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

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; balls, open sets and semicontinuity are taken in the metric space $(\mathcal{P}^{a}_{\rho},W_{a})$ of 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 §metric. By Basic Properties of a Noise-Closed Noise Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets and Bounded Distances §lsc the penalty $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$, and by Basic Properties of a Noise-Closed Noise Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets and Bounded Distances §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 $\mathcal{P}^{a}_{\rho}$ by Noise Intrinsic Test Functions on the Noise Wasserstein Space §continuity. For $\nu',\nu\in\mathcal{P}^{a}_{\rho}$, which form a noise-connected ordered pair (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 $\pi\in\Pi^{a}(\nu',\nu)$ we have $W_{a}(\nu',\nu)^{2}\le I^{a}(\pi)$ by The Noise Wasserstein Distance §distance, hence $W_{a}(\nu',\nu)\le\sqrt{I^{a}(\pi)}$ by 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 Elementary Arithmetic in an Ordered Field §translation, $a\le b$ if and only if $0\le b-a$, and $a-e\le b-e$ if and only if $0\le(b-e)-(a-e)$, where $(b-e)-(a-e)=b-a$; so $a\le b$ if and only if $a-e\le b-e$, and the two applications of Maximum of Two Elements of a Totally Ordered Set select corresponding entries.

Put $O_{1}=\{\nu\in\mathcal{P}^{a}_{\rho}:W_{a}(\nu,\hat{\mu})<\gamma\}$ and $O_{2}=\{\nu\in\mathcal{P}^{a}_{\rho}:\tfrac{\gamma}{2}<W_{a}(\nu,\hat{\mu})\}$. The set $O_{1}$ is open in $(\mathcal{P}^{a}_{\rho},W_{a})$ 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 $\nu\in\mathcal{P}^{a}_{\rho}$ lies in $O_{1}$ or in $O_{2}$, since $W_{a}(\nu,\hat{\mu})\ge\gamma$ implies $W_{a}(\nu,\hat{\mu})>\tfrac{\gamma}{2}$ by Elementary Order Arithmetic in an Ordered Field §halving. 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 $\tfrac{\gamma}{2}<W_{a}(\nu,\hat{\mu})<\gamma$, then The Bump Construction on the Noise Wasserstein Space: the Local Maximum of a Viscosity Subsolution and a Penalised Noise Intrinsic Test Function §annulus 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 $\gamma\le W_{a}(\nu,\hat{\mu})$, then $\nu\notin S_{1}$ and $w(\nu)=v(\nu)$. So $w(\nu)=v(\nu)$ for every $\nu\in\mathcal{D}$ with $\tfrac{\gamma}{2}<W_{a}(\nu,\hat{\mu})$, 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 First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §subsolution); let $C$ be as in Penalty-Subordinate Growth of a Function on the Domain of a Noise Penalty Pair §above for $v$ and $\delta$, let $b$ be as in The Bump Construction on the Noise Wasserstein Space: the Local Maximum of a Viscosity Subsolution and a Penalised Noise Intrinsic Test Function §upper-bound, 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 Elementary Arithmetic in an Ordered Field §scaling, $\lambda+\delta$ 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 Domain of a Noise Penalty Pair §above.

Claim 2. Let $\delta\in\mathbb{R}$ be positive. By claim 1 and The Delta-Envelopes of a Function on the Domain of a Noise 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 a noise 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 $W_{a}(\hat{\nu},\nu)<\tau$,

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

so $v^{-}_{\delta}-\varphi$ has a local maximum at $\hat{\nu}$ relative to $\mathcal{D}$. By continuity of $\varphi$ (Noise Intrinsic Test Functions on the Noise Wasserstein Space §continuity) there is a positive $\sigma_{0}$ with $|\varphi(\nu)-\varphi(\hat{\nu})|<\tfrac{\varepsilon}{2}$ whenever $W_{a}(\hat{\nu},\nu)<\sigma_{0}$. Put $\varepsilon''=\min\{\varepsilon,\sigma_{0},\tau\}$. Applying Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §subsolution to $v$ with $\delta$, $\varphi$, $\hat{\nu}$ and the tolerance $\varepsilon''$ yields $\nu'\in\mathcal{D}_{\Sigma}$, $\pi\in\Pi^{a}(\nu',\hat{\nu})$, $s\in\mathbb{R}$ and $q\in L^{2}(\nu';X^{a})$ satisfying the five conditions there with $v$ and $\varepsilon''$. Since $\varepsilon''\le\varepsilon$ and $v^{-}_{\delta}(\hat{\nu})=w^{-}_{\delta}(\hat{\nu})$, four of the five 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 $W_{a}(\hat{\nu},\nu')\le\sqrt{I^{a}(\pi)}<\varepsilon''$, 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 Properties of the Absolute Value in an Ordered Field §strict-two-sided.

