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

Each result cited is universally quantified over the data in its own statement.

The argument follows the pattern of the published proof (version b62c346b-d736-40af-89ef-52ed5ada0393) of the Wasserstein profile-shift lemma, clause 4; the constant $C_{0}$ is absorbed into the shifting test function. The notation is that of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation. Elementary order and arithmetic of real numbers are used without citation (The Real Numbers: Standing Notation and Background §background). For $\nu,\hat{\mu}\in\mathcal{P}^{a}_{\rho}$, $\pi\in\Pi^{a}(\nu,\hat{\mu})$, $q\in L^{2}(\nu;X^{a})$ and $\eta\in L^{2}(\hat{\mu};X^{a})$ we write $\mathrm{disc}_{\pi}(q,\eta)=\int_{X\times X}|q(x)-\eta(y)|_{a}^{2}\,\pi(dz)$ for the discrepancy, a nonnegative real number which is the integral against $\pi$ of the nonnegative Borel function $z\mapsto|q(x)-\eta(y)|_{a}^{2}$ for any representatives, and which does not depend on the representatives chosen (Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §discrepancy; $\Pi^{a}(\nu,\hat{\mu})\subseteq\Pi(\nu,\hat{\mu})$ by Couplings of Finite Noise Cost and Their Noise Cost §couplings). For every noise penalty pair the score domain is nonempty and contained in the penalty domain (Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty); hence $\mathcal{D}$ is a nonempty subset of $\mathcal{P}^{a}_{\rho}$, and for a noise intrinsic test function $\Theta$ on $\mathcal{D}$ and $\nu\in\mathcal{D}$ the gradient $\nabla\Theta(\nu)\in L^{2}(\nu;X^{a})$ is defined by property (b) of Noise Intrinsic Test Functions on the Noise Wasserstein Space §differentiability. The distance $W_{a}$ is a metric on $\mathcal{P}^{a}_{\rho}$ (The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §metric), symmetric by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §symmetry, and for $\nu,\hat{\mu}\in\mathcal{P}^{a}_{\rho}$ and $\pi\in\Pi^{a}(\nu,\hat{\mu})$ the pair $(\nu,\hat{\mu})$ is noise-connected by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §connected, and $W_{a}(\nu,\hat{\mu})^{2}\le I^{a}(\pi)$ by The Noise Wasserstein Distance §distance. $\mathbb{R}$ carries the absolute-value metric of The Absolute Value Metric on the Real Line, so that continuity in the sense of Continuous Map Between Metric Spaces is the usual $\varepsilon$-$\delta$ statement with $|\Theta(y)-\Theta(x)|$.

Two facts about noise intrinsic test functions are used throughout. By A Toolkit for Penalised Comparison on the Noise Wasserstein Space: Constant Test Functions and Linear Combinations of Test Functions, the Identity as a Unique Noise-Optimal Map, and Discrepancies Along the Push-Forward Under (id, id) and Along Glued Couplings §constants, for $c\in\mathbb{R}$ the constant function $\varphi_{c}$ with value $c$ on $\mathcal{P}^{a}_{\rho}$ is a noise intrinsic test function on $\mathcal{D}$ with $\nabla\varphi_{c}(\mu)=0_{\mu}$ for $\mu\in\mathcal{D}$. By A Toolkit for Penalised Comparison on the Noise Wasserstein Space: Constant Test Functions and Linear Combinations of Test Functions, the Identity as a Unique Noise-Optimal Map, and Discrepancies Along the Push-Forward Under (id, id) and Along Glued Couplings §linear, for noise intrinsic test functions $\varphi,\psi$ on $\mathcal{D}$ and $s,t\in\mathbb{R}$, the function $s\varphi+t\psi$ is a noise intrinsic test function on $\mathcal{D}$ with $\nabla(s\varphi+t\psi)(\mu)=s\nabla\varphi(\mu)+t\nabla\psi(\mu)$ for $\mu\in\mathcal{D}$.

