Proof of The Pointwise Supremum of a Uniformly Subordinate Family of Viscosity Subsolutions on the Wasserstein Space is a Viscosity Subsolution

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 $\pi\in\Pi(\nu,\rho)$ we have $W_{2}(\nu,\rho)\le\sqrt{I(\pi)}$, by The Quadratic Wasserstein Distance on Euclidean Space §distance and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Each $v\in\mathcal{S}$, being a viscosity subsolution, has penalty-subordinate growth from above (Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution). By Basic Properties of a Wasserstein-Closed Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets, Bounded Distances, and Coercive Pairs §lsc the penalty $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$.

Claim 1. Let $\delta\in\mathbb{R}$ be positive and let $C$ be as in the assumption that $\mathcal{S}$ is uniformly subordinate from above, for $\delta$. For $\mu\in\mathcal{D}$, $C+\delta\,\mathcal{E}(\mu)$ is an upper bound of $\{v(\mu):v\in\mathcal{S}\}$, so $u(\mu)\le C+\delta\,\mathcal{E}(\mu)$ by Upper Bound and Least Upper Bound. As $\delta$ was arbitrary, $u$ has penalty-subordinate growth from above by Penalty-Subordinate Growth of a Function on the Penalty Domain §above. For $v\in\mathcal{S}$ and $\mu\in\mathcal{D}$, $v(\mu)$ lies in the set of which $u(\mu)$ is an upper bound, so $v(\mu)\le u(\mu)$. Consequently, for positive $\delta$ the functions $v-\delta\mathcal{E}$ and $u-\delta\mathcal{E}$ on $\mathcal{D}$, both bounded above near each point of $\mathcal{D}$ by The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair §minus, satisfy $v-\delta\mathcal{E}\le u-\delta\mathcal{E}$ pointwise, and Properties of the Upper Semicontinuous Envelope §monotone gives $v^{-}_{\delta}(\nu)\le u^{-}_{\delta}(\nu)$ for $\nu\in\mathcal{D}$.

Claim 2. By claim 1, $u$ has penalty-subordinate growth from above, so its $\delta$-envelopes $u^{-}_{\delta}$ are defined. Let $\delta\in\mathbb{R}$ satisfy $0<\delta<1$, as in Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution, let $\varphi$ be an intrinsic test function on $\mathcal{D}$, let $\hat{\mu}\in\mathcal{D}$ be a point at which the function with value $u^{-}_{\delta}(\mu)-\varphi(\mu)$ at $\mu\in\mathcal{D}$ has a local maximum relative to $\mathcal{D}$, witnessed by a positive radius $\tau$ as in Local Maximum of a Function Relative to a Subset of a Metric Space, and let $\varepsilon\in\mathbb{R}$ be positive. Write $M=u^{-}_{\delta}(\hat{\mu})-\varphi(\hat{\mu})$.

Step 1: a strict maximum. Put $\beta=\tfrac{\varepsilon}{8}$ and let $\psi_{0}(\nu)=W_{2}(\nu,\hat{\mu})^{2}$. Since $\mathcal{D}$ has the map property, $\psi_{0}$ is an intrinsic test function on $\mathcal{D}$ with $H_{\psi_{0}}(\nu)=2I_{d}$ for every $\nu$ and $\nabla\psi_{0}(\hat{\mu})=2(\mathrm{id}-S)$ for any optimal map $S$ from $\hat{\mu}$ to $\hat{\mu}$, by The Squared Wasserstein Distance to a Fixed Measure and Functions of the Mean are Intrinsic Test Functions §distance. The identity map is such an $S$: $\mathrm{id}_{\#}\hat{\mu}=\hat{\mu}$, and by The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §displacement with $S=\mathrm{id}$ the coupling $(\mathrm{id},\mathrm{id})_{\#}\hat{\mu}$ has cost $\lVert\mathrm{id}-\mathrm{id}\rVert_{\hat{\mu}}^{2}=0=W_{2}(\hat{\mu},\hat{\mu})^{2}$, so it is optimal by Optimal Coupling of Two Probability Measures with Finite Second Moment §optimal and $\mathrm{id}$ is an optimal map by Optimal Transport Maps and Uniquely Mapped Pairs of Probability Measures §map. Hence $\nabla\psi_{0}(\hat{\mu})=0$. Let $\tilde{\varphi}=\varphi+\beta\psi_{0}$. By Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §linear, $\tilde{\varphi}$ is an intrinsic test function on $\mathcal{D}$ with

