Proof of Classical Sub- and Supersolutions of a Degenerate Elliptic Equation on the Wasserstein Space are Viscosity Sub- and Supersolutions

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

Claim 1. The function $u$ is continuous by condition (d) of Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian (i), and $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$ by (E1). By Basic Properties of the Delta-Envelopes on the Wasserstein Space (iii), applied for each positive $\delta\in\mathbb{R}$, the function $u$ is bounded above near each point and bounded below near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, and $u^{-}_{\delta}=u-\delta\mathcal{E}$ and $u^{+}_{\delta}=u+\delta\mathcal{E}$ on $\mathcal{D}$.

Claim 2. Let $\delta$, $\varphi$ and $\hat{\mu}$ be as stated, and put $\chi=u-\varphi$. By Sums, Real Multiples and Differences of Test Functions on the Wasserstein Space (ii) the function $\chi$ is a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$ with

$$\nabla\chi(\nu)=\nabla u(\nu)-\nabla\varphi(\nu),\qquad H_{\chi}(\nu)=H_{u}(\nu)-H_{\varphi}(\nu)\qquad\bigl(\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})\bigr).$$

By claim 1, for $\mu\in\mathcal{D}$ one has $u^{-}_{\delta}(\mu)-\varphi(\mu)=u(\mu)-\delta\,\mathcal{E}(\mu)-\varphi(\mu)=\chi(\mu)-\delta\,\mathcal{E}(\mu)$, so $\hat{\mu}$ is a point of $\mathcal{D}$ at which $\chi-\delta\mathcal{E}$ has a local maximum relative to $\mathcal{D}$. By (E2), $\hat{\mu}\in\mathcal{D}_{\Sigma}$, and by First- and Second-Order Conditions at a Penalised Extremum of a Test Function on the Wasserstein Space (i),

$$\nabla u(\hat{\mu})-\nabla\varphi(\hat{\mu})=\delta\,\Sigma(\hat{\mu}),\qquad H_{u}(\hat{\mu})-H_{\varphi}(\hat{\mu})\preceq0_{d}.$$

The first identity gives $\nabla\varphi(\hat{\mu})+\delta\,\Sigma(\hat{\mu})=\nabla u(\hat{\mu})$ by Elementary Identities in a Vector Space, and the second gives $H_{u}(\hat{\mu})\preceq H_{\varphi}(\hat{\mu})$ by claim 1 of Comparison with the Zero Matrix in the Positive Semidefinite Ordering. By claim 1 again, $u^{-}_{\delta}(\hat{\mu})+\delta\,\mathcal{E}(\hat{\mu})=u(\hat{\mu})$. Hence, by The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts (iii),

$$F^{-}_{\delta}\bigl(\hat{\mu},u^{-}_{\delta}(\hat{\mu}),\nabla\varphi(\hat{\mu}),H_{\varphi}(\hat{\mu})\bigr)=F\bigl(\hat{\mu},u(\hat{\mu}),\nabla u(\hat{\mu}),H_{\varphi}(\hat{\mu})\bigr)\le F\bigl(\hat{\mu},u(\hat{\mu}),\nabla u(\hat{\mu}),H_{u}(\hat{\mu})\bigr)\le0 ,$$

the first inequality by Degenerate Elliptic Second-Order Equation Operators on the Wasserstein Space (i) applied to $H_{u}(\hat{\mu})\preceq H_{\varphi}(\hat{\mu})$, and the second because $u$ is a classical subsolution of $F$ on $\mathcal{D}_{\Sigma}$ and $\hat{\mu}\in\mathcal{D}_{\Sigma}$ (Classical Sub- and Supersolutions of a Second-Order Equation on the Wasserstein Space (i)).

Claim 3. By claim 1 the function $u$ is bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, so the envelopes $u^{-}_{\delta}$ are defined. Let $\delta\in\mathbb{R}$ be positive, let $\varphi$ be a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$, let $\hat{\mu}\in\mathcal{D}$ be a point at which the function with value $u^{-}_{\delta}(\mu)-\varphi(\mu)$ at $\mu$ has a local maximum relative to $\mathcal{D}$, and let $\varepsilon\in\mathbb{R}$ be positive. By claim 2, $\hat{\mu}\in\mathcal{D}_{\Sigma}$. Take

$$\nu=\hat{\mu},\qquad\pi=(\mathrm{id},\mathrm{id})_{\#}\hat{\mu},\qquad s=u^{-}_{\delta}(\hat{\mu}),\qquad q=\nabla\varphi(\hat{\mu}),\qquad Y=H_{\varphi}(\hat{\mu}),$$

where $\mathrm{id}$ is the identity map of $\mathbb{R}^{d}$. By Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound (iv), applied with $S=T=\mathrm{id}$, one has $\pi\in\Pi(\hat{\mu},\hat{\mu})$ and

$$I(\pi)=\int_{\mathbb{R}^{d}}\lVert\mathrm{id}-\mathrm{id}\rVert^{2}\,d\hat{\mu}=0 .$$