Notation. For a function $g:\mathcal{D}\to\mathbb{R}$ and a first-order equation operator $G$ over $\mathcal{D}_{\Sigma}$ (The Bundle of Noise Fields over a Set of Measures, First-Order Equation Operators on the Noise Wasserstein Space, and Their Delta-Shifts §operator), say that $(g,G)$ satisfies $(S^{-})$ if the following holds: for every noise intrinsic test function $\varphi$ on $\mathcal{D}$, every $\hat{\mu}\in\mathcal{D}$ at which the function $g-\varphi$ on $\mathcal{D}$ has a local maximum relative to $\mathcal{D}$ (Local Maximum of a Function Relative to a Subset of a Metric Space), and every positive $\varepsilon$, there exist $\nu\in\mathcal{D}_{\Sigma}$, $\pi\in\Pi^{a}(\nu,\hat{\mu})$, $s\in\mathbb{R}$ and $q\in L^{2}(\nu;X^{a})$ with

$$I^{a}(\pi)<\varepsilon^{2},\quad|g(\nu)-g(\hat{\mu})|<\varepsilon,\quad|s-g(\hat{\mu})|<\varepsilon,\quad\mathrm{disc}_{\pi}(q,\nabla\varphi(\hat{\mu}))<\varepsilon^{2},\quad G(\nu,s,q)\le\varepsilon .$$

Say that $(g,G)$ satisfies $(S^{+})$ if the same holds with "local minimum" (Local Minimum of a Function Relative to a Subset of a Metric Space) in place of "local maximum" and with $-\varepsilon\le G(\nu,s,q)$ in place of $G(\nu,s,q)\le\varepsilon$. These conditions involve the pair only through $\mathcal{D}$ and $\mathcal{D}_{\Sigma}$. For a noise penalty pair with domains $\mathcal{D},\mathcal{D}_{\Sigma}$ and $0<\delta$, the $\delta$-shifts $F^{\mp}_{\delta}$ of $F$ are functions $\mathcal{V}^{a}(\mathcal{D}_{\Sigma})\times\mathbb{R}\to\mathbb{R}$ (The Bundle of Noise Fields over a Set of Measures, First-Order Equation Operators on the Noise Wasserstein Space, and Their Delta-Shifts §shifted), that is, first-order equation operators over $\mathcal{D}_{\Sigma}$. If $v$ has penalty-subordinate growth from above relative to that pair, then by Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §subsolution $v$ is a viscosity subsolution of $F$ relative to that pair if and only if $(v^{-}_{\delta},F^{-}_{\delta})$ satisfies $(S^{-})$ for every $\delta\in\mathbb{R}$ with $0<\delta<1$. Likewise, by Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §supersolution, with growth from below, $v$ is a viscosity supersolution if and only if $(v^{+}_{\delta},F^{+}_{\delta})$ satisfies $(S^{+})$ for every such $\delta$.

Step 0. Three auxiliary claims.

Claim A (envelopes and continuous functions). Let $g:\mathcal{D}\to\mathbb{R}$, let $\Theta':\mathcal{P}^{a}_{\rho}\to\mathbb{R}$ be continuous on $\mathcal{P}^{a}_{\rho}$, 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 $x\in\mathcal{D}$. All envelopes are taken in the metric space $(\mathcal{P}^{a}_{\rho},W_{a})$ with $S=\mathcal{D}$.

