Proof of Perron's Method on the Wasserstein Space: Existence of a Viscosity Solution Between a Subsolution and a Supersolution

Each result cited below 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 $W_{2}$ (The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §symmetry, The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §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. For $\nu,\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\pi\in\Pi(\nu,\rho)$, $W_{2}(\nu,\rho)\le\sqrt{I(\pi)}$ by The Quadratic Wasserstein Distance on Euclidean Space §distance and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Being a viscosity subsolution, $f$ has penalty-subordinate growth from above (Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §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 Second-Order Equation on the Wasserstein Space §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 Penalty Domain §above for $g$ and $\delta$, and let $C_{f}$ be as in Penalty-Subordinate Growth of a Function on the Penalty Domain §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 Wasserstein Space is a Viscosity Subsolution §uniform-growth: for positive $\delta$ and $C_{g}$ as in Penalty-Subordinate Growth of a Function on the Penalty Domain §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 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 Wasserstein Space is a Viscosity Subsolution §subsolution shows that $u$ is a viscosity subsolution of $F$ relative to the 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 Second-Order Equation on the Wasserstein Space §supersolution holds, so there are a positive $\delta$, an 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 $\nu\in\mathcal{D}_{\Sigma}$, $\pi\in\Pi(\nu,\hat{\mu})$, $s\in\mathbb{R}$, $q\in L^{2}(\nu;\mathbb{R}^{d})$ and $Y\in\mathcal{S}(d)$ with $I(\pi)<\varepsilon^{2}$, $|u^{+}_{\delta}(\nu)-u^{+}_{\delta}(\hat{\mu})|<\varepsilon$, $|s-u^{+}_{\delta}(\hat{\mu})|<\varepsilon$, discrepancy of $q$ and $\nabla\varphi(\hat{\mu})$ along $\pi$ less than $\varepsilon^{2}$, and $\lVert Y-H_{\varphi}(\hat{\mu})\rVert<\varepsilon$ satisfy $F^{+}_{\delta}(\nu,s,q,Y)<-\varepsilon$.

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 $W_{2}(\hat{\mu},\nu)<\tau$. 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 Penalty Domain Relative to a Penalty Pair §plus) and ordered pointwise by claim 1, so Properties of the Lower Semicontinuous Envelope, by Duality §monotone gives $u^{+}_{\delta}\le g^{+}_{\delta}$ on $\mathcal{D}$.

