ is continuous, because $|(-a)-(-b)|=|a-b|$ (claim 2 of Properties of the Absolute Value in an Ordered Field), and $g=(g-\Theta')-(-\Theta')$ pointwise. The inequality just proved, applied to $g-\Theta'$ and $-\Theta'$, gives $g^{*}(x)\le(g-\Theta')^{*}(x)+\Theta'(x)$, and so equality holds. The proof of (ii) is the same argument with $B_{g}(x)$, suprema and reversed inequalities: $c\in B_{g}(x)$ gives $c-\Theta'(x)-\kappa\in B_{g-\Theta'}(x)$, and so on.

Claim C ($\varepsilon$-form of property (c)). Let $\Theta$ be an intrinsic test function on $\mathcal{D}$, let $\hat{\mu}\in\mathcal{D}$ and let $\kappa\in\mathbb{R}$ be positive. Then there is a positive $\lambda\in\mathbb{R}$ such that $\int_{\mathbb{R}^{d+d}}\lVert\nabla\Theta(\nu)(x)-\nabla\Theta(\hat{\mu})(y)\rVert^{2}\,\pi(dz)<\kappa$ for every $\nu\in\mathcal{D}$ and every $\pi\in\Pi(\nu,\hat{\mu})$ with $I(\pi)<\lambda$.

Proof. Suppose not. Then for each $n\in\mathbb{N}$ there are $\nu_{n}\in\mathcal{D}$ and $\pi_{n}\in\Pi(\nu_{n},\hat{\mu})$ with $I(\pi_{n})<(n+1)^{-1}$ whose discrepancy above is at least $\kappa$. The sequence $(I(\pi_{n}))$ converges to $0$ in the sense of Limit of a Sequence of Real Numbers. Indeed, given a positive $\kappa'$, claim 3 of The Archimedean Property of the Real Numbers gives $N\in\mathbb{N}$ with $0<N^{-1}<\kappa'$, and then $0\le I(\pi_{n})<(n+1)^{-1}\le N^{-1}<\kappa'$ for $n\ge N$. By property (c) of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §gradient-continuity, applied at $\hat{\mu}\in\mathcal{D}$ to the sequences $(\nu_{n})$ in $\mathcal{D}$ and $(\pi_{n})$, the discrepancies converge to $0$, so some of them are less than $\kappa$, a contradiction.

Notation. For a function $g:\mathcal{D}\to\mathbb{R}$ and a second-order equation operator $G$ over $\mathcal{D}_{\Sigma}$ (The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts §operator), say that $(g,G)$ satisfies $(S^{-})$ if the following holds. For every intrinsic test function $\varphi$ on $\mathcal{D}$, every $\hat{\mu}\in\mathcal{D}$ at which $g-\varphi$ has a local maximum relative to $\mathcal{D}$, and every positive $\varepsilon$, there exist $\nu\in\mathcal{D}_{\Sigma}$, $\pi\in\Pi(\nu,\hat{\mu})$, $s\in\mathbb{R}$, $q\in L^{2}(\nu;\mathbb{R}^{d})$ and $Y\in\mathcal{S}(d)$ with $I(\pi)<\varepsilon^{2}$, $|g(\nu)-g(\hat{\mu})|<\varepsilon$, $|s-g(\hat{\mu})|<\varepsilon$, $\int_{\mathbb{R}^{d+d}}\lVert q(x)-\nabla\varphi(\hat{\mu})(y)\rVert^{2}\,\pi(dz)<\varepsilon^{2}$, $\lVert Y-H_{\varphi}(\hat{\mu})\rVert<\varepsilon$ and $G(\nu,s,q,Y)\le\varepsilon$. Say that $(g,G)$ satisfies $(S^{+})$ if the same holds with "local minimum" in place of "local maximum" and with $-\varepsilon\le G(\nu,s,q,Y)$ in place of $G(\nu,s,q,Y)\le\varepsilon$. Let $v$ have penalty-subordinate growth from above relative to a penalty pair with domains $\mathcal{D},\mathcal{D}_{\Sigma}$. By Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution, $v$ is a viscosity subsolution of an operator $F$ relative to that pair if and only if $(v^{-}_{\delta},F^{-}_{\delta})$ satisfies $(S^{-})$ for every $\delta\in(0,1)$, with envelopes and $\delta$-shifts relative to that pair. By Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §supersolution, the analogous statement with growth from below, $(v^{+}_{\delta},F^{+}_{\delta})$ and $(S^{+})$ characterises viscosity supersolutions.