Proof of (i). Fix $x\in\mathcal{D}$ and a positive $\kappa\in\mathbb{R}$. By continuity of $\Theta'$ at $x$ (Continuous Map Between Metric Spaces) there is a positive $r_{0}$ with $|\Theta'(y)-\Theta'(x)|<\kappa$ for every $y\in\mathcal{P}^{a}_{\rho}$ with $W_{a}(x,y)<r_{0}$. Let $c\in A_{g}(x)$, with a positive radius $r$ as in Upper and Lower Semicontinuous Envelopes of a Real-Valued Function, and put $r'=\min\{r,r_{0}/2\}$, which is positive. Let $y\in\mathcal{D}$ with $W_{a}(y,x)\le r'$. Then $W_{a}(y,x)\le r$, so $g(y)\le c$; and $W_{a}(x,y)=W_{a}(y,x)\le r_{0}/2<r_{0}$, so $-\kappa<\Theta'(y)-\Theta'(x)$ by Properties of the Absolute Value in an Ordered Field §strict-two-sided. Hence $g(y)-\Theta'(y)\le c-\Theta'(x)+\kappa$, and $c-\Theta'(x)+\kappa\in A_{g-\Theta'}(x)$. Since $A_{g}(x)$ is nonempty, so is $A_{g-\Theta'}(x)$; as $x$ was arbitrary, $g-\Theta'$ is bounded above near each point of $\mathcal{D}$ and $(g-\Theta')^{*}$ is defined. Its value $(g-\Theta')^{*}(x)=\inf A_{g-\Theta'}(x)$ is a lower bound of $A_{g-\Theta'}(x)$ (Lower Bound and Greatest Lower Bound in a Totally Ordered Set), so $(g-\Theta')^{*}(x)\le c-\Theta'(x)+\kappa$ for every $c\in A_{g}(x)$. Thus $(g-\Theta')^{*}(x)+\Theta'(x)-\kappa$ is a lower bound of $A_{g}(x)$ and hence at most its greatest lower bound $g^{*}(x)$ (Lower Bound and Greatest Lower Bound in a Totally Ordered Set). As $\kappa$ was arbitrary, $(g-\Theta')^{*}(x)+\Theta'(x)\le g^{*}(x)$. This inequality holds for every $g$ bounded above near each point and every continuous $\Theta'$. The function $-\Theta'$ is continuous on $\mathcal{P}^{a}_{\rho}$, because $|(-\Theta'(y))-(-\Theta'(x))|=|\Theta'(y)-\Theta'(x)|$ by Properties of the Absolute Value in an Ordered Field §symmetry; and $g=(g-\Theta')-(-\Theta')$ pointwise on $\mathcal{D}$, with $g-\Theta'$ bounded above near each point. The inequality applied to $g-\Theta'$ and $-\Theta'$ gives $g^{*}(x)-\Theta'(x)\le(g-\Theta')^{*}(x)$, and so equality holds.

Proof of (ii). The argument is the mirror image. Fix $x$, $\kappa$ and $r_{0}$ as before, let $c\in B_{g}(x)$ with radius $r$, and put $r'=\min\{r,r_{0}/2\}$. For $y\in\mathcal{D}$ with $W_{a}(y,x)\le r'$ we have $c\le g(y)$ and $\Theta'(y)-\Theta'(x)<\kappa$ (Properties of the Absolute Value in an Ordered Field §strict-two-sided), so $c-\Theta'(x)-\kappa\le g(y)-\Theta'(y)$, and $c-\Theta'(x)-\kappa\in B_{g-\Theta'}(x)$. Hence $g-\Theta'$ is bounded below near each point of $\mathcal{D}$, and, the supremum $(g-\Theta')_{*}(x)=\sup B_{g-\Theta'}(x)$ being an upper bound of $B_{g-\Theta'}(x)$ (Upper Bound and Least Upper Bound), $c\le(g-\Theta')_{*}(x)+\Theta'(x)+\kappa$ for every $c\in B_{g}(x)$. So $(g-\Theta')_{*}(x)+\Theta'(x)+\kappa$ is an upper bound of $B_{g}(x)$ and is at least its least upper bound $g_{*}(x)$ (Upper Bound and Least Upper Bound). As $\kappa$ was arbitrary, $g_{*}(x)-\Theta'(x)\le(g-\Theta')_{*}(x)$. Applying this inequality to $g-\Theta'$ and the continuous function $-\Theta'$, with $g=(g-\Theta')-(-\Theta')$, gives $(g-\Theta')_{*}(x)+\Theta'(x)\le g_{*}(x)$, and so equality holds.

