Proof of Perron's Method on the Noise Wasserstein Space: Existence of a Viscosity Solution of a First-Order Equation Between a Subsolution and a Supersolution

Each result cited is universally quantified over the data in its own statement, and is applied to the data named at the point of use. The symmetry and the triangle inequality of $W_{a}$ 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 §symmetry, 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 §triangle) and the rules of Elementary Order Arithmetic in an Ordered Field and Elementary Arithmetic in an Ordered Field for adding and scaling inequalities are used without further mention. Any two members of $\mathcal{P}^{a}_{\rho}$, equal or not, form a noise-connected ordered pair 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 $\mathcal{P}^{a}_{\rho}\subseteq\mathcal{P}_{2}(X)$ 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 §moments. For $\nu,\nu_{0}\in\mathcal{P}^{a}_{\rho}$ and $\pi\in\Pi^{a}(\nu,\nu_{0})$, $W_{a}(\nu,\nu_{0})^{2}\le I^{a}(\pi)$ by The Noise Wasserstein Distance §distance, hence $W_{a}(\nu,\nu_{0})\le\sqrt{I^{a}(\pi)}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. For $\nu\in\mathcal{P}^{a}_{\rho}$, $L^{2}(\nu;X^{a})$ is a real vector space (Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields), in which gradients are added, subtracted and scaled. Being a viscosity subsolution, $f$ has penalty-subordinate growth from above (Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §subsolution), and it has growth from below by hypothesis; being a viscosity supersolution, $g$ has penalty-subordinate growth from below (Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §supersolution), and it has growth from above by hypothesis.

Claim 1. The function $f$ is a viscosity subsolution with $f\le f\le g$ on $\mathcal{D}$, by the hypothesis $f\le g$, so $f\in\mathcal{G}$. Let $\nu\in\mathcal{D}$. Since $f\in\mathcal{G}$, $f(\nu)\le u(\nu)$; and $g(\nu)$ is an upper bound of $\{v(\nu):v\in\mathcal{G}\}$, whose least upper bound is $u(\nu)$, so $u(\nu)\le g(\nu)$ (Upper Bound and Least Upper Bound). Let $\delta\in\mathbb{R}$ be positive, let $C_{g}$ be as in Penalty-Subordinate Growth of a Function on the Domain of a Noise Penalty Pair §above for $g$ and $\delta$, and let $C_{f}$ be as in Penalty-Subordinate Growth of a Function on the Domain of a Noise Penalty Pair §below for $f$ and $\delta$. Then $u(\nu)\le g(\nu)\le C_{g}+\delta\,\mathcal{E}(\nu)$ and $-C_{f}-\delta\,\mathcal{E}(\nu)\le f(\nu)\le u(\nu)$ for every $\nu\in\mathcal{D}$. As $\delta$ was arbitrary, $u$ has penalty-subordinate growth from above and from below by those two clauses.

Claim 2, the subsolution property. The set $\mathcal{G}$ is nonempty and consists of viscosity subsolutions. It is uniformly subordinate from above in the sense of The Pointwise Supremum of a Uniformly Subordinate Family of Viscosity Subsolutions on the Noise Wasserstein Space is a Viscosity Subsolution §uniform-growth: for positive $\delta$ and $C_{g}$ as in Penalty-Subordinate Growth of a Function on the Domain of a Noise Penalty Pair §above for $g$ and $\delta$, $v(\nu)\le g(\nu)\le C_{g}+\delta\,\mathcal{E}(\nu)$ for every $v\in\mathcal{G}$ and every $\nu\in\mathcal{D}$. The pair is noise-closed and $\mathcal{D}$ has the noise map property, and the function whose value at $\nu$ is $\sup\{v(\nu):v\in\mathcal{G}\}$ is $u$, so The Pointwise Supremum of a Uniformly Subordinate Family of Viscosity Subsolutions on the Noise Wasserstein Space is a Viscosity Subsolution §subsolution, applied with $\mathcal{G}$ in the role of its $\mathcal{S}$, shows that $u$ is a viscosity subsolution of $F$ relative to the noise penalty pair.