$$\nabla\tilde{\varphi}(\hat{\mu})=\nabla\varphi(\hat{\mu}),\qquad H_{\tilde{\varphi}}(\hat{\mu})=H_{\varphi}(\hat{\mu})+2\beta I_{d},\qquad\tilde{\varphi}(\hat{\mu})=\varphi(\hat{\mu}),$$

the last because $W_{2}(\hat{\mu},\hat{\mu})=0$. Moreover $\lVert I_{d}\rVert\le1$: for $\xi\in\mathbb{R}^{d}$ with $\lVert\xi\rVert\le1$ we have $|\xi\cdot(I_{d}\xi)|=\lVert\xi\rVert^{2}\le1$ 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, so $1$ is an upper bound of the set whose least upper bound is $\lVert I_{d}\rVert$ by Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §norm. By claim 5 of Properties of the Norm of a Symmetric Real Matrix, $\lVert H_{\tilde{\varphi}}(\hat{\mu})-H_{\varphi}(\hat{\mu})\rVert=\lVert2\beta I_{d}\rVert\le2\beta=\tfrac{\varepsilon}{4}$. For $\nu\in\mathcal{D}$ with $W_{2}(\nu,\hat{\mu})<\tau$ the local maximum gives

$$u^{-}_{\delta}(\nu)-\tilde{\varphi}(\nu)\le M-\beta\,W_{2}(\nu,\hat{\mu})^{2}. \tag{1}$$

Step 2: radii. Using Continuous Map Between Metric Spaces for $\varphi$ and $\tilde{\varphi}$ (property (a), Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §continuity), Upper Semicontinuous Function on a Subset of a Metric Space for $u^{-}_{\delta}$, which is upper semicontinuous on $\mathcal{D}$ by Basic Properties of the Delta-Envelopes on the Wasserstein Space §semicontinuity, and property (e) of $\tilde{\varphi}$ (Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §hessian-continuity), choose positive radii $\theta_{0},\theta_{2},\theta_{3},\theta_{4}$ such that for $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$: $W_{2}(\nu,\hat{\mu})<\theta_{0}$ implies $|\varphi(\nu)-\varphi(\hat{\mu})|<1$; $W_{2}(\nu,\hat{\mu})<\theta_{2}$ implies $\lVert H_{\tilde{\varphi}}(\nu)-H_{\tilde{\varphi}}(\hat{\mu})\rVert<\tfrac{\varepsilon}{4}$; $\nu\in\mathcal{D}$ and $W_{2}(\nu,\hat{\mu})<\theta_{3}$ imply $u^{-}_{\delta}(\nu)<u^{-}_{\delta}(\hat{\mu})+\tfrac{\varepsilon}{4}$; $W_{2}(\nu,\hat{\mu})<\theta_{4}$ implies $|\tilde{\varphi}(\nu)-\tilde{\varphi}(\hat{\mu})|<\tfrac{\varepsilon}{8}$.

We also need a radius for the gradients: there is a positive $\theta_{1}$ such that for every $\nu\in\mathcal{D}$ and every $\pi\in\Pi(\nu,\hat{\mu})$ with $I(\pi)<\theta_{1}^{2}$ the discrepancy of $\nabla\tilde{\varphi}(\nu)$ and $\nabla\tilde{\varphi}(\hat{\mu})$ along $\pi$ is less than $(\tfrac{\varepsilon}{4})^{2}$. Suppose not. Let $(h_{n})_{n\in\mathbb{N}}$ be a sequence of positive reals with limit $0$ (Existence of a Sequence of Positive Real Numbers with Limit Zero); for each $n$ there are $\nu_{n}\in\mathcal{D}$ and $\pi_{n}\in\Pi(\nu_{n},\hat{\mu})$ with $I(\pi_{n})<h_{n}^{2}$ and discrepancy $D_{n}\ge(\tfrac{\varepsilon}{4})^{2}$. Since $0\le I(\pi_{n})<h_{n}^{2}$ and $(h_{n}^{2})$ has limit $0$ by claim 2 of Arithmetic of Limits of Real Sequences, $(I(\pi_{n}))$ has limit $0$ by claim 2 of Order Properties of Limits of Real Sequences; property (c) of $\tilde{\varphi}$ on $\mathcal{D}$ (Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §gradient-continuity), at the point $\hat{\mu}\in\mathcal{D}$, then says that $(D_{n})$ has limit $0$, and claim 1 of Order Properties of Limits of Real Sequences gives $(\tfrac{\varepsilon}{4})^{2}\le0$, contradicting claim 5 of Elementary Order Arithmetic in an Ordered Field.