Claim B (one shift at fixed $\delta$). Let $\Theta$ be an intrinsic test function on $\mathcal{D}$, let $g,g':\mathcal{D}\to\mathbb{R}$ satisfy $g'(\mu)=g(\mu)-\Theta(\mu)$ for $\mu\in\mathcal{D}$, and let $G,G'$ be second-order equation operators over $\mathcal{D}_{\Sigma}$ with

$$G'(\nu,r,q,Y)=G\bigl(\nu,\ r+\Theta(\nu),\ q+\nabla\Theta(\nu),\ Y+H_{\Theta}(\nu)\bigr)$$

for all $(\nu,q)\in\mathcal{V}(\mathcal{D}_{\Sigma})$, $r\in\mathbb{R}$, $Y\in\mathcal{S}(d)$. Then $(g,G)$ satisfies $(S^{-})$ if and only if $(g',G')$ does, and $(g,G)$ satisfies $(S^{+})$ if and only if $(g',G')$ does.

Proof, "only if" for $(S^{-})$. Fix an intrinsic test function $\varphi$ on $\mathcal{D}$, a point $\hat{\mu}\in\mathcal{D}$ at which $g'-\varphi$ has a local maximum relative to $\mathcal{D}$, and a positive $\varepsilon$. By Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §linear (with $a=b=1$), $\varphi+\Theta$ is an intrinsic test function on $\mathcal{D}$ with $\nabla(\varphi+\Theta)(\hat{\mu})=\nabla\varphi(\hat{\mu})+\nabla\Theta(\hat{\mu})$ and $H_{\varphi+\Theta}(\hat{\mu})=H_{\varphi}(\hat{\mu})+H_{\Theta}(\hat{\mu})$. Since $g-(\varphi+\Theta)$ and $g'-\varphi$ agree at every point of $\mathcal{D}$, $g-(\varphi+\Theta)$ has a local maximum at $\hat{\mu}$. Order of choice. First, by continuity of $\Theta$ (property (a), Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §continuity) at $\hat{\mu}$, choose a positive $\eta$ with $|\Theta(\nu)-\Theta(\hat{\mu})|<\varepsilon/2$ whenever $W_{2}(\hat{\mu},\nu)<\eta$. Second, by Claim C with $\kappa=\varepsilon^{2}/4$, choose a positive $\lambda$. Third, by property (e) of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §hessian-continuity for $\Theta$ at $\hat{\mu}$ with $\varepsilon/2$, choose a positive $\theta$ with $\lVert H_{\Theta}(\nu)-H_{\Theta}(\hat{\mu})\rVert<\varepsilon/2$ whenever $W_{2}(\nu,\hat{\mu})<\theta$ (the Hessians there are the translation Hessians, Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §hessian). Finally put $\varepsilon'=\min\{\varepsilon/2,1,\eta,\lambda,\theta\}$, which is positive. Apply $(S^{-})$ for $(g,G)$ to $\varphi+\Theta$, $\hat{\mu}$ and $\varepsilon'$. This gives $\nu\in\mathcal{D}_{\Sigma}$, $\pi\in\Pi(\nu,\hat{\mu})$, $s$, $q$ and $Y$ satisfying the six conditions with $\varepsilon'$, $g$, $G$ and $\varphi+\Theta$. Put

$$s'=s-\Theta(\nu),\qquad q'=q-\nabla\Theta(\nu)\in L^{2}(\nu;\mathbb{R}^{d}),\qquad Y'=Y-H_{\Theta}(\nu)\in\mathcal{S}(d).$$