Claim B ($\varepsilon$-form of property (c)). Let $\Theta$ be a noise 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 $\mathrm{disc}_{\pi}(\nabla\Theta(\nu),\nabla\Theta(\hat{\mu}))<\kappa$ for every $\nu\in\mathcal{D}$ and every $\pi\in\Pi^{a}(\nu,\hat{\mu})$ with $I^{a}(\pi)<\lambda$.

Proof. Suppose not. Then for every positive $\lambda$ there are $\nu\in\mathcal{D}$ and $\pi\in\Pi^{a}(\nu,\hat{\mu})$ with $I^{a}(\pi)<\lambda$ and $\kappa\le\mathrm{disc}_{\pi}(\nabla\Theta(\nu),\nabla\Theta(\hat{\mu}))$. Taking $\lambda=(n+1)^{-1}$ for each $n\in\mathbb{N}$, we obtain a sequence $(\nu_{n})_{n\in\mathbb{N}}$ in $\mathcal{D}$ (Sequence in a Set) and couplings $\pi_{n}\in\Pi^{a}(\nu_{n},\hat{\mu})$ with $I^{a}(\pi_{n})<(n+1)^{-1}$ and $\kappa\le\mathrm{disc}_{\pi_{n}}(\nabla\Theta(\nu_{n}),\nabla\Theta(\hat{\mu}))$. The sequence $(I^{a}(\pi_{n}))_{n\in\mathbb{N}}$ converges to $0$ in the sense of Limit of a Sequence of Real Numbers: 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 for $n\ge N$ one has $0\le I^{a}(\pi_{n})<(n+1)^{-1}<N^{-1}<\kappa'$, the noise cost being nonnegative (Couplings of Finite Noise Cost and Their Noise Cost §cost). So $(\pi_{n})_{n\in\mathbb{N}}$ is a sequence of couplings of vanishing noise cost from $(\nu_{n})_{n\in\mathbb{N}}$ to $\hat{\mu}$ (Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §couplings; $\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$). By property (c) of Noise Intrinsic Test Functions on the Noise Wasserstein Space §gradient-continuity, applied at $\hat{\mu}\in\mathcal{D}$, and by Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §strong, the discrepancies $\mathrm{disc}_{\pi_{n}}(\nabla\Theta(\nu_{n}),\nabla\Theta(\hat{\mu}))$ converge to $0$; by Limit of a Sequence of Real Numbers with $\kappa$ in place of $\varepsilon$, some of them are less than $\kappa$, a contradiction.

Claim C (one shift at fixed $\delta$). Let $\Theta$ be a noise 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 first-order equation operators over $\mathcal{D}_{\Sigma}$ with

$$G'(\nu,r,q)=G\bigl(\nu,\ r+\Theta(\nu),\ q+\nabla\Theta(\nu)\bigr)\qquad\text{for all }(\nu,q)\in\mathcal{V}^{a}(\mathcal{D}_{\Sigma}),\ r\in\mathbb{R}.$$

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 $(S^{-})$. Suppose $(g,G)$ satisfies $(S^{-})$. Fix a noise 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 A Toolkit for Penalised Comparison on the Noise Wasserstein Space: Constant Test Functions and Linear Combinations of Test Functions, the Identity as a Unique Noise-Optimal Map, and Discrepancies Along the Push-Forward Under (id, id) and Along Glued Couplings §linear with $s=t=1$, $\varphi+\Theta$ is a noise intrinsic test function on $\mathcal{D}$ with $\nabla(\varphi+\Theta)(\hat{\mu})=\nabla\varphi(\hat{\mu})+\nabla\Theta(\hat{\mu})$. The functions $g-(\varphi+\Theta)$ and $g'-\varphi$ agree at every point of $\mathcal{D}$, so $g-(\varphi+\Theta)$ has a local maximum at $\hat{\mu}$ relative to $\mathcal{D}$.