Let $C$ be as in the assumption that $\mathcal{S}$ is uniformly subordinate from above, for the positive number $\tfrac{\delta}{2}$ (claim 8 of Elementary Order Arithmetic in an Ordered Field). Put

$$\rho=\tfrac12\min\{\tau,\theta_{0}\},\qquad\sigma=\tfrac12\min\{\rho,\theta_{1},\theta_{2},\theta_{3},\theta_{4},\varepsilon\},\qquad\eta=\min\Bigl\{\tfrac{\beta\sigma^{2}}{3},\tfrac{\varepsilon}{24}\Bigr\},$$

all positive by claim 2 of Elementary Properties of the Minimum of Two Elements (applied repeatedly) and claim 8 of Elementary Order Arithmetic in an Ordered Field, and let $K=\{\nu\in\mathcal{D}:W_{2}(\nu,\hat{\mu})\le\rho\}$. Since $\rho<\tau$ and $\rho<\theta_{0}$, every $\nu\in K$ satisfies (1) and $|\varphi(\nu)-\varphi(\hat{\mu})|<1$.

Step 3: a member of the family and its maximiser. Let $\theta_{\eta}$ be a positive radius with $|\tilde{\varphi}(\nu)-\tilde{\varphi}(\hat{\mu})|<\eta$ whenever $W_{2}(\nu,\hat{\mu})<\theta_{\eta}$, and put $\eta'=\min\{\eta,\tfrac12\theta_{\eta},\rho\}$. By Properties of the Upper Semicontinuous Envelope §approximation, applied to $u-\delta\mathcal{E}$ on $\mathcal{D}$ at $\hat{\mu}$ with $\eta'$, there is $z\in\mathcal{D}$ with $W_{2}(z,\hat{\mu})\le\eta'$ and $|u(z)-\delta\,\mathcal{E}(z)-u^{-}_{\delta}(\hat{\mu})|<\eta'\le\eta$. Then $z\in K$ and $|\tilde{\varphi}(z)-\tilde{\varphi}(\hat{\mu})|<\eta$. By claim 3 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ there is $v\in\mathcal{S}$ with $u(z)-\eta<v(z)$.

Let $g:\mathcal{D}\to\mathbb{R}$ be $g(\nu)=v^{-}_{\delta}(\nu)-\tilde{\varphi}(\nu)$. It is upper semicontinuous on $\mathcal{D}$: at $\nu_{0}\in\mathcal{D}$ and for positive $\varepsilon_{0}$, Basic Properties of the Delta-Envelopes on the Wasserstein Space §semicontinuity and Upper Semicontinuous Function on a Subset of a Metric Space give a radius within which $v^{-}_{\delta}(\nu)<v^{-}_{\delta}(\nu_{0})+\tfrac{\varepsilon_{0}}{2}$, continuity of $\tilde{\varphi}$ a radius within which $-\tilde{\varphi}(\nu)<-\tilde{\varphi}(\nu_{0})+\tfrac{\varepsilon_{0}}{2}$ (claim 3 of Properties of the Absolute Value in an Ordered Field), and within the lesser radius the two add to $g(\nu)<g(\nu_{0})+\varepsilon_{0}$. Since $v(\nu)\le C+\tfrac{\delta}{2}\,\mathcal{E}(\nu)$ for every $\nu\in\mathcal{D}$, $0\le\tfrac{\delta}{2}\le\delta$ and $\delta-\tfrac{\delta}{2}=\tfrac{\delta}{2}$, Basic Properties of the Delta-Envelopes on the Wasserstein Space §bound, applied to $v$ with $\tfrac{\delta}{2}$ in the role of the weight there written $\eta$ and with the constant $C$, gives $v^{-}_{\delta}(\nu)\le C-\tfrac{\delta}{2}\,\mathcal{E}(\nu)$ for every $\nu\in\mathcal{D}$; and for $\nu\in K$, $-\tilde{\varphi}(\nu)\le-\varphi(\nu)<1-\varphi(\hat{\mu})$, since $\beta\psi_{0}(\nu)\ge0$. Hence

$$g(\nu)\le\bigl(C+1-\varphi(\hat{\mu})\bigr)-\tfrac{\delta}{2}\,\mathcal{E}(\nu)\qquad(\nu\in K). \tag{3}$$

