Proof of The Bump Construction on the Wasserstein Space: the Local Maximum of a Viscosity Subsolution and a Penalised Intrinsic Test Function

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. 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 $\mathcal{D}$, and by Basic Properties of a Wasserstein-Closed Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets, Bounded Distances, and Coercive Pairs §bounded-below there is $e_{0}\in\mathbb{R}$ with $e_{0}\le\mathcal{E}(\nu)$ for $\nu\in\mathcal{D}$. The function $\psi$ is continuous on $\mathcal{P}_{2}(\mathbb{R}^{d})$ by Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §continuity. For $\nu',\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\pi\in\Pi(\nu',\nu)$ we have $W_{2}(\nu',\nu)\le\sqrt{I(\pi)}$ by The Quadratic Wasserstein Distance on Euclidean Space §distance and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. For $a,b,e\in\mathbb{R}$ one has $\max\{a,b\}-e=\max\{a-e,b-e\}$: by claim 3 of Elementary Arithmetic in an Ordered Field, $a\le b$ if and only if $a-e\le b-e$, so the two applications of Maximum of Two Elements of a Totally Ordered Set select corresponding entries.

Put $O_{1}=\{\nu:W_{2}(\nu,\hat{\mu})<\gamma\}$ and $O_{2}=\{\nu:\tfrac{\gamma}{2}<W_{2}(\nu,\hat{\mu})\}$. The set $O_{1}$ is open in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ by Open Ball in a Metric Space is Open; $O_{2}$ is the complement of the closed ball of radius $\tfrac{\gamma}{2}$ about $\hat{\mu}$, which is closed by claim 3 of Elementary Properties of the Closed Ball in a Metric Space, so $O_{2}$ is open by Closed Subset of a Topological Space. Every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ lies in $O_{1}$ or in $O_{2}$, since $W_{2}(\nu,\hat{\mu})\ge\gamma$ implies $W_{2}(\nu,\hat{\mu})>\tfrac{\gamma}{2}$ by claim 8 of Elementary Order Arithmetic in an Ordered Field. Put $S_{1}=\mathcal{D}\cap O_{1}$ and $S_{2}=\mathcal{D}\cap O_{2}$.

Claim 1. If $\nu\in S_{1}$ then $v(\nu)\le\max\{\psi(\nu)-\lambda\,\mathcal{E}(\nu),v(\nu)\}=w(\nu)$ by claim 1 of Elementary Properties of the Maximum of Two Elements; if $\nu\in\mathcal{D}\setminus S_{1}$ then $w(\nu)=v(\nu)$ by definition. If $\nu\in\mathcal{D}$ and $\tfrac{\gamma}{2}<W_{2}(\nu,\hat{\mu})<\gamma$, then the annulus condition gives $\psi(\nu)-\lambda\,\mathcal{E}(\nu)\le v(\nu)$, so $w(\nu)=v(\nu)$ by Maximum of Two Elements of a Totally Ordered Set; if $\nu\in\mathcal{D}$ and $\gamma\le W_{2}(\nu,\hat{\mu})$, then $\nu\notin S_{1}$ and $w(\nu)=v(\nu)$. So $w(\nu)=v(\nu)$ for every $\nu\in\mathcal{D}$ with $\tfrac{\gamma}{2}<W_{2}(\nu,\hat{\mu})$, that is, $w=v$ on $S_{2}$.

For the growth, let $\delta\in\mathbb{R}$ be positive. The subsolution $v$ has penalty-subordinate growth from above (Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution); let $C$ be as in Penalty-Subordinate Growth of a Function on the Penalty Domain §above for $v$ and $\delta$, let $b$ be as in the upper-bound hypothesis, and put $C'=\max\{C,\,b-(\lambda+\delta)e_{0}\}$. Let $\nu\in\mathcal{D}$. Then $v(\nu)\le C+\delta\,\mathcal{E}(\nu)\le C'+\delta\,\mathcal{E}(\nu)$ by claim 1 of Elementary Properties of the Maximum of Two Elements. If $\nu\in S_{1}$, then $\psi(\nu)\le b$, and $(\lambda+\delta)e_{0}\le(\lambda+\delta)\,\mathcal{E}(\nu)$ by claim 5 of Elementary Arithmetic in an Ordered Field, $\lambda+\delta$ being positive; hence