Order of choice. First, by property (a) of Noise Intrinsic Test Functions on the Noise Wasserstein Space §continuity, $\Theta$ is continuous at $\hat{\mu}$; choose a positive $\eta$ with $|\Theta(\nu)-\Theta(\hat{\mu})|<\varepsilon/2$ for every $\nu\in\mathcal{P}^{a}_{\rho}$ with $W_{a}(\hat{\mu},\nu)<\eta$ (Continuous Map Between Metric Spaces). Second, by Claim B for $\Theta$ at $\hat{\mu}$ with $\kappa=\varepsilon^{2}/4$, choose a positive $\lambda$. Third, put $\varepsilon'=\min\{\varepsilon/2,1,\eta,\lambda\}$, which is positive, being one of these four positive numbers. Only then apply $(S^{-})$ for $(g,G)$ to the test function $\varphi+\Theta$, the point $\hat{\mu}$ and $\varepsilon'$. This gives $\nu\in\mathcal{D}_{\Sigma}$, $\pi\in\Pi^{a}(\nu,\hat{\mu})$, $s\in\mathbb{R}$ and $q\in L^{2}(\nu;X^{a})$ with

$$I^{a}(\pi)<\varepsilon'^{2},\quad|g(\nu)-g(\hat{\mu})|<\varepsilon',\quad|s-g(\hat{\mu})|<\varepsilon',\quad\mathrm{disc}_{\pi}\bigl(q,\nabla(\varphi+\Theta)(\hat{\mu})\bigr)<\varepsilon'^{2},\quad G(\nu,s,q)\le\varepsilon'.$$

Since $\nu\in\mathcal{D}_{\Sigma}\subseteq\mathcal{D}$, $\nabla\Theta(\nu)\in L^{2}(\nu;X^{a})$ is defined, and we put

$$s'=s-\Theta(\nu),\qquad q'=q-\nabla\Theta(\nu)\in L^{2}(\nu;X^{a}).$$

Since $0<\varepsilon'\le1$, $\varepsilon'^{2}\le\varepsilon'$. From $W_{a}(\nu,\hat{\mu})^{2}\le I^{a}(\pi)<\varepsilon'^{2}$ and claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field (both $W_{a}(\nu,\hat{\mu})$ and $\varepsilon'$ being nonnegative) we get $W_{a}(\hat{\mu},\nu)=W_{a}(\nu,\hat{\mu})<\varepsilon'\le\eta$, so $|\Theta(\nu)-\Theta(\hat{\mu})|<\varepsilon/2$. Also $I^{a}(\pi)<\varepsilon'^{2}\le\varepsilon'\le\lambda$, and $\varepsilon'^{2}\le(\varepsilon/2)^{2}=\varepsilon^{2}/4$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. We verify the five conditions of $(S^{-})$ for $(g',G')$, $\varphi$, $\hat{\mu}$, $\varepsilon$ and $(\nu,\pi,s',q')$.

(1) $I^{a}(\pi)<\varepsilon'^{2}\le\varepsilon^{2}/4<\varepsilon^{2}$.

(2) $g'(\nu)-g'(\hat{\mu})=(g(\nu)-g(\hat{\mu}))+(-(\Theta(\nu)-\Theta(\hat{\mu})))$. By the triangle inequality and symmetry of the absolute value (Properties of the Absolute Value in an Ordered Field §triangle, Properties of the Absolute Value in an Ordered Field §symmetry), 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 $\varepsilon'+\varepsilon/2\le\varepsilon$.