Claim 2, the supersolution property. Suppose, seeking a contradiction, that $u$ is not a viscosity supersolution. By claim 1 the penalty-subordinate growth from below required by Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §supersolution holds, so there are $\delta\in\mathbb{R}$ with $0<\delta<1$, a noise intrinsic test function $\varphi$ on $\mathcal{D}$, a point $\hat{\mu}\in\mathcal{D}$ at which the function with value $u^{+}_{\delta}(\mu)-\varphi(\mu)$ at $\mu\in\mathcal{D}$ has a local minimum relative to $\mathcal{D}$, and a positive $\varepsilon$, such that, the order of $\mathbb{R}$ being total:

$(\ast)$ every $\nu\in\mathcal{D}_{\Sigma}$, $\pi\in\Pi^{a}(\nu,\hat{\mu})$, $s\in\mathbb{R}$ and $q\in L^{2}(\nu;X^{a})$ with $I^{a}(\pi)<\varepsilon^{2}$, $|u^{+}_{\delta}(\nu)-u^{+}_{\delta}(\hat{\mu})|<\varepsilon$, $|s-u^{+}_{\delta}(\hat{\mu})|<\varepsilon$ and $\int_{X\times X}|q(x)-\nabla\varphi(\hat{\mu})(y)|_{a}^{2}\,\pi(dz)<\varepsilon^{2}$ satisfy $F^{+}_{\delta}(\nu,s,q)<-\varepsilon$.

Let $\tau$ be a positive radius witnessing the local minimum (Local Minimum of a Function Relative to a Subset of a Metric Space): $u^{+}_{\delta}(\hat{\mu})-\varphi(\hat{\mu})\le u^{+}_{\delta}(\nu)-\varphi(\nu)$ for $\nu\in\mathcal{D}$ with $W_{a}(\hat{\mu},\nu)<\tau$. The functions $u+\delta\mathcal{E}$ and $g+\delta\mathcal{E}$ on $\mathcal{D}$ are bounded below near each point (The Delta-Envelopes of a Function on the Domain of a Noise Penalty Pair §plus) and ordered pointwise by claim 1, so Properties of the Lower Semicontinuous Envelope, by Duality §monotone, in the metric space $(\mathcal{P}^{a}_{\rho},W_{a})$ with $S=\mathcal{D}$, gives $u^{+}_{\delta}\le g^{+}_{\delta}$ on $\mathcal{D}$. The data below are chosen in the order: $\sigma_{1},\varepsilon_{1}$ (Step 1, inside an argument by contradiction); then $\Gamma$, the radii $\theta_{a},\theta_{c},\theta_{d}$, then $\gamma,\alpha,\kappa,c_{0}$ and $\psi$ (Step 2); the coupling $\pi$ and its modification are chosen afresh for each $\nu$ in the test condition (Step 3).