$$\psi(\nu)-\lambda\,\mathcal{E}(\nu)\le b-(\lambda+\delta)\,\mathcal{E}(\nu)+\delta\,\mathcal{E}(\nu)\le b-(\lambda+\delta)e_{0}+\delta\,\mathcal{E}(\nu)\le C'+\delta\,\mathcal{E}(\nu),$$

and $w(\nu)\le C'+\delta\,\mathcal{E}(\nu)$ by claim 3 of Elementary Properties of the Maximum of Two Elements. Otherwise $w(\nu)=v(\nu)\le C'+\delta\,\mathcal{E}(\nu)$. As $\delta$ was arbitrary, $w$ has penalty-subordinate growth from above by Penalty-Subordinate Growth of a Function on the Penalty Domain §above.

Claim 2. Let $\delta\in\mathbb{R}$ be positive. By claim 1 and The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair §minus, $W=w-\delta\mathcal{E}$ and $v-\delta\mathcal{E}$ are bounded above near each point of $\mathcal{D}$, with $w^{-}_{\delta}=W^{*}$ and $v^{-}_{\delta}=(v-\delta\mathcal{E})^{*}$.

Suppose $S_{1}$ is nonempty. Let $P,Q:S_{1}\to\mathbb{R}$ be $P(\nu)=\psi(\nu)-(\lambda+\delta)\,\mathcal{E}(\nu)$ and $Q(\nu)=v(\nu)-\delta\,\mathcal{E}(\nu)$. For $\nu\in S_{1}$ the identity above gives $W(\nu)=\max\{P(\nu),Q(\nu)\}$, since $(\psi(\nu)-\lambda\,\mathcal{E}(\nu))-\delta\,\mathcal{E}(\nu)=P(\nu)$; so $W|_{S_{1}}=P\vee Q$ in the notation of The Semicontinuous Envelopes of a Pointwise Maximum and of a Pointwise Minimum. The function $P$ is upper semicontinuous on $S_{1}$, $\lambda+\delta$ being positive: for $\nu_{0}\in S_{1}$ and positive $\varepsilon_{0}$, lower semicontinuity of $\mathcal{E}$ at $\nu_{0}$ with $\tfrac{\varepsilon_{0}}{2}(\lambda+\delta)^{-1}$ and continuity of $\psi$ at $\nu_{0}$ with $\tfrac{\varepsilon_{0}}{2}$ give a radius within which $-(\lambda+\delta)\mathcal{E}(\nu)<-(\lambda+\delta)\mathcal{E}(\nu_{0})+\tfrac{\varepsilon_{0}}{2}$ and $\psi(\nu)<\psi(\nu_{0})+\tfrac{\varepsilon_{0}}{2}$, and the sum is $P(\nu)<P(\nu_{0})+\varepsilon_{0}$. Taking $\varepsilon_{0}=1$ and half the radius so obtained shows that $P$ is bounded above near each point of $S_{1}$, and then $P^{*}=P$ by Properties of the Upper Semicontinuous Envelope §fixed. By Local Bounds, Semicontinuous Envelopes and Local Extrema Are Unchanged on the Trace of an Open Set §near-bounds and Local Bounds, Semicontinuous Envelopes and Local Extrema Are Unchanged on the Trace of an Open Set §upper, applied to $v-\delta\mathcal{E}$ on $\mathcal{D}$ with the open set $O_{1}$, $Q$ is bounded above near each point of $S_{1}$ and $Q^{*}=v^{-}_{\delta}$ on $S_{1}$. By The Semicontinuous Envelopes of a Pointwise Maximum and of a Pointwise Minimum §upper, $(P\vee Q)^{*}=\max\{P,v^{-}_{\delta}\}$ on $S_{1}$, and by Local Bounds, Semicontinuous Envelopes and Local Extrema Are Unchanged on the Trace of an Open Set §upper applied to $W$ with $O_{1}$, $W^{*}=(W|_{S_{1}})^{*}$ on $S_{1}$. Hence $w^{-}_{\delta}(\nu)=\max\{\psi(\nu)-(\lambda+\delta)\mathcal{E}(\nu),v^{-}_{\delta}(\nu)\}$ for $\nu\in S_{1}$. (If $S_{1}$ is empty there is nothing to prove.)