On the other hand, for $\nu\in K$ we have $v^{-}_{\delta}(\nu)\le u^{-}_{\delta}(\nu)$ by claim 1, and $\nu$ satisfies (1), so

$$g(\nu)\le M-\beta\,W_{2}(\nu,\hat{\mu})^{2}\le M\qquad(\nu\in K). \tag{4}$$

By Basic Properties of the Delta-Envelopes on the Wasserstein Space §semicontinuity, $v(z)-\delta\,\mathcal{E}(z)\le v^{-}_{\delta}(z)$, so, by the choice of $v$, of $z$ and $\tilde{\varphi}(\hat{\mu})=\varphi(\hat{\mu})$,

$$g(z)\ge v(z)-\delta\,\mathcal{E}(z)-\tilde{\varphi}(z)>u(z)-\delta\,\mathcal{E}(z)-\eta-\tilde{\varphi}(z)>u^{-}_{\delta}(\hat{\mu})-2\eta-\bigl(\varphi(\hat{\mu})+\eta\bigr)=M-3\eta . \tag{5}$$

We perturb $g$ by a Borwein--Preiss argument on a complete subset of $K$. The data are chosen in the order $\ell$, $B$, $X$, $\varepsilon'$, $(c_{k})_{k\in\mathbb{N}}$, and then $\hat{\nu}$ and $(\nu_{k})_{k\in\mathbb{N}}$. Put

$$\ell=\max\Bigl\{\mathcal{E}(z),\ \tfrac{2}{\delta}\bigl(C+1-\varphi(\hat{\mu})-M+3\eta\bigr)\Bigr\}.$$

Then $\mathcal{E}(z)\le\ell$ by claim 1 of Elementary Properties of the Maximum of Two Elements, which also gives $\tfrac{2}{\delta}\bigl(C+1-\varphi(\hat{\mu})-M+3\eta\bigr)\le\ell$; so by (3) every $\nu\in K$ with $\mathcal{E}(\nu)>\ell$ satisfies $g(\nu)<\bigl(C+1-\varphi(\hat{\mu})\bigr)-\tfrac{\delta}{2}\,\ell\le M-3\eta$, since $\tfrac{\delta}{2}$ is positive; we record this as

$$g(\nu)<M-3\eta\qquad\text{for }\nu\in K\text{ with }\mathcal{E}(\nu)>\ell. \tag{6}$$

Write $\mathcal{D}_{\ell}=\{\nu\in\mathcal{D}:\mathcal{E}(\nu)\le\ell\}$, let $B\in\mathbb{R}$ be as in Wasserstein-Closed Penalty Pairs §bounded for $\ell$, so that $M_{2}(\nu)\le B$ for every $\nu\in\mathcal{D}_{\ell}$, and let $X=\{\nu\in\mathcal{D}_{\ell}:W_{2}(\nu,\hat{\mu})\le\rho\}=K\cap\mathcal{D}_{\ell}$, which contains $z$. By Basic Properties of a Wasserstein-Closed Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets, Bounded Distances, and Coercive Pairs §complete, $(\mathcal{D}_{\ell},W_{2})$ is a complete metric space. The restriction of $W_{2}$ to $X\times X$, again written $W_{2}$, is a metric on $X$, the conditions of Metric Space being inherited from $\mathcal{D}_{\ell}$, and $(X,W_{2})$ is complete in the sense of Complete Metric Space: a Cauchy sequence $(\mu_{n})_{n\in\mathbb{N}}$ in $(X,W_{2})$ is one in $(\mathcal{D}_{\ell},W_{2})$, the distances being the same, so it converges in $(\mathcal{D}_{\ell},W_{2})$ to some $\mu\in\mathcal{D}_{\ell}$ (Convergent Sequence in a Metric Space); for every positive $t$ there is $n$ with $W_{2}(\mu_{n},\mu)<t$, whence $W_{2}(\mu,\hat{\mu})\le W_{2}(\mu,\mu_{n})+W_{2}(\mu_{n},\hat{\mu})<t+\rho$; were $W_{2}(\mu,\hat{\mu})>\rho$, the choice $t=W_{2}(\mu,\hat{\mu})-\rho$ would give $W_{2}(\mu,\hat{\mu})<W_{2}(\mu,\hat{\mu})$; so $W_{2}(\mu,\hat{\mu})\le\rho$, $\mu\in X$, and $(\mu_{n})_{n\in\mathbb{N}}$ converges to $\mu$ in $(X,W_{2})$.