Step 1: a strict gap at $\hat{\mu}$. Suppose $u^{+}_{\delta}(\hat{\mu})=g^{+}_{\delta}(\hat{\mu})$. For $\nu\in\mathcal{D}$ with $W_{a}(\hat{\mu},\nu)<\tau$, $g^{+}_{\delta}(\hat{\mu})-\varphi(\hat{\mu})=u^{+}_{\delta}(\hat{\mu})-\varphi(\hat{\mu})\le u^{+}_{\delta}(\nu)-\varphi(\nu)\le g^{+}_{\delta}(\nu)-\varphi(\nu)$, so $g^{+}_{\delta}-\varphi$ has a local minimum at $\hat{\mu}$ relative to $\mathcal{D}$. By continuity of $\varphi$ (Noise Intrinsic Test Functions on the Noise Wasserstein Space §continuity) there is a positive $\sigma_{1}\le\tau$ with $|\varphi(\nu)-\varphi(\hat{\mu})|<\varepsilon$ whenever $W_{a}(\hat{\mu},\nu)<\sigma_{1}$; put $\varepsilon_{1}=\min\{\varepsilon,\sigma_{1}\}$. Applying Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §supersolution to $g$ with $\delta$, $\varphi$, $\hat{\mu}$ and $\varepsilon_{1}$ 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 the four closeness conditions for $g$ at tolerance $\varepsilon_{1}$ and $-\varepsilon_{1}\le F^{+}_{\delta}(\nu,s,q)$. Since $\varepsilon_{1}\le\varepsilon$ and $g^{+}_{\delta}(\hat{\mu})=u^{+}_{\delta}(\hat{\mu})$, three of the closeness conditions of $(\ast)$ follow (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field for the cost and the discrepancy). For the fourth, $\nu\in\mathcal{D}$ and $W_{a}(\hat{\mu},\nu)\le\sqrt{I^{a}(\pi)}<\varepsilon_{1}\le\sigma_{1}\le\tau$, so $u^{+}_{\delta}(\nu)\le g^{+}_{\delta}(\nu)<g^{+}_{\delta}(\hat{\mu})+\varepsilon_{1}\le u^{+}_{\delta}(\hat{\mu})+\varepsilon$ and $u^{+}_{\delta}(\nu)\ge u^{+}_{\delta}(\hat{\mu})+\varphi(\nu)-\varphi(\hat{\mu})>u^{+}_{\delta}(\hat{\mu})-\varepsilon$, whence $|u^{+}_{\delta}(\nu)-u^{+}_{\delta}(\hat{\mu})|<\varepsilon$ by Properties of the Absolute Value in an Ordered Field §strict-two-sided. Then $(\ast)$ gives $F^{+}_{\delta}(\nu,s,q)<-\varepsilon\le-\varepsilon_{1}$, a contradiction. Hence $u^{+}_{\delta}(\hat{\mu})<g^{+}_{\delta}(\hat{\mu})$, and $\Gamma=g^{+}_{\delta}(\hat{\mu})-u^{+}_{\delta}(\hat{\mu})$ is positive.

Step 2: radii, parameters and the test function. Choose positive radii $\theta_{a},\theta_{c},\theta_{d}$ as follows. By continuity of $\varphi$: $|\varphi(\nu)-\varphi(\hat{\mu})|<\min\{\tfrac{\varepsilon}{4},\tfrac{\Gamma}{4}\}$ whenever $\nu\in\mathcal{P}^{a}_{\rho}$ and $W_{a}(\nu,\hat{\mu})<\theta_{a}$. By Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §semicontinuity, $g^{+}_{\delta}$ is lower semicontinuous on $\mathcal{D}$, so by Lower Semicontinuous Function on a Subset of a Metric Space: $g^{+}_{\delta}(\hat{\mu})-\tfrac{\Gamma}{4}<g^{+}_{\delta}(\nu)$ whenever $\nu\in\mathcal{D}$ and $W_{a}(\nu,\hat{\mu})<\theta_{c}$. Finally, for every $\nu\in\mathcal{D}$ and $\pi\in\Pi^{a}(\nu,\hat{\mu})$ with $I^{a}(\pi)<\theta_{d}^{2}$ the discrepancy $\int_{X\times X}|\nabla\varphi(\nu)(x)-\nabla\varphi(\hat{\mu})(y)|_{a}^{2}\,\pi(dz)$ is less than $(\tfrac{\varepsilon}{2})^{2}$: otherwise, with a sequence $(h_{n})$ of positive reals of limit $0$ (Existence of a Sequence of Positive Real Numbers with Limit Zero), there would be $\nu_{n}\in\mathcal{D}$ and $\pi_{n}\in\Pi^{a}(\nu_{n},\hat{\mu})$ with $0\le I^{a}(\pi_{n})<h_{n}^{2}$ and discrepancies $D_{n}\ge(\tfrac{\varepsilon}{2})^{2}$; then $(I^{a}(\pi_{n}))$ has limit $0$ by Arithmetic of Limits of Real Sequences §products and claim 2 of Order Properties of Limits of Real Sequences, so $(\pi_{n})$ is a sequence of couplings of vanishing noise cost from $(\nu_{n})$ to $\hat{\mu}$ (Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §couplings), and $(D_{n})$ has limit $0$ by Noise Intrinsic Test Functions on the Noise Wasserstein Space §gradient-continuity at $\hat{\mu}\in\mathcal{D}$ and Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §strong; claim 1 of Order Properties of Limits of Real Sequences would give $(\tfrac{\varepsilon}{2})^{2}\le0$, contradicting Elementary Order Arithmetic in an Ordered Field §positive-products. Put