Suppose $S_{2}$ is nonempty. By claim 1, $W|_{S_{2}}=(v-\delta\mathcal{E})|_{S_{2}}$, so two applications of Local Bounds, Semicontinuous Envelopes and Local Extrema Are Unchanged on the Trace of an Open Set §upper with the open set $O_{2}$ give $w^{-}_{\delta}(\nu)=(W|_{S_{2}})^{*}(\nu)=((v-\delta\mathcal{E})|_{S_{2}})^{*}(\nu)=v^{-}_{\delta}(\nu)$ for $\nu\in S_{2}$.

Claim 3. By claim 1, $w$ has penalty-subordinate growth from above, so its $\delta$-envelopes $w^{-}_{\delta}$ are defined. Let $\delta\in\mathbb{R}$ satisfy $0<\delta<1$, let $\varphi$ be an intrinsic test function on $\mathcal{D}$, let $\hat{\nu}\in\mathcal{D}$ be a point at which the function with value $w^{-}_{\delta}(\nu)-\varphi(\nu)$ at $\nu\in\mathcal{D}$ has a local maximum relative to $\mathcal{D}$, with witnessing radius $\tau$ (Local Maximum of a Function Relative to a Subset of a Metric Space), and let $\varepsilon\in\mathbb{R}$ be positive. By claim 1 and Properties of the Upper Semicontinuous Envelope §monotone, $v^{-}_{\delta}(\nu)\le w^{-}_{\delta}(\nu)$ for $\nu\in\mathcal{D}$. Exactly one of the following cases occurs.

Case 1: $w^{-}_{\delta}(\hat{\nu})=v^{-}_{\delta}(\hat{\nu})$. For $\nu\in\mathcal{D}$ with $W_{2}(\hat{\nu},\nu)<\tau$,

$$v^{-}_{\delta}(\nu)-\varphi(\nu)\le w^{-}_{\delta}(\nu)-\varphi(\nu)\le w^{-}_{\delta}(\hat{\nu})-\varphi(\hat{\nu})=v^{-}_{\delta}(\hat{\nu})-\varphi(\hat{\nu}),$$

so $v^{-}_{\delta}-\varphi$ has a local maximum at $\hat{\nu}$ relative to $\mathcal{D}$. By continuity of $\varphi$ (Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §continuity) there is a positive $\sigma_{0}$ with $|\varphi(\nu)-\varphi(\hat{\nu})|<\tfrac{\varepsilon}{2}$ whenever $W_{2}(\hat{\nu},\nu)<\sigma_{0}$. Put $\varepsilon''=\min\{\varepsilon,\sigma_{0},\tau\}$. Applying Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution to $v$ with $\delta$, $\varphi$, $\hat{\nu}$ and the tolerance $\varepsilon''$ yields $\nu'\in\mathcal{D}_{\Sigma}$, $\pi\in\Pi(\nu',\hat{\nu})$, $s$, $q$ and $Y$ satisfying the six conditions there with $v$ and $\varepsilon''$. Since $\varepsilon''\le\varepsilon$ and $v^{-}_{\delta}(\hat{\nu})=w^{-}_{\delta}(\hat{\nu})$, five of the six conditions required for $w$ with tolerance $\varepsilon$ follow at once (for the cost and the discrepancy by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field). For the remaining one: $\nu'\in\mathcal{D}$ and $W_{2}(\hat{\nu},\nu')\le\sqrt{I(\pi)}<\varepsilon''$, so the local maximum and the choice of $\sigma_{0}$ give $w^{-}_{\delta}(\nu')-w^{-}_{\delta}(\hat{\nu})\le\varphi(\nu')-\varphi(\hat{\nu})<\tfrac{\varepsilon}{2}$, while $w^{-}_{\delta}(\nu')\ge v^{-}_{\delta}(\nu')>v^{-}_{\delta}(\hat{\nu})-\varepsilon''\ge w^{-}_{\delta}(\hat{\nu})-\varepsilon$. So $|w^{-}_{\delta}(\nu')-w^{-}_{\delta}(\hat{\nu})|<\varepsilon$ by claim 9 of Properties of the Absolute Value in an Ordered Field.