The restriction of $g$ to $X$ is upper semicontinuous on $X$, the radii supplied by the upper semicontinuity of $g$ on $\mathcal{D}$ serving at each point of $X\subseteq\mathcal{D}$, and it is bounded above by $M$ by (4). Hence $\{g(\nu):\nu\in X\}$, nonempty since $z\in X$, has a least upper bound (The Real Numbers: Standing Notation and Background §bounds), which is at most $M$ by Upper Bound and Least Upper Bound, and by (5), $g(z)>M-3\eta\ge\sup_{\nu\in X}g(\nu)-3\eta$. For $\nu,\nu'\in X$ let $\kappa(\nu,\nu')=W_{2}(\nu,\nu')^{2}$. Then $\kappa(\nu,\nu)=0$, and $0\le\kappa(\nu,\nu')\le4B$ by Basic Properties of a Wasserstein-Closed Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets, Bounded Distances, and Coercive Pairs §diameter, applied with $\ell$ and $B$; in particular $4B\ge\kappa(z,z)=0$. For $\nu'\in X$ the function $\kappa(\cdot,\nu')$ is the restriction to $X$ of the function with value $W_{2}(\nu,\nu')^{2}$ at $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, which is an intrinsic test function on $\mathcal{D}$ by The Squared Wasserstein Distance to a Fixed Measure and Functions of the Mean are Intrinsic Test Functions §distance and so continuous on $\mathcal{P}_{2}(\mathbb{R}^{d})$ (Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §continuity); by Continuous Map Between Metric Spaces and claim 3 of Properties of the Absolute Value in an Ordered Field it is therefore lower semicontinuous on $X$ in the sense of Lower Semicontinuous Function on a Subset of a Metric Space. For every positive $t$, the positive number $\tfrac{t^{2}}{4}$ has the property that $\kappa(\nu,\nu')\le\tfrac{t^{2}}{4}$ implies $W_{2}(\nu,\nu')^{2}<t^{2}$ and hence $W_{2}(\nu,\nu')<t$, by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Put

$$\varepsilon'=\frac{\varepsilon}{32(1+\rho)},\qquad c_{k}=\varepsilon'\bigl(\tfrac12\bigr)^{k}\quad(k\in\mathbb{N}),$$

all positive; by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric and Elementary Properties of Series of Real Numbers §linearity the series $\sum_{k=1}^{\infty}c_{k}$ converges, with sum $\varepsilon'$.

We apply A Smooth Variational Principle of Borwein-Preiss Type with a Gauge on a Complete Metric Space to the nonempty complete metric space $(X,W_{2})$, the restriction of $g$ to $X$ in the role of the function there written $f$, the gauge $\kappa$ in the role of the function there written $g$, the bound $4B$, the weights $(c_{k})_{k\in\mathbb{N}}$, the tolerance $3\eta$ in the role of its $\varepsilon$, and the point $z$ in the role of its $x_{1}$. It provides $\hat{\nu}\in X$ and a sequence $(\nu_{k})_{k\in\mathbb{N}}$ in $X$ with $\nu_{1}=z$. Since $M_{2}(\nu_{k})\le B$ for every $k$, A Convergent Series of Squared Wasserstein Distances to Measures of Bounded Second Moment is an Intrinsic Test Function §convergence, with $Q=\mathcal{D}$, $B$, $(\nu_{k})_{k\in\mathbb{N}}$ and $(c_{k})_{k\in\mathbb{N}}$, defines $\psi:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ by $\psi(\nu)=\sum_{k=1}^{\infty}c_{k}\,W_{2}(\nu,\nu_{k})^{2}$, and $\psi(\nu)\ge0$ for every $\nu$, its partial sums being nonnegative (Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates). The function of the theorem is $g-\psi$ on $X$, and by its clauses A Smooth Variational Principle of Borwein-Preiss Type with a Gauge on a Complete Metric Space §value and A Smooth Variational Principle of Borwein-Preiss Type with a Gauge on a Complete Metric Space §maximum,

$$g(\hat{\nu})-\psi(\hat{\nu})\ge g(z)>M-3\eta,\qquad g(\nu)-\psi(\nu)<g(\hat{\nu})-\psi(\hat{\nu})\quad\text{for }\nu\in X\text{ with }\nu\ne\hat{\nu}, \tag{7}$$

the first by (5).