Step 1: a strict gap at $\hat{\mu}$. Suppose $u^{+}_{\delta}(\hat{\mu})=g^{+}_{\delta}(\hat{\mu})$. For $\nu\in\mathcal{D}$ with $W_{2}(\hat{\mu},\nu)<\tau$, $g^{+}_{\delta}(\hat{\mu})-\varphi(\hat{\mu})=u^{+}_{\delta}(\hat{\mu})-\varphi(\hat{\mu})\le u^{+}_{\delta}(\nu)-\varphi(\nu)\le g^{+}_{\delta}(\nu)-\varphi(\nu)$, so $g^{+}_{\delta}-\varphi$ has a local minimum at $\hat{\mu}$ relative to $\mathcal{D}$. By continuity of $\varphi$ (Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §continuity) there is a positive $\sigma_{1}\le\tau$ with $|\varphi(\nu)-\varphi(\hat{\mu})|<\varepsilon$ whenever $W_{2}(\hat{\mu},\nu)<\sigma_{1}$; put $\varepsilon_{1}=\min\{\varepsilon,\sigma_{1}\}$. Applying Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §supersolution to $g$ with $\delta$, $\varphi$, $\hat{\mu}$ and $\varepsilon_{1}$ gives $\nu\in\mathcal{D}_{\Sigma}$, $\pi\in\Pi(\nu,\hat{\mu})$, $s$, $q$ and $Y$ with the five closeness conditions for $g$ at tolerance $\varepsilon_{1}$ and $-\varepsilon_{1}\le F^{+}_{\delta}(\nu,s,q,Y)$. Since $\varepsilon_{1}\le\varepsilon$ and $g^{+}_{\delta}(\hat{\mu})=u^{+}_{\delta}(\hat{\mu})$, four 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 fifth, $\nu\in\mathcal{D}$ and $W_{2}(\hat{\mu},\nu)\le\sqrt{I(\pi)}<\varepsilon_{1}\le\sigma_{1}\le\tau$, 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 claim 9 of Properties of the Absolute Value in an Ordered Field. Then $(\ast)$ gives $F^{+}_{\delta}(\nu,s,q,Y)<-\varepsilon\le-\varepsilon_{1}$, 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 and parameters. Choose positive radii $\theta_{a},\theta_{b},\theta_{c},\theta_{d}$ as follows. By continuity of $\varphi$: $|\varphi(\nu)-\varphi(\hat{\mu})|<\min\{\tfrac{\varepsilon}{4},\tfrac{\Gamma}{4}\}$ whenever $W_{2}(\nu,\hat{\mu})<\theta_{a}$. By Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §hessian-continuity: $\lVert H_{\varphi}(\nu)-H_{\varphi}(\hat{\mu})\rVert<\tfrac{\varepsilon}{2}$ whenever $W_{2}(\nu,\hat{\mu})<\theta_{b}$. By Basic Properties of the Delta-Envelopes on the Wasserstein Space §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 $W_{2}(\nu,\hat{\mu})<\theta_{c}$. Finally, for every $\nu\in\mathcal{D}$ and $\pi\in\Pi(\nu,\hat{\mu})$ with $I(\pi)<\theta_{d}^{2}$ the discrepancy of $\nabla\varphi(\nu)$ and $\nabla\varphi(\hat{\mu})$ along $\pi$ 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 $\pi_{n}\in\Pi(\nu_{n},\hat{\mu})$ with $0\le I(\pi_{n})<h_{n}^{2}$ and discrepancies $D_{n}\ge(\tfrac{\varepsilon}{2})^{2}$; then $(I(\pi_{n}))$ has limit $0$ by claim 2 of Arithmetic of Limits of Real Sequences and claim 2 of Order Properties of Limits of Real Sequences, so $(D_{n})$ has limit $0$ by Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §gradient-continuity at $\hat{\mu}\in\mathcal{D}$, and claim 1 of Order Properties of Limits of Real Sequences would give $(\tfrac{\varepsilon}{2})^{2}\le0$, contradicting claim 5 of Elementary Order Arithmetic in an Ordered Field. Put

all of $\gamma,\eta,\kappa$ positive by claim 2 of Elementary Properties of the Minimum of Two Elements. By claim 1 of that lemma and claim 5 of Elementary Arithmetic in an Ordered Field,

Let $\psi_{0}(\nu)=W_{2}(\nu,\hat{\mu})^{2}$; as $\mathcal{D}$ has the map property, $\psi_{0}$ is an intrinsic test function on $\mathcal{D}$ with $\nabla\psi_{0}(\nu)=2(\mathrm{id}-S_{\nu})$ for $\nu\in\mathcal{D}$ and any optimal map $S_{\nu}$ from $\nu$ to $\hat{\mu}$, and $H_{\psi_{0}}=2I_{d}$, by The Squared Wasserstein Distance to a Fixed Measure and Functions of the Mean are Intrinsic Test Functions §distance. The constant function with value $c_{0}+\kappa$ is $\phi\circ m$ for the constant $\phi:\mathbb{R}^{d}\to\mathbb{R}$ with that value, which is of class $C^{2}$ with vanishing gradient and Hessian by Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §quadratic (with $M=0_{d}$, $q=0_{\mathbb{R}^{d}}$, $c=c_{0}+\kappa$); so by The Squared Wasserstein Distance to a Fixed Measure and Functions of the Mean are Intrinsic Test Functions §mean it is an intrinsic test function on $\mathcal{D}$ with gradient the class of the zero map, which is the zero element of $L^{2}(\nu;\mathbb{R}^{d})$, and translation Hessian $0_{d}$. Let

By Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §linear, applied twice, $\psi$ is an intrinsic test function on $\mathcal{D}$ with

and $\psi(\hat{\mu})=\varphi(\hat{\mu})+c_{0}+\kappa=u^{+}_{\delta}(\hat{\mu})+\kappa$. Moreover $\lVert I_{d}\rVert\le1$: $|\xi\cdot(I_{d}\xi)|=\lVert\xi\rVert^{2}\le1$ for $\lVert\xi\rVert\le1$ by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, and $\lVert I_{d}\rVert$ is the least upper bound of these numbers by Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §norm; hence $\lVert2\eta I_{d}\rVert\le2\eta$ by claim 5 of Properties of the Norm of a Symmetric Real Matrix.

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

Upper bound. Let $\nu\in\mathcal{D}$ with $W_{2}(\nu,\hat{\mu})<\gamma$. As $\gamma\le\theta_{a}$, $\varphi(\nu)<\varphi(\hat{\mu})+\tfrac{\Gamma}{4}$ (claim 3 of Properties of the Absolute Value in an Ordered Field), and $0\le\eta\,W_{2}(\nu,\hat{\mu})^{2}$; 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. Let $\nu\in\mathcal{D}$ with $\tfrac{\gamma}{2}<W_{2}(\nu,\hat{\mu})<\gamma$. 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 Wasserstein Space §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, $\kappa=\eta(\tfrac{\gamma}{2})^{2}\le\eta\,W_{2}(\nu,\hat{\mu})^{2}$, so $\psi(\nu)\le\varphi(\nu)+c_{0}$ and $\psi(\nu)-\delta\,\mathcal{E}(\nu)\le u(\nu)$.

Test condition. Let $\nu\in\mathcal{D}_{\Sigma}$ with $W_{2}(\nu,\hat{\mu})<\gamma$ and $u(\nu)<\psi(\nu)-\delta\,\mathcal{E}(\nu)$. By Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment there is $\pi\in\Pi(\nu,\hat{\mu})$ with $I(\pi)=W_{2}(\nu,\hat{\mu})^{2}$. We test $(\ast)$ with $(\nu,\pi,\psi(\nu),\nabla\psi(\nu),H_{\psi}(\nu))$.

First, $I(\pi)<\gamma^{2}\le\varepsilon^{2}$ 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 Wasserstein Space §semicontinuity, the hypothesis on $\nu$ and $0\le\eta W_{2}(\nu,\hat{\mu})^{2}$,

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 claim 9 of Properties of the Absolute Value in an Ordered Field.

Thirdly, by claim 5 of Properties of the Absolute Value in an Ordered Field and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, $|\psi(\nu)-u^{+}_{\delta}(\hat{\mu})|\le|\varphi(\nu)-\varphi(\hat{\mu})|+\kappa+\eta W_{2}(\nu,\hat{\mu})^{2}<\tfrac{\varepsilon}{4}+\tfrac{\varepsilon}{16}+\eta\gamma^{2}<\varepsilon$.

Fourthly, the discrepancy. Let $S_{\nu}$ be an optimal map from $\nu$ to $\hat{\mu}$, which exists by The Map Property of a Set of Probability Measures §map-property. In the real Hilbert space $L^{2}(\pi;\mathbb{R}^{d})$ of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields, read with $d+d$ in place of $q$ and $d$ in place of $r$, let $A$ and $B$ be the classes of $z\mapsto\nabla\varphi(\nu)(x)-\nabla\varphi(\hat{\mu})(y)$ and of $z\mapsto2\eta\,(x-S_{\nu}(x))$, for representatives of the gradients; both maps are Borel and square-integrable against $\pi$, the first by The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures §well-defined, the second because, $\nu$ being the first marginal of $\pi$ (Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling), the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward and claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ give