Since $0<\varepsilon'\le1$ we have $\varepsilon'^{2}\le\varepsilon'$. From $W_{2}(\nu,\hat{\mu})^{2}\le I(\pi)<\varepsilon'^{2}$ and claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field we get $W_{2}(\nu,\hat{\mu})<\varepsilon'\le\min\{\eta,\theta\}$. Also $I(\pi)<\varepsilon'^{2}\le\varepsilon'\le\lambda$, and $\varepsilon'^{2}\le(\varepsilon/2)^{2}$ by claim 2 of that lemma. We verify the six conditions for $g'$, $G'$, $\varphi$, $\varepsilon$ and $(\nu,\pi,s',q',Y')$. (1) $I(\pi)<\varepsilon'^{2}\le\varepsilon^{2}/4<\varepsilon^{2}$. (2) $g'(\nu)-g'(\hat{\mu})=(g(\nu)-g(\hat{\mu}))-(\Theta(\nu)-\Theta(\hat{\mu}))$. By the triangle inequality (claims 2 and 5 of Properties of the Absolute Value in an Ordered Field), its absolute value is less than $\varepsilon'+\varepsilon/2\le\varepsilon$. (3) Likewise $s'-g'(\hat{\mu})=(s-g(\hat{\mu}))-(\Theta(\nu)-\Theta(\hat{\mu}))$ has absolute value less than $\varepsilon$. (4) Choose representatives with $q'=q-\nabla\Theta(\nu)$ and $\nabla(\varphi+\Theta)(\hat{\mu})=\nabla\varphi(\hat{\mu})+\nabla\Theta(\hat{\mu})$ pointwise; discrepancies do not depend on representatives, and their integrands are nonnegative Borel functions (The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures §well-defined). For $z\in\mathbb{R}^{d+d}$ put $a(z)=q(x)-\nabla(\varphi+\Theta)(\hat{\mu})(y)$ and $b(z)=\nabla\Theta(\nu)(x)-\nabla\Theta(\hat{\mu})(y)$. Then $q'(x)-\nabla\varphi(\hat{\mu})(y)=a(z)-b(z)$. By claims 5 and 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, $\lVert a(z)-b(z)\rVert\le\lVert a(z)\rVert+\lVert b(z)\rVert$. Hence, by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and $(\alpha+\beta)^{2}\le2\alpha^{2}+2\beta^{2}$ (the difference being $(\alpha-\beta)^{2}\ge0$), $\lVert a(z)-b(z)\rVert^{2}\le2\lVert a(z)\rVert^{2}+2\lVert b(z)\rVert^{2}$. Integrating against $\pi$ (linearity and monotonicity, Linearity and Monotonicity of the Lebesgue Integral §nonnegative) and using $I(\pi)<\lambda$ with Claim C (note $\nu\in\mathcal{D}_{\Sigma}\subseteq\mathcal{D}$), we get

$$\int_{\mathbb{R}^{d+d}}\lVert q'(x)-\nabla\varphi(\hat{\mu})(y)\rVert^{2}\,\pi(dz)<2\varepsilon'^{2}+2\cdot\frac{\varepsilon^{2}}{4}\le\frac{\varepsilon^{2}}{2}+\frac{\varepsilon^{2}}{2}=\varepsilon^{2}.$$

(5) $Y'-H_{\varphi}(\hat{\mu})=(Y-H_{\varphi+\Theta}(\hat{\mu}))-(H_{\Theta}(\nu)-H_{\Theta}(\hat{\mu}))$. By the triangle inequality for the metric $d_{\mathcal{S}(d)}(A,B)=\lVert A-B\rVert$ (Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §norm, The Set of Symmetric Real Matrices is a Metric Space, applied to the points $Y'$, $Y-H_{\Theta}(\hat{\mu})$, $H_{\varphi}(\hat{\mu})$), $\lVert Y'-H_{\varphi}(\hat{\mu})\rVert<\varepsilon'+\varepsilon/2\le\varepsilon$. (6) $G'(\nu,s',q',Y')=G(\nu,s,q,Y)\le\varepsilon'\le\varepsilon$. So $(g',G')$ satisfies $(S^{-})$. For $(S^{+})$ the argument is identical, with "local minimum" in place of "local maximum" and with (6) replaced by $-\varepsilon\le-\varepsilon'\le G(\nu,s,q,Y)=G'(\nu,s',q',Y')$.