By A Convergent Series of Squared Wasserstein Distances to Measures of Bounded Second Moment is an Intrinsic Test Function §test, $\psi$ is an intrinsic test function on $\mathcal{D}$, so by Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §linear with $a=b=1$ so is $\hat{\varphi}=\tilde{\varphi}+\psi$, with $\nabla\hat{\varphi}(\hat{\nu})=\nabla\tilde{\varphi}(\hat{\nu})+\nabla\psi(\hat{\nu})$ and, by A Convergent Series of Squared Wasserstein Distances to Measures of Bounded Second Moment is an Intrinsic Test Function §hessian, $H_{\hat{\varphi}}(\hat{\nu})=H_{\tilde{\varphi}}(\hat{\nu})+2\varepsilon' I_{d}$. For each $k$ let $S_{k}$ be an optimal map from $\hat{\nu}$ to $\nu_{k}$, which exists because $\hat{\nu}\in\mathcal{D}$ and $\mathcal{D}$ has the map property (The Map Property of a Set of Probability Measures §map-property). Since $\hat{\nu},\nu_{k}\in K$, $W_{2}(\hat{\nu},\nu_{k})\le W_{2}(\hat{\nu},\hat{\mu})+W_{2}(\hat{\mu},\nu_{k})\le2\rho$, so by A Convergent Series of Squared Wasserstein Distances to Measures of Bounded Second Moment is an Intrinsic Test Function §gradient, Elementary Properties of Series of Real Numbers §order and Elementary Properties of Series of Real Numbers §linearity,

$$\lVert\nabla\psi(\hat{\nu})\rVert_{\hat{\nu}}\le2\sum_{k=1}^{\infty}c_{k}\,W_{2}(\hat{\nu},\nu_{k})\le4\rho\,\varepsilon'=\frac{\rho\,\varepsilon}{8(1+\rho)}<\frac{\varepsilon}{8}. \tag{8}$$

By claim 5 of Properties of the Norm of a Symmetric Real Matrix and $\lVert I_{d}\rVert\le1$ (Step 1),

$$\lVert H_{\hat{\varphi}}(\hat{\nu})-H_{\tilde{\varphi}}(\hat{\nu})\rVert=\lVert2\varepsilon' I_{d}\rVert\le2\varepsilon'=\frac{\varepsilon}{16(1+\rho)}<\frac{\varepsilon}{8}. \tag{9}$$

Step 4: the maximiser is close to $\hat{\mu}$. By (7) and $\psi(\hat{\nu})\ge0$, $g(\hat{\nu})\ge g(\hat{\nu})-\psi(\hat{\nu})>M-3\eta$; on the other hand $\hat{\nu}\in X\subseteq K$, so (4) gives $g(\hat{\nu})\le M-\beta\,W_{2}(\hat{\nu},\hat{\mu})^{2}$. Therefore $\beta\,W_{2}(\hat{\nu},\hat{\mu})^{2}<3\eta\le\beta\sigma^{2}$, whence $W_{2}(\hat{\nu},\hat{\mu})^{2}<\sigma^{2}$ and $W_{2}(\hat{\nu},\hat{\mu})<\sigma$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. In particular $W_{2}(\hat{\nu},\hat{\mu})$ is less than each of $\rho,\theta_{1},\theta_{2},\theta_{3},\theta_{4}$ and $\varepsilon$, each of which is at least $2\sigma$. Consequently

$$u^{-}_{\delta}(\hat{\mu})-\tfrac{\varepsilon}{4}<v^{-}_{\delta}(\hat{\nu})<u^{-}_{\delta}(\hat{\mu})+\tfrac{\varepsilon}{4}: \tag{2}$$

the right inequality from $v^{-}_{\delta}(\hat{\nu})\le u^{-}_{\delta}(\hat{\nu})$ and the choice of $\theta_{3}$; the left from $v^{-}_{\delta}(\hat{\nu})=g(\hat{\nu})+\tilde{\varphi}(\hat{\nu})>M-3\eta+\tilde{\varphi}(\hat{\mu})-\tfrac{\varepsilon}{8}=u^{-}_{\delta}(\hat{\mu})-3\eta-\tfrac{\varepsilon}{8}$, the choice of $\theta_{4}$ and $3\eta\le\tfrac{\varepsilon}{8}$.