Case 2: $v^{-}_{\delta}(\hat{\nu})<w^{-}_{\delta}(\hat{\nu})$. By claim 2, $\hat{\nu}\notin S_{2}$, so $\hat{\nu}\in S_{1}$, and $w^{-}_{\delta}(\hat{\nu})=\max\{P(\hat{\nu}),v^{-}_{\delta}(\hat{\nu})\}$ equals $P(\hat{\nu})$ by claim 2 of Elementary Properties of the Maximum of Two Elements, the other value being excluded. Let $\tau'=\min\{\tau,\gamma-W_{2}(\hat{\nu},\hat{\mu})\}$, positive. For $\nu\in\mathcal{D}$ with $W_{2}(\hat{\nu},\nu)<\tau'$ we have $\nu\in S_{1}$, so by claim 2 and claim 1 of Elementary Properties of the Maximum of Two Elements

$$P(\nu)-\varphi(\nu)\le w^{-}_{\delta}(\nu)-\varphi(\nu)\le w^{-}_{\delta}(\hat{\nu})-\varphi(\hat{\nu})=P(\hat{\nu})-\varphi(\hat{\nu}).$$

Let $\chi=\psi-\varphi$, an intrinsic test function on $\mathcal{D}$ with $\nabla\chi(\nu)=\nabla\psi(\nu)-\nabla\varphi(\nu)$ for $\nu\in\mathcal{D}$ and $H_{\chi}=H_{\psi}-H_{\varphi}$, by Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §difference. The display says that $\chi-(\lambda+\delta)\mathcal{E}$ has a local maximum at $\hat{\nu}$ relative to $\mathcal{D}$. Since $\lambda+\delta$ is positive, Penalty Pairs with Regular Penalised Maxima §regular gives $\hat{\nu}\in\mathcal{D}_{\Sigma}$, and then First- and Second-Order Conditions at a Penalised Extremum of an Intrinsic Test Function on the Wasserstein Space §maximum, applied with $Q=\mathcal{D}$, gives

$$\nabla\psi(\hat{\nu})-\nabla\varphi(\hat{\nu})=(\lambda+\delta)\,\Sigma(\hat{\nu}),\qquad H_{\psi}(\hat{\nu})-H_{\varphi}(\hat{\nu})\preceq(\lambda+\delta)H_{\mathcal{E}}(\hat{\nu}).$$

The first identity, rearranged in the vector space $L^{2}(\hat{\nu};\mathbb{R}^{d})$ (Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields), reads $\nabla\varphi(\hat{\nu})+\delta\,\Sigma(\hat{\nu})=\nabla\psi(\hat{\nu})-\lambda\,\Sigma(\hat{\nu})$. The second says, by The Positive Semidefinite Ordering on Symmetric Matrices, that $(\lambda+\delta)H_{\mathcal{E}}(\hat{\nu})-(H_{\psi}(\hat{\nu})-H_{\varphi}(\hat{\nu}))$ is positive semidefinite. This matrix equals $\bigl(H_{\varphi}(\hat{\nu})+\delta H_{\mathcal{E}}(\hat{\nu})\bigr)-\bigl(H_{\psi}(\hat{\nu})-\lambda H_{\mathcal{E}}(\hat{\nu})\bigr)$, the two being equal entrywise by Difference of Real Matrices and the field axioms, and both matrices in parentheses lie in $\mathcal{S}(d)$ by claim 1 of The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure; so, by The Positive Semidefinite Ordering on Symmetric Matrices again,

$$H_{\psi}(\hat{\nu})-\lambda H_{\mathcal{E}}(\hat{\nu})\preceq H_{\varphi}(\hat{\nu})+\delta H_{\mathcal{E}}(\hat{\nu}).$$