"If". By Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §difference, $-\Theta$ is an intrinsic test function on $\mathcal{D}$ with $\nabla(-\Theta)=-\nabla\Theta$ on $\mathcal{D}$ and $H_{-\Theta}=-H_{\Theta}$. We have $g(\mu)=g'(\mu)-(-\Theta)(\mu)$, and $G(\nu,r,q,Y)=G'(\nu,r+(-\Theta)(\nu),q+\nabla(-\Theta)(\nu),Y+H_{-\Theta}(\nu))$. So the "only if" part, applied to $(g',G')$, $(g,G)$ and $-\Theta$, gives the converse.

Step 1 (composition). Let $k_{0}:\mathbb{R}^{d}\to\mathbb{R}$ be the constant function $0$. With $\mathbb{R}^{d}$ open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, claim 2 of Differences and Constants for Functions of Class $C^2$ on a Euclidean Open Set says that $k_{0}$ is of class $C^{2}$ with $Dk_{0}(a)=0_{\mathbb{R}^{d}}$ and $D^{2}k_{0}(a)=0_{d}$ for every $a$. By The Squared Wasserstein Distance to a Fixed Measure and Functions of the Mean are Intrinsic Test Functions §mean with $Q=\mathcal{D}$ and $\phi=k_{0}$, the function $\mu\mapsto k_{0}(m(\mu))=0$, which is the zero function $0$, is an intrinsic test function on $\mathcal{D}$. By the formulas of that clause, $\nabla0(\mu)=Dk_{0}(m(\mu))$ for $\mu\in\mathcal{D}$ and $H_{0}(\mu)=D^{2}k_{0}(m(\mu))$. So its gradient $\nabla0(\mu)$ is the class of the constant map with value $0_{\mathbb{R}^{d}}$, that is, the zero element of $L^{2}(\mu;\mathbb{R}^{d})$, for $\mu\in\mathcal{D}$, and $H_{0}(\mu)=0_{d}$. Hence, by Viscosity Subsolutions, Supersolutions and Solutions Relative to a Penalty Pair and a Profile §shifted, $F^{0}(\nu,r,q,Y)=F(\nu,r+0,q+0,Y+0_{d})=F(\nu,r,q,Y)$, so $F^{0}=F$. Now let $\Phi,\Theta$ be intrinsic test functions on $\mathcal{D}$. By Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §linear with $a=b=1$, $\Phi+\Theta$ is one too, with $\nabla(\Phi+\Theta)=\nabla\Phi+\nabla\Theta$ on $\mathcal{D}$ and $H_{\Phi+\Theta}=H_{\Phi}+H_{\Theta}$. The operator $F^{\Phi}$ is a second-order equation operator over $\mathcal{D}_{\Sigma}$ (Viscosity Subsolutions, Supersolutions and Solutions Relative to a Penalty Pair and a Profile §shifted), so $(F^{\Phi})^{\Theta}$ is defined. Applying that definition twice, for $(\nu,q)\in\mathcal{V}(\mathcal{D}_{\Sigma})$, $r\in\mathbb{R}$, $Y\in\mathcal{S}(d)$ we get

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

Step 2 (shift). Let $\delta\in(0,1)$. Suppose $v$ and $v-\Theta$ have growth from above relative to $P$. By The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair §minus, $v-\delta\mathcal{E}$ is bounded above near each point and $v^{-}_{\delta}=(v-\delta\mathcal{E})^{*}$. Since $(v-\Theta)-\delta\mathcal{E}=(v-\delta\mathcal{E})-\Theta$ pointwise and $\Theta$ is continuous (property (a)), Claim A(i) gives $(v-\Theta)^{-}_{\delta}=v^{-}_{\delta}-\Theta$ on $\mathcal{D}$. By The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts §shifted, applied to the operator $F^{\Theta}$ over $\mathcal{D}_{\Sigma}$, and by Viscosity Subsolutions, Supersolutions and Solutions Relative to a Penalty Pair and a Profile §shifted,