$$\gamma=\min\{\tau,\varepsilon,\theta_{a},\theta_{c},\theta_{d}\},\qquad\alpha=\min\Bigl\{\tfrac{\varepsilon}{8\gamma},\tfrac{\varepsilon}{4\gamma^{2}},\tfrac{\Gamma}{\gamma^{2}}\Bigr\},\qquad\kappa=\tfrac{\alpha\gamma^{2}}{4},\qquad c_{0}=u^{+}_{\delta}(\hat{\mu})-\varphi(\hat{\mu}),$$

all of $\gamma,\alpha,\kappa$ positive by claim 2 of Elementary Properties of the Minimum of Two Elements. By claim 1 of that lemma and Elementary Arithmetic in an Ordered Field §scaling,

$$2\alpha\gamma\le\tfrac{\varepsilon}{4},\qquad\alpha\gamma^{2}\le\tfrac{\varepsilon}{4},\qquad\kappa\le\tfrac{\varepsilon}{16},\qquad\kappa\le\tfrac{\Gamma}{4}.$$

Let $\psi_{0}:\mathcal{P}^{a}_{\rho}\to\mathbb{R}$ be $\psi_{0}(\nu)=W_{a}(\nu,\hat{\mu})^{2}$. As $\mathcal{D}$ has the noise map property, Squared Noise Wasserstein Distances, Their Convergent Series and Linear Combinations are Noise Intrinsic Test Functions §distance, with $\mathcal{D}$ in the role of its $Q$ and $\hat{\mu}$ in the role of its $\nu_{0}$, shows that $\psi_{0}$ is a noise intrinsic test function on $\mathcal{D}$ with $\nabla\psi_{0}(\nu)=2(\mathrm{id}-S_{\nu})$ for $\nu\in\mathcal{D}$ and any noise-optimal map $S_{\nu}$ from $\nu$ to $\hat{\mu}$. 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, with $\mathcal{D}$ in the role of its $Q$ and $c=c_{0}+\kappa$, the constant function $\varphi_{c}$ with value $c_{0}+\kappa$ is a noise intrinsic test function on $\mathcal{D}$ with $\nabla\varphi_{c}(\nu)=0_{\nu}$ for $\nu\in\mathcal{D}$. Let

$$\psi(\nu)=\varphi(\nu)+c_{0}+\kappa-\alpha\,W_{a}(\nu,\hat{\mu})^{2}\qquad\bigl(\nu\in\mathcal{P}^{a}_{\rho}\bigr).$$

By Squared Noise Wasserstein Distances, Their Convergent Series and Linear Combinations are Noise Intrinsic Test Functions §linear, applied first with $\varphi_{1}=\varphi$, $\varphi_{2}=\varphi_{c}$ and $s=t=1$, then with $\varphi_{1}=\varphi+\varphi_{c}$, $\varphi_{2}=\psi_{0}$, $s=1$ and $t=-\alpha$ (each time with $\mathcal{D}$ in the role of its $Q$), $\psi$ is a noise intrinsic test function on $\mathcal{D}$ with

$$\nabla\psi(\nu)=\nabla\varphi(\nu)+0_{\nu}-2\alpha\,(\mathrm{id}-S_{\nu})=\nabla\varphi(\nu)+2\alpha\,(S_{\nu}-\mathrm{id})\qquad(\nu\in\mathcal{D}),$$

