Proof of The Structure Estimate at a Maximiser for Operators within a Cost Defect of a Reference Operator

Each result cited is universally quantified over the data in its own statement, and is applied to the data named here. We write $F^{-}_{1,\delta}$ for the first $\delta$-shift of $F_{1}$ and $F^{+}_{2,\delta}$ for the second $\delta$-shift of $F_{2}$, relative to the penalty pair (The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts §shifted). We write $W$ for $W_{2}$, $M=M(\delta,\alpha)$ and $\Psi=\Psi_{\delta,\alpha}$. For $n\in\mathbb{N}$ let $\epsilon_{n}$ be the multiplicative inverse of the positive real attached to $n$ (The Real Numbers: Standing Notation and Background §numbers), so that $0<\epsilon_{n}\le1$; the real sequences $(\epsilon_{n})_{n}$ and $(\epsilon_{n}^{2})_{n}$ converge to $0$ by The Archimedean Property of the Real Numbers and Arithmetic of Limits of Real Sequences. The distance on $\mathcal{S}(d)$ is $d_{\mathcal{S}(d)}(P,P')=\lVert P-P'\rVert$, a metric (Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §matrices).

Step 0 (Norms along a coupling). Let $\nu,\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, $\pi\in\Pi(\nu,\mu)$, $q\in L^{2}(\nu;\mathbb{R}^{d})$ and $\eta\in L^{2}(\mu;\mathbb{R}^{d})$. The maps $q\circ\mathrm{pr}_{1}$ and $\eta\circ\mathrm{pr}_{2}$ (for representatives) are Borel, and by the change-of-variables formula and $(\mathrm{pr}_{1})_{\#}\pi=\nu$, $(\mathrm{pr}_{2})_{\#}\pi=\mu$ their classes in the real Hilbert space $L^{2}(\pi;\mathbb{R}^{d})$ (Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields) have norms $\lVert q\rVert_{\nu}$ and $\lVert\eta\rVert_{\mu}$, while the norm of their difference is the square root of the discrepancy of $q$ and $\eta$ along $\pi$. The triangle inequality The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle gives

With $q$ and $\eta$ the classes of $\mathrm{id}$ (Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients §identity), the discrepancy is $I(\pi)$ (Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §cost) and $\lVert\mathrm{id}\rVert_{\nu}^{2}=M_{2}(\nu)$, so $\sqrt{M_{2}(\nu)}\le\sqrt{I(\pi)}+\sqrt{M_{2}(\mu)}$.

Step 0' (Transfer to the reference operator, and an elementary bound). Let $(\nu,q)\in\mathcal{V}(\mathcal{D}_{\Sigma})$, $r\in\mathbb{R}$, $Y\in\mathcal{S}(d)$ and $\delta'>0$. By The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts §shifted, $F^{-}_{1,\delta'}(\nu,r,q,Y)=F_{1}(\nu,r',q',Y')$ and $F^{-}_{\delta'}(\nu,r,q,Y)=F(\nu,r',q',Y')$ with the same $r'=r+\delta'\mathcal{E}(\nu)$, $q'=q+\delta'\Sigma(\nu)\in L^{2}(\nu;\mathbb{R}^{d})$ and $Y'=Y+\delta'H_{\mathcal{E}}(\nu)\in\mathcal{S}(d)$; as $(\nu,q')\in\mathcal{V}(\mathcal{D}_{\Sigma})$, the first hypothesis on $F_{1}$ at $(\nu,r',q',Y')$ gives the first inequality below, and in the same way the hypothesis on $F_{2}$ gives the second:

Moreover $t\le1+t^{2}$ for every real $t$ (E): if $t\le1$ this follows from $0\le t^{2}$ (claim 2 of Nonnegativity of Squares in an Ordered Field); if $1<t$, then $0<t$ and $t=t\cdot1<t\cdot t\le1+t^{2}$ (claim 10 of Elementary Order Arithmetic in an Ordered Field).