(4) Fix representatives $f_{q}$, $h$, $f$, $k$, measurable maps into $X^{a}$, of $q$, $\nabla\Theta(\nu)$, $\nabla\varphi(\hat{\mu})$ and $\nabla\Theta(\hat{\mu})$. By The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations the pointwise combinations $f_{q}-h$ and $f+k$ represent $q'$ and $\nabla(\varphi+\Theta)(\hat{\mu})$, and since discrepancies do not depend on representatives (Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §discrepancy) the three discrepancies $\mathrm{disc}_{\pi}(q',\nabla\varphi(\hat{\mu}))$, $\mathrm{disc}_{\pi}(q,\nabla(\varphi+\Theta)(\hat{\mu}))$ and $\mathrm{disc}_{\pi}(\nabla\Theta(\nu),\nabla\Theta(\hat{\mu}))$ are the integrals against $\pi$ of the nonnegative Borel functions $z\mapsto|f_{q}(x)-h(x)-f(y)|_{a}^{2}$, $z\mapsto|\alpha(z)|_{a}^{2}$ and $z\mapsto|\beta(z)|_{a}^{2}$, where $\alpha(z)=f_{q}(x)-(f(y)+k(y))$ and $\beta(z)=h(x)-k(y)$. In the real inner product space $X^{a}$ (The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert), $f_{q}(x)-h(x)-f(y)=\alpha(z)-\beta(z)$ by the vector space axioms, and The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle gives $|\alpha(z)-\beta(z)|_{a}\le|\alpha(z)|_{a}+|\beta(z)|_{a}$, both sides being nonnegative. By claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and the real inequality $(p+r)^{2}\le2p^{2}+2r^{2}$, which holds because $2p^{2}+2r^{2}-(p+r)^{2}=(p-r)^{2}\ge0$ (claim 2 of Nonnegativity of Squares in an Ordered Field),

$$|f_{q}(x)-h(x)-f(y)|_{a}^{2}\le2\,|\alpha(z)|_{a}^{2}+2\,|\beta(z)|_{a}^{2}\qquad\text{for every }z\in X\times X.$$

Integrating against $\pi$ (monotonicity, additivity and nonnegative multiples, Linearity and Monotonicity of the Lebesgue Integral §nonnegative), and using $\mathrm{disc}_{\pi}(q,\nabla(\varphi+\Theta)(\hat{\mu}))<\varepsilon'^{2}$ together with Claim B ($\nu\in\mathcal{D}$, $\pi\in\Pi^{a}(\nu,\hat{\mu})$, $I^{a}(\pi)<\lambda$), we get

$$\mathrm{disc}_{\pi}\bigl(q',\nabla\varphi(\hat{\mu})\bigr)\le2\,\mathrm{disc}_{\pi}\bigl(q,\nabla(\varphi+\Theta)(\hat{\mu})\bigr)+2\,\mathrm{disc}_{\pi}\bigl(\nabla\Theta(\nu),\nabla\Theta(\hat{\mu})\bigr)<2\varepsilon'^{2}+2\cdot\frac{\varepsilon^{2}}{4}\le\frac{\varepsilon^{2}}{2}+\frac{\varepsilon^{2}}{2}=\varepsilon^{2}.$$

(5) $G'(\nu,s',q')=G(\nu,s'+\Theta(\nu),q'+\nabla\Theta(\nu))=G(\nu,s,q)\le\varepsilon'\le\varepsilon$, using the identity $q'+\nabla\Theta(\nu)=q$ in the vector space $L^{2}(\nu;X^{a})$.

So $(g',G')$ satisfies $(S^{-})$. For $(S^{+})$ the argument is identical, with "local minimum" in place of "local maximum" throughout and with (5) replaced by $-\varepsilon\le-\varepsilon'\le G(\nu,s,q)=G'(\nu,s',q')$.

"If". By A Toolkit for Penalised Comparison on the Noise Wasserstein Space: Constant Test Functions and Linear Combinations of Test Functions, the Identity as a Unique Noise-Optimal Map, and Discrepancies Along the Push-Forward Under (id, id) and Along Glued Couplings §linear with $\varphi=\psi=\Theta$, $s=-1$ and $t=0$, the function $\tilde{\Theta}=-\Theta$ is a noise intrinsic test function on $\mathcal{D}$ with $\nabla\tilde{\Theta}(\mu)=-\nabla\Theta(\mu)$ for $\mu\in\mathcal{D}$. We have $g(\mu)=g'(\mu)-\tilde{\Theta}(\mu)$ for $\mu\in\mathcal{D}$, and for $(\nu,q)\in\mathcal{V}^{a}(\mathcal{D}_{\Sigma})$ and $r\in\mathbb{R}$,

$$G'\bigl(\nu,\ r+\tilde{\Theta}(\nu),\ q+\nabla\tilde{\Theta}(\nu)\bigr)=G\bigl(\nu,\ r-\Theta(\nu)+\Theta(\nu),\ q-\nabla\Theta(\nu)+\nabla\Theta(\nu)\bigr)=G(\nu,r,q).$$