$\mathrm{id}-S_{\nu}$ denoting $-(S_{\nu}-\mathrm{id})$; and $\psi(\hat{\mu})=\varphi(\hat{\mu})+c_{0}+\kappa=u^{+}_{\delta}(\hat{\mu})+\kappa$, since $W_{a}(\hat{\mu},\hat{\mu})=0$ 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 §identity. For $\nu\in\mathcal{D}$, $S_{\nu}$ exists: $(\nu,\hat{\mu})$ is uniquely noise-mapped by The Noise Map Property of a Set of Probability Measures §map-property, which provides a noise-optimal map by Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §uniquely-mapped. Moreover, by The Noise-Optimal Map: Transport Cost, Stability Along Nearly Optimal Couplings and Stability Under Perturbation of the Source §cost, $\lVert S_{\nu}-\mathrm{id}\rVert_{\nu}^{2}=W_{a}(\nu,\hat{\mu})^{2}$, and, by bilinearity of the inner product $\langle\cdot,\cdot\rangle_{\nu}$ of the real Hilbert space $L^{2}(\nu;X^{a})$, $\lVert2\alpha(S_{\nu}-\mathrm{id})\rVert_{\nu}^{2}=(2\alpha)^{2}\lVert S_{\nu}-\mathrm{id}\rVert_{\nu}^{2}=\bigl(2\alpha\,W_{a}(\nu,\hat{\mu})\bigr)^{2}$; as both $\lVert2\alpha(S_{\nu}-\mathrm{id})\rVert_{\nu}$ and $2\alpha\,W_{a}(\nu,\hat{\mu})$ are nonnegative, claim 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives

$$\lVert\nabla\psi(\nu)-\nabla\varphi(\nu)\rVert_{\nu}=2\alpha\,W_{a}(\nu,\hat{\mu})\qquad(\nu\in\mathcal{D}). \tag{1}$$

Step 3: the bump. We apply The Bump Construction on the Noise Wasserstein Space: the Local Maximum of a Viscosity Subsolution and a Penalised Noise Intrinsic Test Function with $v=u$, which is a viscosity subsolution by the first part of claim 2, with $\lambda=\delta$, the point $\hat{\mu}\in\mathcal{P}^{a}_{\rho}$, the radius $\gamma$ and the noise intrinsic test function $\psi$ on $\mathcal{D}$; the pair is noise-closed with regular penalised maxima. We verify its three hypotheses.

Upper bound (The Bump Construction on the Noise Wasserstein Space: the Local Maximum of a Viscosity Subsolution and a Penalised Noise Intrinsic Test Function §upper-bound). Let $\nu\in\mathcal{D}$ with $W_{a}(\nu,\hat{\mu})<\gamma$. As $\gamma\le\theta_{a}$, $\varphi(\nu)<\varphi(\hat{\mu})+\tfrac{\Gamma}{4}$ (Properties of the Absolute Value in an Ordered Field §bounds), and $0\le\alpha\,W_{a}(\nu,\hat{\mu})^{2}$; so $\psi(\nu)\le\varphi(\nu)+c_{0}+\kappa<\varphi(\hat{\mu})+c_{0}+\kappa+\tfrac{\Gamma}{4}$, and $b=\varphi(\hat{\mu})+c_{0}+\kappa+\tfrac{\Gamma}{4}$ serves.

Annulus condition (The Bump Construction on the Noise Wasserstein Space: the Local Maximum of a Viscosity Subsolution and a Penalised Noise Intrinsic Test Function §annulus). Let $\nu\in\mathcal{D}$ with $\tfrac{\gamma}{2}<W_{a}(\nu,\hat{\mu})<\gamma$. As $\gamma\le\tau$, the local minimum gives $u^{+}_{\delta}(\nu)\ge u^{+}_{\delta}(\hat{\mu})+\varphi(\nu)-\varphi(\hat{\mu})=\varphi(\nu)+c_{0}$, and $u^{+}_{\delta}(\nu)\le u(\nu)+\delta\,\mathcal{E}(\nu)$ by Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §semicontinuity; so $u(\nu)\ge\varphi(\nu)+c_{0}-\delta\,\mathcal{E}(\nu)$. By claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, $\kappa=\alpha(\tfrac{\gamma}{2})^{2}\le\alpha\,W_{a}(\nu,\hat{\mu})^{2}$, so $\psi(\nu)\le\varphi(\nu)+c_{0}$ and $\psi(\nu)-\delta\,\mathcal{E}(\nu)\le u(\nu)$.