$$(F^{\Theta})^{-}_{\delta}(\nu,r,q,Y)=F\bigl(\nu,\ r+\delta\mathcal{E}(\nu)+\Theta(\nu),\ q+\delta\Sigma(\nu)+\nabla\Theta(\nu),\ Y+\delta H_{\mathcal{E}}(\nu)+H_{\Theta}(\nu)\bigr)=F^{-}_{\delta}\bigl(\nu,\ r+\Theta(\nu),\ q+\nabla\Theta(\nu),\ Y+H_{\Theta}(\nu)\bigr).$$

So Claim B applies with $g=v^{-}_{\delta}$, $g'=(v-\Theta)^{-}_{\delta}$, $G=F^{-}_{\delta}$ and $G'=(F^{\Theta})^{-}_{\delta}$: $(v^{-}_{\delta},F^{-}_{\delta})$ satisfies $(S^{-})$ if and only if $((v-\Theta)^{-}_{\delta},(F^{\Theta})^{-}_{\delta})$ does. As this holds for every $\delta\in(0,1)$, the characterisation in the Notation paragraph gives: $v$ is a viscosity subsolution of $F$ relative to $P$ if and only if $v-\Theta$ is a viscosity subsolution of $F^{\Theta}$ relative to $P$. For supersolutions, with growth from below, the argument is the same. Use The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair §plus and Claim A(ii): $(v-\Theta)^{+}_{\delta}=((v+\delta\mathcal{E})-\Theta)_{*}=v^{+}_{\delta}-\Theta$. The identity $(F^{\Theta})^{+}_{\delta}(\nu,r,q,Y)=F^{+}_{\delta}(\nu,r+\Theta(\nu),q+\nabla\Theta(\nu),Y+H_{\Theta}(\nu))$ follows in the same way from the second formula of The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts §shifted. Then apply Claim B for $(S^{+})$.

Step 3 (invariance). By Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §difference, $\Phi'-\Phi$ is an intrinsic test function on $\mathcal{D}$. Put $v=u-\Phi$. Then $v-(\Phi'-\Phi)=u-\Phi'$ pointwise, so $v$ and $v-(\Phi'-\Phi)$ have growth from above. Step 2 holds for an arbitrary second-order equation operator over $\mathcal{D}_{\Sigma}$. Apply it with $F^{\Phi}$ in place of $F$ and $\Theta=\Phi'-\Phi$, and use Step 1: $(F^{\Phi})^{\Phi'-\Phi}=F^{\Phi+(\Phi'-\Phi)}=F^{\Phi'}$, since $\Phi+(\Phi'-\Phi)=\Phi'$ pointwise. This shows that $u-\Phi$ is a viscosity subsolution of $F^{\Phi}$ relative to $P$ if and only if $u-\Phi'$ is a viscosity subsolution of $F^{\Phi'}$ relative to $P$. By Viscosity Subsolutions, Supersolutions and Solutions Relative to a Penalty Pair and a Profile §subsolution this is the asserted equivalence. Supersolutions are handled identically, using growth from below, the supersolution half of Step 2 and Viscosity Subsolutions, Supersolutions and Solutions Relative to a Penalty Pair and a Profile §supersolution. For the "in particular", take $\Phi'=0$, an intrinsic test function by Step 1. Then $u-0=u$ and $F^{0}=F$, so being a viscosity subsolution (supersolution) relative to $P$ and the profile $0$ means being one of $F$ relative to $P$.