Step 5: the witnesses for $v$. The function $\mathcal{D}\to\mathbb{R}$ with value $v^{-}_{\delta}(\nu)-\hat{\varphi}(\nu)=g(\nu)-\psi(\nu)$ has a local maximum at $\hat{\nu}$ relative to $\mathcal{D}$, with radius $\rho-W_{2}(\hat{\nu},\hat{\mu})$, positive since $W_{2}(\hat{\nu},\hat{\mu})<\rho$: if $\nu\in\mathcal{D}$ and $W_{2}(\hat{\nu},\nu)<\rho-W_{2}(\hat{\nu},\hat{\mu})$ then $W_{2}(\nu,\hat{\mu})<\rho$, so $\nu\in K$; if $\mathcal{E}(\nu)\le\ell$, then $\nu\in X$ and $g(\nu)-\psi(\nu)\le g(\hat{\nu})-\psi(\hat{\nu})$ by (7), with equality when $\nu=\hat{\nu}$; if $\mathcal{E}(\nu)>\ell$, then by $\psi(\nu)\ge0$, (6) and (7), $g(\nu)-\psi(\nu)\le g(\nu)<M-3\eta<g(\hat{\nu})-\psi(\hat{\nu})$. Put $\varepsilon''=\min\{\tfrac{\varepsilon}{8},\sigma\}$. Applying Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution to the viscosity subsolution $v$, with the same $\delta$, which satisfies $0<\delta<1$, the intrinsic test function $\hat{\varphi}$ on $\mathcal{D}$, the point $\hat{\nu}$ and the tolerance $\varepsilon''$, we obtain $\nu'\in\mathcal{D}_{\Sigma}$, $\pi'\in\Pi(\nu',\hat{\nu})$, $s\in\mathbb{R}$, $q\in L^{2}(\nu';\mathbb{R}^{d})$ and $Y\in\mathcal{S}(d)$ with

$$I(\pi')<\varepsilon''^{2},\quad|v^{-}_{\delta}(\nu')-v^{-}_{\delta}(\hat{\nu})|<\varepsilon'',\quad|s-v^{-}_{\delta}(\hat{\nu})|<\varepsilon'',$$ $$\int_{\mathbb{R}^{d+d}}\lVert q(x)-\nabla\hat{\varphi}(\hat{\nu})(y)\rVert^{2}\,\pi'(dz)<\varepsilon''^{2},\quad\lVert Y-H_{\hat{\varphi}}(\hat{\nu})\rVert<\varepsilon'',\quad F^{-}_{\delta}(\nu',s,q,Y)\le\varepsilon'' .$$

We next replace $\pi'$ by a coupling along which $q$ is compared with $\nabla\tilde{\varphi}(\hat{\nu})$. Let $\Delta=(\mathrm{id},\mathrm{id})_{\#}\hat{\nu}\in\Pi(\hat{\nu},\hat{\nu})$; by The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §displacement with $S=\mathrm{id}$, $I(\Delta)=\lVert\mathrm{id}-\mathrm{id}\rVert_{\hat{\nu}}^{2}=0$. Let $\varsigma'$ be a gluing of $\pi'$ and $\Delta$ (Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling §glued) and $\pi''=(\mathrm{q}_{1},\mathrm{q}_{3})_{\#}\varsigma'$, which belongs to $\Pi(\nu',\hat{\nu})$ with $\sqrt{I(\pi'')}\le\sqrt{I(\pi')}+\sqrt{I(\Delta)}=\sqrt{I(\pi')}<\varepsilon''$ by Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling §composite and claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. By A Triangle Inequality for Discrepancies Along a Composite Coupling §triangle, applied with $q$, $\nabla\hat{\varphi}(\hat{\nu})$ and $\nabla\tilde{\varphi}(\hat{\nu})$, by The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures §diagonal, by $\nabla\hat{\varphi}(\hat{\nu})-\nabla\tilde{\varphi}(\hat{\nu})=\nabla\psi(\hat{\nu})$ (Step 3) and by (8),

$$\sqrt{\int_{\mathbb{R}^{d+d}}\lVert q(x)-\nabla\tilde{\varphi}(\hat{\nu})(y)\rVert^{2}\,\pi''(dz)}<\varepsilon''+\lVert\nabla\psi(\hat{\nu})\rVert_{\hat{\nu}}<\varepsilon''+\tfrac{\varepsilon}{8}. \tag{10}$$

Step 6: transport to $\hat{\mu}$. By Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment there is $\gamma\in\Pi(\hat{\nu},\hat{\mu})$ with $I(\gamma)=W_{2}(\hat{\nu},\hat{\mu})^{2}$. Let $\varsigma$ be a gluing of $\pi''$ and $\gamma$ (Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling §glued) and $\pi=(\mathrm{q}_{1},\mathrm{q}_{3})_{\#}\varsigma$, which belongs to $\Pi(\nu',\hat{\mu})$ with $\sqrt{I(\pi)}\le\sqrt{I(\pi'')}+\sqrt{I(\gamma)}<\varepsilon''+W_{2}(\hat{\nu},\hat{\mu})<2\sigma$ by Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling §composite and claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. We check the six conditions of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution for $u$ at $\hat{\mu}$ with $\varphi$ and tolerance $\varepsilon$, with witnesses $\nu'$, $\pi$, $s$, $q$, $Y$.