Test condition (The Bump Construction on the Noise Wasserstein Space: the Local Maximum of a Viscosity Subsolution and a Penalised Noise Intrinsic Test Function §condition). Let $\nu\in\mathcal{D}_{\Sigma}$ with $W_{a}(\nu,\hat{\mu})<\gamma$ and $u(\nu)<\psi(\nu)-\delta\,\mathcal{E}(\nu)$; then $\nu\in\mathcal{D}$ (Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair). 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 §optimal there is a noise-optimal coupling $\pi\in\Pi^{a}(\nu,\hat{\mu})$, so that $I^{a}(\pi)=W_{a}(\nu,\hat{\mu})^{2}$ (Noise-Optimal Couplings §optimal) and $\sqrt{I^{a}(\pi)}=W_{a}(\nu,\hat{\mu})$ (Existence and Uniqueness of the Nonnegative Square Root). Let $\Delta=(\mathrm{id},\mathrm{id})_{\#}\nu$, which 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 §diagonal belongs to $\Pi^{a}(\nu,\nu)$ with $I^{a}(\Delta)=0$; let $\varsigma$ be a gluing of $\Delta$ and $\pi$ (Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs §glued, with $\nu,\nu,\hat{\mu}$ in place of its $\mu,\lambda,\nu$), and let $\pi_{1}=(q_{1},q_{3})_{\#}\varsigma$, which by Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs §noise-triangle belongs to $\Pi^{a}(\nu,\hat{\mu})$ with $\sqrt{I^{a}(\pi_{1})}\le\sqrt{I^{a}(\Delta)}+\sqrt{I^{a}(\pi)}=W_{a}(\nu,\hat{\mu})$. We test $(\ast)$ with $(\nu,\pi_{1},\psi(\nu),\nabla\psi(\nu))$.

First, $\sqrt{I^{a}(\pi_{1})}\le W_{a}(\nu,\hat{\mu})<\gamma\le\varepsilon$, so $I^{a}(\pi_{1})<\varepsilon^{2}$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

Secondly, by Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §semicontinuity, the hypothesis on $\nu$ and $0\le\alpha W_{a}(\nu,\hat{\mu})^{2}$,

$$u^{+}_{\delta}(\nu)\le u(\nu)+\delta\,\mathcal{E}(\nu)<\psi(\nu)\le\varphi(\nu)+c_{0}+\kappa=u^{+}_{\delta}(\hat{\mu})+\bigl(\varphi(\nu)-\varphi(\hat{\mu})\bigr)+\kappa<u^{+}_{\delta}(\hat{\mu})+\tfrac{\varepsilon}{4}+\tfrac{\varepsilon}{16},$$

while the local minimum gives $u^{+}_{\delta}(\nu)\ge u^{+}_{\delta}(\hat{\mu})+\varphi(\nu)-\varphi(\hat{\mu})>u^{+}_{\delta}(\hat{\mu})-\tfrac{\varepsilon}{4}$; so $|u^{+}_{\delta}(\nu)-u^{+}_{\delta}(\hat{\mu})|<\varepsilon$ by Properties of the Absolute Value in an Ordered Field §strict-two-sided.

Thirdly, by Properties of the Absolute Value in an Ordered Field §triangle and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, $|\psi(\nu)-u^{+}_{\delta}(\hat{\mu})|\le|\varphi(\nu)-\varphi(\hat{\mu})|+\kappa+\alpha W_{a}(\nu,\hat{\mu})^{2}<\tfrac{\varepsilon}{4}+\tfrac{\varepsilon}{16}+\alpha\gamma^{2}<\varepsilon$.