Step 4 (pair). First, $H_{\mathcal{E}'}=H_{\mathcal{E}}+H_{\Psi}$ on $\mathcal{D}$. Let $\mu\in\mathcal{D}$. For $a\in\mathbb{R}^{d}$, $(\tau_{a})_{\#}\mu\in\mathcal{D}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §translation, so $e'_{\mu}(a)=\mathcal{E}'((\tau_{a})_{\#}\mu)=e_{\mu}(a)+\psi_{\mu}(a)$, where $\psi_{\mu}(a)=\Psi((\tau_{a})_{\#}\mu)$. Thus $e_{\mu}=e'_{\mu}-\psi_{\mu}$. Here $e'_{\mu}$ is of class $C^{2}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §translation for $P'$, and $\psi_{\mu}$ is of class $C^{2}$ by property (d) of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §translation-c2. So claim 1 of Differences and Constants for Functions of Class $C^2$ on a Euclidean Open Set gives $D^{2}e_{\mu}(0_{\mathbb{R}^{d}})=D^{2}e'_{\mu}(0_{\mathbb{R}^{d}})-D^{2}\psi_{\mu}(0_{\mathbb{R}^{d}})$, that is, $H_{\mathcal{E}}(\mu)=H_{\mathcal{E}'}(\mu)-H_{\Psi}(\mu)$ (Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §hessian, Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §hessian).

Now let $\delta\in(0,1)$, and write $F'^{\mp}_{\delta}$ and $v'^{\mp}_{\delta}$ for the $\delta$-shifts and envelopes relative to $P'$. By Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §linear with $\varphi=\chi=\Psi$ and $(a,b)=(\delta,0)$, respectively $(-\delta,0)$, the functions $\Theta_{\pm}=\pm\delta\Psi$ are intrinsic test functions on $\mathcal{D}$ with $\nabla\Theta_{\pm}=\pm\delta\nabla\Psi$ on $\mathcal{D}$ and $H_{\Theta_{\pm}}=\pm\delta H_{\Psi}$. By The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts §shifted for $P'$ and for $P$, the hypotheses on $\mathcal{E}',\Sigma'$ and the identity for $H_{\mathcal{E}'}$,

$$F'^{-}_{\delta}(\nu,r,q,Y)=F\bigl(\nu,\ r+\delta\mathcal{E}(\nu)+\delta\Psi(\nu),\ q+\delta\Sigma(\nu)+\delta\nabla\Psi(\nu),\ Y+\delta H_{\mathcal{E}}(\nu)+\delta H_{\Psi}(\nu)\bigr)=F^{-}_{\delta}\bigl(\nu,\ r+\Theta_{+}(\nu),\ q+\nabla\Theta_{+}(\nu),\ Y+H_{\Theta_{+}}(\nu)\bigr),$$

and likewise $F'^{+}_{\delta}(\nu,r,q,Y)=F^{+}_{\delta}(\nu,r+\Theta_{-}(\nu),q+\nabla\Theta_{-}(\nu),Y+H_{\Theta_{-}}(\nu))$. If $v$ has growth from above relative to $P$ and to $P'$, then $v-\delta\mathcal{E}'=(v-\delta\mathcal{E})-\Theta_{+}$ pointwise. So The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair §minus and Claim A(i) ($\Theta_{+}$ being continuous by property (a)) give $v'^{-}_{\delta}=v^{-}_{\delta}-\Theta_{+}$. If $v$ has growth from below relative to both pairs, then $v+\delta\mathcal{E}'=(v+\delta\mathcal{E})-\Theta_{-}$, and The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair §plus and Claim A(ii) give $v'^{+}_{\delta}=v^{+}_{\delta}-\Theta_{-}$. Claim B with $\Theta=\Theta_{+}$, $g=v^{-}_{\delta}$, $g'=v'^{-}_{\delta}$, $G=F^{-}_{\delta}$, $G'=F'^{-}_{\delta}$ gives: $(v^{-}_{\delta},F^{-}_{\delta})$ satisfies $(S^{-})$ if and only if $(v'^{-}_{\delta},F'^{-}_{\delta})$ does. Claim B with $\Theta=\Theta_{-}$ gives the same for $(S^{+})$ and the plus objects. As $\delta\in(0,1)$ is arbitrary, the characterisation in the Notation paragraph, applied to $P$ and to $P'$ (which have the same domains, so $(S^{\pm})$ refers to the same test functions and local extrema), gives both equivalences of clause 4.