First, $\sqrt{I(\pi)}<2\sigma\le\varepsilon$, so $I(\pi)<\varepsilon^{2}$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

Secondly, $\nu'\in\mathcal{D}$ and $W_{2}(\nu',\hat{\mu})\le\sqrt{I(\pi)}<2\sigma\le\theta_{3}$, so $u^{-}_{\delta}(\nu')<u^{-}_{\delta}(\hat{\mu})+\tfrac{\varepsilon}{4}$; and by claim 1 and (2), $u^{-}_{\delta}(\nu')\ge v^{-}_{\delta}(\nu')>v^{-}_{\delta}(\hat{\nu})-\varepsilon''>u^{-}_{\delta}(\hat{\mu})-\tfrac{\varepsilon}{4}-\tfrac{\varepsilon}{8}$. 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 (2), $|s-u^{-}_{\delta}(\hat{\mu})|\le|s-v^{-}_{\delta}(\hat{\nu})|+|v^{-}_{\delta}(\hat{\nu})-u^{-}_{\delta}(\hat{\mu})|<\tfrac{\varepsilon}{8}+\tfrac{\varepsilon}{4}<\varepsilon$.

Fourthly, since $\hat{\nu}\in\mathcal{D}$ and $I(\gamma)=W_{2}(\hat{\nu},\hat{\mu})^{2}<\theta_{1}^{2}$ (claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field), the choice of $\theta_{1}$ bounds the discrepancy of $\nabla\tilde{\varphi}(\hat{\nu})$ and $\nabla\tilde{\varphi}(\hat{\mu})=\nabla\varphi(\hat{\mu})$ along $\gamma$ by $(\tfrac{\varepsilon}{4})^{2}$. By A Triangle Inequality for Discrepancies Along a Composite Coupling §triangle, applied with $q$, $\nabla\tilde{\varphi}(\hat{\nu})$ and $\nabla\varphi(\hat{\mu})$ along $\pi''$ and $\gamma$, by (10) and claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field,

$$\sqrt{\int_{\mathbb{R}^{d+d}}\lVert q(x)-\nabla\varphi(\hat{\mu})(y)\rVert^{2}\,\pi(dz)}<\varepsilon''+\tfrac{\varepsilon}{8}+\tfrac{\varepsilon}{4}<\varepsilon ,$$

so the discrepancy of $q$ and $\nabla\varphi(\hat{\mu})$ along $\pi$ is less than $\varepsilon^{2}$.

Fifthly, all matrices below lie in $\mathcal{S}(d)$, differences of symmetric matrices being symmetric by claim 1 of The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure, and claim 5 of Properties of the Norm of a Symmetric Real Matrix gives

$$\lVert Y-H_{\varphi}(\hat{\mu})\rVert\le\lVert Y-H_{\hat{\varphi}}(\hat{\nu})\rVert+\lVert H_{\hat{\varphi}}(\hat{\nu})-H_{\tilde{\varphi}}(\hat{\nu})\rVert+\lVert H_{\tilde{\varphi}}(\hat{\nu})-H_{\tilde{\varphi}}(\hat{\mu})\rVert+\lVert H_{\tilde{\varphi}}(\hat{\mu})-H_{\varphi}(\hat{\mu})\rVert<\tfrac{\varepsilon}{8}+\tfrac{\varepsilon}{8}+\tfrac{\varepsilon}{4}+\tfrac{\varepsilon}{4}<\varepsilon ,$$

the second term by (9), the third by the choice of $\theta_{2}$ and the last by Step 1.

Finally, $F^{-}_{\delta}(\nu',s,q,Y)\le\varepsilon''\le\varepsilon$.

As $\delta$, $\varphi$, $\hat{\mu}$ and $\varepsilon$ were arbitrary and $u$ has penalty-subordinate growth from above, $u$ is a viscosity subsolution of $F$ relative to the penalty pair.