So the "only if" part, applied with $(g',G')$ in place of $(g,G)$, $(g,G)$ in place of $(g',G')$ and $\tilde{\Theta}$ in place of $\Theta$, gives the converse, for $(S^{-})$ and for $(S^{+})$.

Step 1. The shifting test functions. Let $\delta\in\mathbb{R}$ with $0<\delta<1$. By A Toolkit for Penalised Comparison on the Noise Wasserstein Space: Constant Test Functions and Linear Combinations of Test Functions, the Identity as a Unique Noise-Optimal Map, and Discrepancies Along the Push-Forward Under (id, id) and Along Glued Couplings §linear, applied with $\varphi=\Psi$, $\psi=\varphi_{C_{0}}$ (a noise intrinsic test function on $\mathcal{D}$ with zero gradients by A Toolkit for Penalised Comparison on the Noise Wasserstein Space: Constant Test Functions and Linear Combinations of Test Functions, the Identity as a Unique Noise-Optimal Map, and Discrepancies Along the Push-Forward Under (id, id) and Along Glued Couplings §constants) and $s=t=\delta$, respectively $s=t=-\delta$, the functions

$$\Theta_{+}(\mu)=\delta\Psi(\mu)+\delta C_{0},\qquad\Theta_{-}(\mu)=-\delta\Psi(\mu)-\delta C_{0}\qquad(\mu\in\mathcal{P}^{a}_{\rho})$$

are noise intrinsic test functions on $\mathcal{D}$ with $\nabla\Theta_{\pm}(\mu)=\pm\delta\nabla\Psi(\mu)\pm\delta\,0_{\mu}=\pm\delta\nabla\Psi(\mu)$ for $\mu\in\mathcal{D}$. By property (a) of Noise Intrinsic Test Functions on the Noise Wasserstein Space §continuity, $\Theta_{+}$ and $\Theta_{-}$ are continuous on $\mathcal{P}^{a}_{\rho}$.

Step 2. The $\delta$-shifts. Write $F^{\mp}_{\delta}$ for the $\delta$-shifts relative to $P$ and $F'^{\mp}_{\delta}$ for those relative to $P'$. Let $(\nu,q)\in\mathcal{V}^{a}(\mathcal{D}_{\Sigma})$ and $r\in\mathbb{R}$. Since $\nu\in\mathcal{D}_{\Sigma}\subseteq\mathcal{D}$, the hypotheses give $\mathcal{E}'(\nu)=\mathcal{E}(\nu)+\Psi(\nu)+C_{0}$ and $\Sigma'(\nu)=\Sigma(\nu)+\nabla\Psi(\nu)$. By Step 1, $\Theta_{+}(\nu)=\delta\Psi(\nu)+\delta C_{0}$, $\nabla\Theta_{+}(\nu)=\delta\nabla\Psi(\nu)$, $\Theta_{-}(\nu)=-\delta\Psi(\nu)-\delta C_{0}$ and $\nabla\Theta_{-}(\nu)=-\delta\nabla\Psi(\nu)$, so

$$\delta\mathcal{E}'(\nu)=\delta\mathcal{E}(\nu)+\Theta_{+}(\nu),\qquad\delta\Sigma'(\nu)=\delta\Sigma(\nu)+\nabla\Theta_{+}(\nu),$$ $$-\delta\mathcal{E}'(\nu)=-\delta\mathcal{E}(\nu)+\Theta_{-}(\nu),\qquad-\delta\Sigma'(\nu)=-\delta\Sigma(\nu)+\nabla\Theta_{-}(\nu).$$