the middle equality by The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §displacement with $S=S_{\nu}$ and the last by the optimality of $(\mathrm{id},S_{\nu})_{\#}\nu$ (Optimal Transport Maps and Uniquely Mapped Pairs of Probability Measures §map, Optimal Coupling of Two Probability Measures with Finite Second Moment §optimal). By the formula for $\nabla\psi(\nu)$ and Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §vector-space, $z\mapsto\nabla\psi(\nu)(x)-\nabla\varphi(\hat{\mu})(y)$ represents $A-B$, and the discrepancy of $\nabla\psi(\nu)$ and $\nabla\varphi(\hat{\mu})$ along $\pi$ is $\lVert A-B\rVert_{\pi}^{2}$. Since $I(\pi)<\gamma^{2}\le\theta_{d}^{2}$ and $\nu\in\mathcal{D}$, $\lVert A\rVert_{\pi}<\tfrac{\varepsilon}{2}$ by the choice of $\theta_{d}$ and claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; and $\lVert B\rVert_{\pi}=2\eta W_{2}(\nu,\hat{\mu})<2\eta\gamma\le\tfrac{\varepsilon}{4}$. By the triangle inequality of claim 1 of The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity, $\lVert A-B\rVert_{\pi}\le\lVert A\rVert_{\pi}+\lVert B\rVert_{\pi}<\varepsilon$, so the discrepancy is less than $\varepsilon^{2}$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

Fifthly, by claim 5 of Properties of the Norm of a Symmetric Real Matrix, with differences of symmetric matrices symmetric by claim 1 of The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure, $\lVert H_{\psi}(\nu)-H_{\varphi}(\hat{\mu})\rVert\le\lVert H_{\varphi}(\nu)-H_{\varphi}(\hat{\mu})\rVert+\lVert2\eta I_{d}\rVert<\tfrac{\varepsilon}{2}+2\eta<\varepsilon$.

Therefore $(\ast)$ gives $F^{+}_{\delta}(\nu,\psi(\nu),\nabla\psi(\nu),H_{\psi}(\nu))<-\varepsilon<0$, and the test condition holds.

Let $w$ be the function of The Bump Construction on the Wasserstein Space: the Local Maximum of a Viscosity Subsolution and a Penalised Intrinsic Test Function for these data. By The Bump Construction on the Wasserstein Space: the Local Maximum of a Viscosity Subsolution and a Penalised Intrinsic Test Function §subsolution it is a viscosity subsolution, and by The Bump Construction on the Wasserstein Space: the Local Maximum of a Viscosity Subsolution and a Penalised Intrinsic Test Function §growth, $u\le w$ 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 $W_{2}(\nu,\hat{\mu})<\gamma$ and $w(\nu)=\max\{\psi(\nu)-\delta\,\mathcal{E}(\nu),u(\nu)\}$. By Basic Properties of the Delta-Envelopes on the Wasserstein Space §semicontinuity and the choice of $\theta_{c}$,

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

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 $W_{2}(\nu,\hat{\mu})<\gamma$, 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 Penalty Domain Relative to a 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 $\hat{\mu}$ with $\bigl(u(\nu_{k})+\delta\,\mathcal{E}(\nu_{k})\bigr)$ converging to $u^{+}_{\delta}(\hat{\mu})$. By claim 1 of Continuity Between Metric Spaces is Equivalent to Sequential Continuity, $(\psi(\nu_{k}))$ converges to $\psi(\hat{\mu})$. Let $\varepsilon_{0}$ be positive and choose $k$ with $W_{2}(\nu_{k},\hat{\mu})<\gamma$, $\psi(\hat{\mu})-\varepsilon_{0}<\psi(\nu_{k})$ 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, claim 3 of Properties of the Absolute Value in an Ordered Field). 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 penalty pair by Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §solution.