Fourthly, the discrepancy. Since $\nu\in\mathcal{D}$ and $I^{a}(\pi)=W_{a}(\nu,\hat{\mu})^{2}<\gamma^{2}\le\theta_{d}^{2}$ (claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field), the choice of $\theta_{d}$ gives $\int_{X\times X}|\nabla\varphi(\nu)(x)-\nabla\varphi(\hat{\mu})(y)|_{a}^{2}\,\pi(dz)<(\tfrac{\varepsilon}{2})^{2}$, so the square root of this discrepancy is less than $\tfrac{\varepsilon}{2}$ (claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field). 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 §diagonal, the discrepancy of $\nabla\psi(\nu)$ and $\nabla\varphi(\nu)$ along $\Delta$ is $\lVert\nabla\psi(\nu)-\nabla\varphi(\nu)\rVert_{\nu}^{2}$, whose nonnegative square root is $2\alpha W_{a}(\nu,\hat{\mu})$ by (1) and Existence and Uniqueness of the Nonnegative Square Root. 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 §discrepancy-gluing, applied with $\nu,\nu,\hat{\mu}$ as its $\nu,\lambda,\mu$, with $\Delta$, $\pi$ and $\varsigma$ as its $\pi_{12}$, $\pi_{23}$ and $\sigma$, and with $\nabla\psi(\nu)$, $\nabla\varphi(\nu)$ and $\nabla\varphi(\hat{\mu})$ as its $q$, $\eta$ and $\theta$,

$$\sqrt{\int_{X\times X}|\nabla\psi(\nu)(x)-\nabla\varphi(\hat{\mu})(y)|_{a}^{2}\,\pi_{1}(dz)}<2\alpha\,W_{a}(\nu,\hat{\mu})+\tfrac{\varepsilon}{2}\le2\alpha\gamma+\tfrac{\varepsilon}{2}\le\tfrac{\varepsilon}{4}+\tfrac{\varepsilon}{2}<\varepsilon,$$

so the discrepancy of $\nabla\psi(\nu)$ and $\nabla\varphi(\hat{\mu})$ along $\pi_{1}$ is less than $\varepsilon^{2}$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

Therefore $(\ast)$, with $\pi_{1}$ in the role of its $\pi$, $s=\psi(\nu)$ and $q=\nabla\psi(\nu)\in L^{2}(\nu;X^{a})$, gives $F^{+}_{\delta}(\nu,\psi(\nu),\nabla\psi(\nu))<-\varepsilon<0$, and the test condition holds.

Let $w$ be the function of The Bump Construction on the Noise Wasserstein Space: the Local Maximum of a Viscosity Subsolution and a Penalised Noise Intrinsic Test Function for these data. By The Bump Construction on the Noise Wasserstein Space: the Local Maximum of a Viscosity Subsolution and a Penalised Noise Intrinsic Test Function §subsolution it is a viscosity subsolution, and by The Bump Construction on the Noise Wasserstein Space: the Local Maximum of a Viscosity Subsolution and a Penalised Noise Intrinsic Test Function §domination, $u\le w$ on $\mathcal{D}$.

Step 4: $w$ belongs to $\mathcal{G}$. By claim 1, $f\le u\le w$. Let $\nu\in\mathcal{D}$. If $w(\nu)=u(\nu)$ then $w(\nu)\le g(\nu)$ by claim 1. Otherwise $W_{a}(\nu,\hat{\mu})<\gamma$ and $w(\nu)=\max\{\psi(\nu)-\delta\,\mathcal{E}(\nu),u(\nu)\}$. By Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §semicontinuity and the choice of $\theta_{c}$,

$$g(\nu)\ge g^{+}_{\delta}(\nu)-\delta\,\mathcal{E}(\nu)>g^{+}_{\delta}(\hat{\mu})-\tfrac{\Gamma}{4}-\delta\,\mathcal{E}(\nu)=u^{+}_{\delta}(\hat{\mu})+\tfrac{3\Gamma}{4}-\delta\,\mathcal{E}(\nu),$$

whereas, using $0\le\alpha W_{a}(\nu,\hat{\mu})^{2}$, $|\varphi(\nu)-\varphi(\hat{\mu})|<\tfrac{\Gamma}{4}$ and $\kappa\le\tfrac{\Gamma}{4}$,

$$\psi(\nu)-\delta\,\mathcal{E}(\nu)\le u^{+}_{\delta}(\hat{\mu})+\bigl(\varphi(\nu)-\varphi(\hat{\mu})\bigr)+\kappa-\delta\,\mathcal{E}(\nu)<u^{+}_{\delta}(\hat{\mu})+\tfrac{\Gamma}{2}-\delta\,\mathcal{E}(\nu).$$