By The Bundle of Noise Fields over a Set of Measures, First-Order Equation Operators on the Noise Wasserstein Space, and Their Delta-Shifts §shifted for $P'$, then for $P$, and the vector space operations of $L^{2}(\nu;X^{a})$,

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

Step 3. The envelopes. Write $v^{\mp}_{\delta}$ for the $\delta$-envelopes relative to $P$ and $v'^{\mp}_{\delta}$ for those relative to $P'$. Suppose $v$ has penalty-subordinate growth from above relative to $P$ and to $P'$. By The Delta-Envelopes of a Function on the Domain of a Noise Penalty Pair §minus, $v-\delta\mathcal{E}$ is bounded above near each point of $\mathcal{D}$, $v^{-}_{\delta}=(v-\delta\mathcal{E})^{*}$ and $v'^{-}_{\delta}=(v-\delta\mathcal{E}')^{*}$. Pointwise on $\mathcal{D}$, $v-\delta\mathcal{E}'=(v-\delta\mathcal{E})-\Theta_{+}$, so Claim A(i), with $g=v-\delta\mathcal{E}$ and the continuous function $\Theta'=\Theta_{+}$, gives

$$v'^{-}_{\delta}(\mu)=v^{-}_{\delta}(\mu)-\Theta_{+}(\mu)\qquad(\mu\in\mathcal{D}).$$

Suppose $v$ has penalty-subordinate growth from below relative to $P$ and to $P'$. By The Delta-Envelopes of a Function on the Domain of a Noise Penalty Pair §plus, $v+\delta\mathcal{E}$ is bounded below near each point of $\mathcal{D}$, $v^{+}_{\delta}=(v+\delta\mathcal{E})_{*}$ and $v'^{+}_{\delta}=(v+\delta\mathcal{E}')_{*}$. Pointwise on $\mathcal{D}$, $v+\delta\mathcal{E}'=(v+\delta\mathcal{E})-\Theta_{-}$, so Claim A(ii), with $\Theta'=\Theta_{-}$, gives $v'^{+}_{\delta}(\mu)=v^{+}_{\delta}(\mu)-\Theta_{-}(\mu)$ for $\mu\in\mathcal{D}$.

Step 4. Conclusion. Subsolutions: suppose $v$ has penalty-subordinate growth from above relative to $P$ and to $P'$, and let $0<\delta<1$. By Steps 2 and 3, Claim C applies with $\Theta=\Theta_{+}$, $g=v^{-}_{\delta}$, $g'=v'^{-}_{\delta}$, $G=F^{-}_{\delta}$ and $G'=F'^{-}_{\delta}$, and gives: $(v^{-}_{\delta},F^{-}_{\delta})$ satisfies $(S^{-})$ if and only if $(v'^{-}_{\delta},F'^{-}_{\delta})$ does. The pairs $P$ and $P'$ have the same domains $\mathcal{D}$ and $\mathcal{D}_{\Sigma}$, so $(S^{-})$ refers to the same test functions, local maxima and score-domain points for both. As $\delta$ was arbitrary, the characterisation in the Notation paragraph, applied to $P$ and to $P'$, shows that $v$ is a viscosity subsolution of $F$ relative to $P$ if and only if it is one relative to $P'$.

Supersolutions: suppose $v$ has penalty-subordinate growth from below relative to $P$ and to $P'$, and let $0<\delta<1$. By Steps 2 and 3, Claim C applies with $\Theta=\Theta_{-}$, $g=v^{+}_{\delta}$, $g'=v'^{+}_{\delta}$, $G=F^{+}_{\delta}$ and $G'=F'^{+}_{\delta}$, and gives: $(v^{+}_{\delta},F^{+}_{\delta})$ satisfies $(S^{+})$ if and only if $(v'^{+}_{\delta},F'^{+}_{\delta})$ does. As before, the characterisation in the Notation paragraph shows that $v$ is a viscosity supersolution of $F$ relative to $P$ if and only if it is one relative to $P'$. $\blacksquare$