The number $\varepsilon^{2}$ is positive by claim 5 of Elementary Order Arithmetic in an Ordered Field, so $I(\pi)<\varepsilon^{2}$. Since $\nu=\hat{\mu}$ and $s=u^{-}_{\delta}(\hat{\mu})$, the differences $u^{-}_{\delta}(\nu)-u^{-}_{\delta}(\hat{\mu})$ and $s-u^{-}_{\delta}(\hat{\mu})$ are $0$, of absolute value $0<\varepsilon$ by claim 1 of Properties of the Absolute Value in an Ordered Field. By The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures (ii) the discrepancy of $q$ and $\nabla\varphi(\hat{\mu})$ along $\pi$ is $\lVert q-\nabla\varphi(\hat{\mu})\rVert_{\hat{\mu}}^{2}=0<\varepsilon^{2}$, the norm of the zero element vanishing by Elementary Identities in a Real Inner Product Space (iii). Also $Y-H_{\varphi}(\hat{\mu})=0_{d}$ by Difference of Real Matrices, so $\lVert Y-H_{\varphi}(\hat{\mu})\rVert=0<\varepsilon$ by claim 4 of Properties of the Norm of a Symmetric Real Matrix. Finally, claim 2 gives $F^{-}_{\delta}(\nu,s,q,Y)\le0\le\varepsilon$. Every requirement of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space (i) is therefore met, and $u$ is a viscosity subsolution of $F$ relative to the penalty pair.

Claim 4. Let $\delta$, $\varphi$ and $\hat{\mu}$ be as stated and put $\chi=u-\varphi$ as in claim 2. By claim 1, $u^{+}_{\delta}(\mu)-\varphi(\mu)=\chi(\mu)+\delta\,\mathcal{E}(\mu)$ for $\mu\in\mathcal{D}$, so $\hat{\mu}$ is a point of $\mathcal{D}$ at which $\chi+\delta\mathcal{E}$ has a local minimum relative to $\mathcal{D}$. By (E2), $\hat{\mu}\in\mathcal{D}_{\Sigma}$, and by First- and Second-Order Conditions at a Penalised Extremum of a Test Function on the Wasserstein Space (ii),

$$\nabla u(\hat{\mu})-\nabla\varphi(\hat{\mu})=-\delta\,\Sigma(\hat{\mu}),\qquad 0_{d}\preceq H_{u}(\hat{\mu})-H_{\varphi}(\hat{\mu}).$$

The first identity gives $\nabla\varphi(\hat{\mu})-\delta\,\Sigma(\hat{\mu})=\nabla u(\hat{\mu})$ by Elementary Identities in a Vector Space, and the second gives $H_{\varphi}(\hat{\mu})\preceq H_{u}(\hat{\mu})$ by claim 2 of Comparison with the Zero Matrix in the Positive Semidefinite Ordering. By claim 1, $u^{+}_{\delta}(\hat{\mu})-\delta\,\mathcal{E}(\hat{\mu})=u(\hat{\mu})$. Hence, by The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts (iii),

$$F^{+}_{\delta}\bigl(\hat{\mu},u^{+}_{\delta}(\hat{\mu}),\nabla\varphi(\hat{\mu}),H_{\varphi}(\hat{\mu})\bigr)=F\bigl(\hat{\mu},u(\hat{\mu}),\nabla u(\hat{\mu}),H_{\varphi}(\hat{\mu})\bigr)\ge F\bigl(\hat{\mu},u(\hat{\mu}),\nabla u(\hat{\mu}),H_{u}(\hat{\mu})\bigr)\ge0 ,$$

the first inequality by Degenerate Elliptic Second-Order Equation Operators on the Wasserstein Space (i) applied to $H_{\varphi}(\hat{\mu})\preceq H_{u}(\hat{\mu})$, and the second because $u$ is a classical supersolution of $F$ on $\mathcal{D}_{\Sigma}$ and $\hat{\mu}\in\mathcal{D}_{\Sigma}$ (Classical Sub- and Supersolutions of a Second-Order Equation on the Wasserstein Space (ii)).

Claim 5. With the data of claim 3, taken now at a local minimum of the function with value $u^{+}_{\delta}(\mu)-\varphi(\mu)$ at $\mu$ and with $s=u^{+}_{\delta}(\hat{\mu})$, the same computations give $I(\pi)<\varepsilon^{2}$, $|u^{+}_{\delta}(\nu)-u^{+}_{\delta}(\hat{\mu})|=0<\varepsilon$, $|s-u^{+}_{\delta}(\hat{\mu})|=0<\varepsilon$, discrepancy $0<\varepsilon^{2}$ and $\lVert Y-H_{\varphi}(\hat{\mu})\rVert=0<\varepsilon$, while claim 4 gives $-\varepsilon\le0\le F^{+}_{\delta}(\nu,s,q,Y)$, the first inequality by claim 4 of Elementary Order Arithmetic in an Ordered Field. Every requirement of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space (ii) is met, so $u$ is a viscosity supersolution of $F$ relative to the penalty pair.

Claim 6. A classical solution of $F$ on $\mathcal{D}_{\Sigma}$ is both a classical subsolution and a classical supersolution there, by Classical Sub- and Supersolutions of a Second-Order Equation on the Wasserstein Space (iii); by claims 3 and 5 it is then both a viscosity subsolution and a viscosity supersolution of $F$ relative to the penalty pair, hence a viscosity solution by Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space (iii), the required local bounds holding by claim 1.