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, with $P$ as in Claim 2 for this $\delta$, $w^{-}_{\delta}(\hat{\nu})=\max\{P(\hat{\nu}),v^{-}_{\delta}(\hat{\nu})\}$ equals $P(\hat{\nu})$ by claim 2 of Elementary Properties of the Maximum of Two Elements, the other value being excluded. Let $\tau'=\min\{\tau,\gamma-W_{a}(\hat{\nu},\hat{\mu})\}$, positive. For $\nu\in\mathcal{D}$ with $W_{a}(\hat{\nu},\nu)<\tau'$ 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 $\chi=\psi-\varphi$. 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 §linear, applied with $\mathcal{D}$ in the role of its set $Q$ (not the function $Q$ above), $\psi$ in the role of its $\varphi$, $\varphi$ in the role of its $\psi$, $s=1$ and $t=-1$, shows that $\chi$ is a noise intrinsic test function on $\mathcal{D}$ with $\nabla\chi(\nu)=\nabla\psi(\nu)-\nabla\varphi(\nu)$ for $\nu\in\mathcal{D}$. The display says that the function $\chi-(\lambda+\delta)\mathcal{E}$ on $\mathcal{D}$ has a local maximum at $\hat{\nu}$ relative to $\mathcal{D}$, with radius $\tau'$. Since $\lambda+\delta$ is positive, Noise Penalty Pairs with Regular Penalised Maxima §regular gives $\hat{\nu}\in\mathcal{D}_{\Sigma}$, and then The First-Order Condition at a Penalised Extremum of a Noise Intrinsic Test Function on the Noise Wasserstein Space §maximum, applied with $\mathcal{D}$ in the role of its set $Q$, the weight $\lambda+\delta$ in the role of its $\delta$, and $\hat{\nu}$ in the role of its $\hat{\mu}$, gives

$$\nabla\psi(\hat{\nu})-\nabla\varphi(\hat{\nu})=(\lambda+\delta)\,\Sigma(\hat{\nu})\qquad\text{in }L^{2}(\hat{\nu};X^{a}).$$

Rearranged in the real vector space $L^{2}(\hat{\nu};X^{a})$ (Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields), this reads

$$\nabla\varphi(\hat{\nu})+\delta\,\Sigma(\hat{\nu})=\nabla\psi(\hat{\nu})-\lambda\,\Sigma(\hat{\nu}). \tag{1}$$

We check the test condition at $\hat{\nu}$. 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, $v(\hat{\nu})-\delta\,\mathcal{E}(\hat{\nu})\le v^{-}_{\delta}(\hat{\nu})<P(\hat{\nu})=\psi(\hat{\nu})-(\lambda+\delta)\mathcal{E}(\hat{\nu})$, so $v(\hat{\nu})<\psi(\hat{\nu})-\lambda\,\mathcal{E}(\hat{\nu})$. As $\hat{\nu}\in\mathcal{D}_{\Sigma}$ and $W_{a}(\hat{\nu},\hat{\mu})<\gamma$, The Bump Construction on the Noise Wasserstein Space: the Local Maximum of a Viscosity Subsolution and a Penalised Noise Intrinsic Test Function §condition and The Bundle of Noise Fields over a Set of Measures, First-Order Equation Operators on the Noise Wasserstein Space, and Their Delta-Shifts §shifted give

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

We take as witnesses $\nu=\hat{\nu}\in\mathcal{D}_{\Sigma}$, $\pi=\Delta=(\mathrm{id},\mathrm{id})_{\#}\hat{\nu}$, $s=w^{-}_{\delta}(\hat{\nu})$ and $q=\nabla\varphi(\hat{\nu})\in L^{2}(\hat{\nu};X^{a})$. 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, applied with $\hat{\nu}$ as its $\nu$, $\Delta\in\Pi^{a}(\hat{\nu},\hat{\nu})$ and $I^{a}(\Delta)=0<\varepsilon^{2}$. By The Bundle of Noise Fields over a Set of Measures, First-Order Equation Operators on the Noise Wasserstein Space, and Their Delta-Shifts §shifted, by $s+\delta\,\mathcal{E}(\hat{\nu})=P(\hat{\nu})+\delta\,\mathcal{E}(\hat{\nu})=\psi(\hat{\nu})-\lambda\,\mathcal{E}(\hat{\nu})$ and by (1),

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

The remaining conditions hold with vanishing left-hand sides: $|w^{-}_{\delta}(\hat{\nu})-w^{-}_{\delta}(\hat{\nu})|=0$, $|s-w^{-}_{\delta}(\hat{\nu})|=0$, and the discrepancy of $q$ and $\nabla\varphi(\hat{\nu})$ along $\Delta$ equals $\lVert q-\nabla\varphi(\hat{\nu})\rVert_{\hat{\nu}}^{2}=0$ 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 §diagonal; 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 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 §subsolution.