Step 1 (A maximising pair and admitted matrices). By Existence, Penalty Bounds and Monotonicity of the Maximum of the Wasserstein-Doubled Difference of Bounded Functions §maximiser there is $(\hat{\mu},\hat{\nu})\in\mathcal{D}\times\mathcal{D}$ with $\Psi(\hat{\mu},\hat{\nu})=M$. Apply Intrinsic Test Functions at a Maximiser of the Wasserstein-Doubled Difference with the present $\delta$, $\alpha$, $u$, $v$, $b$, $b'$ and $(\hat{\mu},\hat{\nu})$; its hypotheses are among ours. It provides $\rho^{*},\sigma^{*}\in\mathcal{D}$ and $\mathbb{X},\mathbb{Y}\in\mathcal{S}(d)$; let $S$, $S'$ be the optimal maps named there, and put

By Intrinsic Test Functions at a Maximiser of the Wasserstein-Doubled Difference §maximiser, $\Psi(\rho^{*},\sigma^{*})=M\ge0$, so Existence, Penalty Bounds and Monotonicity of the Maximum of the Wasserstein-Doubled Difference of Bounded Functions §penalty, read with the present $e_{0}$, gives $\delta|\mathcal{E}(\rho^{*})|\le|b|+|b'|+\delta|e_{0}|\le B$ and likewise $\delta|\mathcal{E}(\sigma^{*})|\le B$, as $0<\delta<1$ gives $\delta|e_{0}|\le|e_{0}|$ (claim 5 of Elementary Arithmetic in an Ordered Field); by Intrinsic Test Functions at a Maximiser of the Wasserstein-Doubled Difference §admitted, $(\mathbb{X},\mathbb{Y})$ is admitted at $\alpha$, so $\lVert\mathbb{X}\rVert\le6\alpha$ and $\lVert\mathbb{Y}\rVert\le6\alpha$ (The Second-Order Structure Condition at Optimally Coupled Pairs on the Lift of the Wasserstein Space §admitted). As $s_{*}-t_{*}=M+\tfrac{\alpha}{2}W(\rho^{*},\sigma^{*})^{2}$ and $0\le M$, we have $M\le s_{*}-t_{*}$ and $t_{*}\le s_{*}$. Since $u^{-}_{\delta}\le b-\delta\mathcal{E}\le b-\delta e_{0}$ and $v^{+}_{\delta}\ge b'+\delta\mathcal{E}\ge b'+\delta e_{0}$ on $\mathcal{D}$ (The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §bounded and claim 5 of Elementary Arithmetic in an Ordered Field),

and as $0<\delta<1$ gives $|\delta e_{0}|\le|e_{0}|$ (claims 4 and 5 of Properties of the Absolute Value in an Ordered Field), $|s_{*}|\le|b|+|b'|+|e_{0}|\le B$ and $|t_{*}|\le B$ (claim 6 of Properties of the Absolute Value in an Ordered Field).

Bounds not involving $u$, $v$ beyond $b$, $b'$. From $\tfrac{\alpha}{2}W(\rho^{*},\sigma^{*})^{2}=(s_{*}-t_{*})-M\le s_{*}-t_{*}\le|s_{*}|+|t_{*}|\le2B$ (claims 3 and 5 of Properties of the Absolute Value in an Ordered Field) we get $\alpha W(\rho^{*},\sigma^{*})^{2}\le4B$. By The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class §cost, $\lVert\mathrm{id}-S\rVert_{\rho^{*}}^{2}=W(\rho^{*},\sigma^{*})^{2}$ and $\lVert\mathrm{id}-S'\rVert_{\sigma^{*}}^{2}=W(\sigma^{*},\rho^{*})^{2}=W(\rho^{*},\sigma^{*})^{2}$, by symmetry of $W$ (The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §metric); as $V'_{*}=(-\alpha)(\mathrm{id}-S')$ and $|\pm\alpha|=\alpha$, homogeneity of the norm (Elementary Identities in a Real Inner Product Space §homogeneity, in the real Hilbert spaces of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields) gives