We check the test condition at $\hat{\nu}$. By Basic Properties of the Delta-Envelopes on the Wasserstein Space §semicontinuity, $v(\hat{\nu})-\delta\,\mathcal{E}(\hat{\nu})\le v^{-}_{\delta}(\hat{\nu})<P(\hat{\nu})=\psi(\hat{\nu})-(\lambda+\delta)\mathcal{E}(\hat{\nu})$, so $v(\hat{\nu})<\psi(\hat{\nu})-\lambda\,\mathcal{E}(\hat{\nu})$. As $\hat{\nu}\in\mathcal{D}_{\Sigma}$ and $W_{2}(\hat{\nu},\hat{\mu})<\gamma$, the test condition and The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts §shifted give

$$F\bigl(\hat{\nu},\ \psi(\hat{\nu})-\lambda\,\mathcal{E}(\hat{\nu}),\ \nabla\psi(\hat{\nu})-\lambda\,\Sigma(\hat{\nu}),\ H_{\psi}(\hat{\nu})-\lambda H_{\mathcal{E}}(\hat{\nu})\bigr)=F^{+}_{\lambda}\bigl(\hat{\nu},\psi(\hat{\nu}),\nabla\psi(\hat{\nu}),H_{\psi}(\hat{\nu})\bigr)\le0 .$$

We take as witnesses $\nu=\hat{\nu}\in\mathcal{D}_{\Sigma}$, $\pi=(\mathrm{id},\mathrm{id})_{\#}\hat{\nu}$, $s=w^{-}_{\delta}(\hat{\nu})$, $q=\nabla\varphi(\hat{\nu})$ and $Y=H_{\varphi}(\hat{\nu})$. By The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §displacement with $S=\mathrm{id}$, $\pi\in\Pi(\hat{\nu},\hat{\nu})$ and $I(\pi)=\lVert\mathrm{id}-\mathrm{id}\rVert_{\hat{\nu}}^{2}=0<\varepsilon^{2}$. By The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts §shifted, and since $w^{-}_{\delta}(\hat{\nu})+\delta\,\mathcal{E}(\hat{\nu})=P(\hat{\nu})+\delta\,\mathcal{E}(\hat{\nu})=\psi(\hat{\nu})-\lambda\,\mathcal{E}(\hat{\nu})$,

$$F^{-}_{\delta}\bigl(\hat{\nu},s,q,Y\bigr)=F\bigl(\hat{\nu},\ \psi(\hat{\nu})-\lambda\,\mathcal{E}(\hat{\nu}),\ \nabla\psi(\hat{\nu})-\lambda\,\Sigma(\hat{\nu}),\ H_{\varphi}(\hat{\nu})+\delta H_{\mathcal{E}}(\hat{\nu})\bigr)\le F\bigl(\hat{\nu},\ \psi(\hat{\nu})-\lambda\,\mathcal{E}(\hat{\nu}),\ \nabla\psi(\hat{\nu})-\lambda\,\Sigma(\hat{\nu}),\ H_{\psi}(\hat{\nu})-\lambda H_{\mathcal{E}}(\hat{\nu})\bigr)\le0\le\varepsilon,$$

the first inequality by Degenerate Elliptic Second-Order Equation Operators on the Wasserstein Space §elliptic applied with $H_{\psi}(\hat{\nu})-\lambda H_{\mathcal{E}}(\hat{\nu})\preceq H_{\varphi}(\hat{\nu})+\delta H_{\mathcal{E}}(\hat{\nu})$ (the pair $(\hat{\nu},\nabla\psi(\hat{\nu})-\lambda\Sigma(\hat{\nu}))$ lying in $\mathcal{V}(\mathcal{D}_{\Sigma})$, both fields belonging to $L^{2}(\hat{\nu};\mathbb{R}^{d})$). The remaining conditions hold with vanishing left-hand sides: $|w^{-}_{\delta}(\hat{\nu})-w^{-}_{\delta}(\hat{\nu})|=0$, $|s-w^{-}_{\delta}(\hat{\nu})|=0$, the discrepancy of $q$ and $\nabla\varphi(\hat{\nu})$ along $\pi$ equals $\lVert q-\nabla\varphi(\hat{\nu})\rVert_{\hat{\nu}}^{2}=0$ by The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures §diagonal, and $\lVert Y-H_{\varphi}(\hat{\nu})\rVert=\lVert0_{d}\rVert=0$ by claim 4 of Properties of the Norm of a Symmetric Real Matrix; each is less than the corresponding positive bound.

In both cases witnesses exist. As $\delta$, $\varphi$, $\hat{\nu}$ and $\varepsilon$ were arbitrary, $w$ is a viscosity subsolution of $F$ relative to the penalty pair.