As $\tfrac{\Gamma}{2}<\tfrac{3\Gamma}{4}$, $\psi(\nu)-\delta\,\mathcal{E}(\nu)<g(\nu)$; with $u(\nu)\le g(\nu)$ this gives $w(\nu)\le g(\nu)$ by claim 3 of Elementary Properties of the Maximum of Two Elements. Hence $w\in\mathcal{G}$.

Step 5: the contradiction. Since $w\in\mathcal{G}$, $w\le u$; with Step 3, $w=u$. For $\nu\in\mathcal{D}$ with $W_{a}(\nu,\hat{\mu})<\gamma$, claim 1 of Elementary Properties of the Maximum of Two Elements then gives $u(\nu)=w(\nu)\ge\psi(\nu)-\delta\,\mathcal{E}(\nu)$, that is $\psi(\nu)\le u(\nu)+\delta\,\mathcal{E}(\nu)$. By The Delta-Envelopes of a Function on the Domain of a Noise Penalty Pair §plus, $u^{+}_{\delta}$ is the lower semicontinuous envelope of $u+\delta\mathcal{E}$ on $\mathcal{D}$, and Properties of the Lower Semicontinuous Envelope, by Duality §approximation provides a sequence $(\nu_{k})_{k\in\mathbb{N}}$ in $\mathcal{D}$ converging to $\hat{\mu}$ in $(\mathcal{P}^{a}_{\rho},W_{a})$ with $\bigl(u(\nu_{k})+\delta\,\mathcal{E}(\nu_{k})\bigr)$ converging to $u^{+}_{\delta}(\hat{\mu})$. Since $\psi$ is continuous on $\mathcal{P}^{a}_{\rho}$ (Noise Intrinsic Test Functions on the Noise Wasserstein Space §continuity), claim 1 of Continuity Between Metric Spaces is Equivalent to Sequential Continuity shows that $(\psi(\nu_{k}))$ converges to $\psi(\hat{\mu})$. Let $\varepsilon_{0}$ be positive and choose $k$ with $W_{a}(\nu_{k},\hat{\mu})<\gamma$, $\psi(\hat{\mu})-\varepsilon_{0}<\psi(\nu_{k})$ and $u(\nu_{k})+\delta\,\mathcal{E}(\nu_{k})<u^{+}_{\delta}(\hat{\mu})+\varepsilon_{0}$, each holding from some index on (Convergent Sequence in a Metric Space, Properties of the Absolute Value in an Ordered Field §bounds). Then $\psi(\hat{\mu})-\varepsilon_{0}<\psi(\nu_{k})\le u(\nu_{k})+\delta\,\mathcal{E}(\nu_{k})<u^{+}_{\delta}(\hat{\mu})+\varepsilon_{0}$, so $\psi(\hat{\mu})\le u^{+}_{\delta}(\hat{\mu})+2\varepsilon_{0}$; as $\varepsilon_{0}$ was arbitrary, $\psi(\hat{\mu})\le u^{+}_{\delta}(\hat{\mu})$ by Comparison of Real Numbers with Arbitrary Positive Slack §slack-above. But $\psi(\hat{\mu})=u^{+}_{\delta}(\hat{\mu})+\kappa$, so $\kappa\le0$, contradicting $0<\kappa$.

Hence $u$ is a viscosity supersolution. Being also a viscosity subsolution, and having penalty-subordinate growth from above and from below by claim 1, $u$ is a viscosity solution of $F$ relative to the noise penalty pair by Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §solution.