so by (E), $\lVert V_{*}\rVert_{\rho^{*}}\le1+4\alpha B$ and $\lVert V'_{*}\rVert_{\sigma^{*}}\le1+4\alpha B$. Next, $\delta|\mathcal{E}(\rho^{*})|\le B$ gives $\mathcal{E}(\rho^{*})\le|\mathcal{E}(\rho^{*})|\le\delta^{-1}B$, so $M_{2}(\rho^{*})\le m$ by the choice of $m$; (E) with $t=\sqrt{M_{2}(\rho^{*})}$, whose square is $M_{2}(\rho^{*})$, gives $\sqrt{M_{2}(\rho^{*})}\le1+m$, and likewise $\sqrt{M_{2}(\sigma^{*})}\le1+m$.

Step 2 (Viscosity data on the subsolution side). Let $n\in\mathbb{N}$. By Intrinsic Test Functions at a Maximiser of the Wasserstein-Doubled Difference §subsolution with $\varepsilon=\epsilon_{n}$ there are $\rho_{n}\in\mathcal{D}$, an intrinsic test function $\varphi_{n}$ on $\mathcal{D}$ with $u^{-}_{\delta}-\varphi_{n}$ having a local maximum relative to $\mathcal{D}$ at $\rho_{n}$, and $\pi_{n}\in\Pi(\rho_{n},\rho^{*})$ with

As $u$ is a viscosity subsolution of $F_{1}$, Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution, read for $F_{1}$, with $\varphi_{n}$, $\rho_{n}$ and $\epsilon_{n}$ gives $\nu_{n}\in\mathcal{D}_{\Sigma}$, $\gamma_{n}\in\Pi(\nu_{n},\rho_{n})$, $s_{n}\in\mathbb{R}$, $q_{n}\in L^{2}(\nu_{n};\mathbb{R}^{d})$ and $X_{n}\in\mathcal{S}(d)$ with

By (T) and $h_{1}(\nu_{n})\le H$, $F^{-}_{\delta}(\nu_{n},s_{n},q_{n},X_{n})\le\epsilon_{n}+h_{1}(\nu_{n})\le\epsilon_{n}+H$. Let $\beta_{n}$ be a gluing of $\gamma_{n}$ and $\pi_{n}$ (Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling §glued) and $\kappa_{n}=(\mathrm{q}_{1},\mathrm{q}_{3})_{\#}\beta_{n}\in\Pi(\nu_{n},\rho^{*})$. By Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling §composite and A Triangle Inequality for Discrepancies Along a Composite Coupling §triangle, applied with $q_{n}$, $\nabla\varphi_{n}(\rho_{n})$ and $V_{*}$, and taking square roots (claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field),

and by the triangle inequalities of $|\cdot|$ (claim 5 of Properties of the Absolute Value in an Ordered Field) and of the metric $d_{\mathcal{S}(d)}$,

Hence $(\kappa_{n})_{n}$ is a sequence of couplings of vanishing cost from $(\nu_{n})_{n}$ to $\rho^{*}$, $(q_{n})_{n}$ converges strongly to $V_{*}$ along it, $(s_{n})_{n}$ converges to $s_{*}$ and $(X_{n})_{n}$ to $\mathbb{X}$ (squares of the bounds $2\epsilon_{n}$ being $4\epsilon_{n}^{2}$, and Arithmetic of Limits of Real Sequences). By Step 0 with $\kappa_{n}$, $\sqrt{M_{2}(\nu_{n})}<\sqrt{M_{2}(\rho^{*})}+2$ and $\lVert q_{n}\rVert_{\nu_{n}}<\lVert V_{*}\rVert_{\rho^{*}}+2$; also $|s_{n}|<B+2$ and $\lVert X_{n}\rVert\le d_{\mathcal{S}(d)}(X_{n},\mathbb{X})+\lVert\mathbb{X}\rVert<6\alpha+2$. Finally $\nu_{n}\in\mathcal{D}_{\Sigma}\subseteq\mathcal{D}$, so $e_{0}\le\mathcal{E}(\nu_{n})$, and $-B-2<u^{-}_{\delta}(\nu_{n})\le b-\delta\mathcal{E}(\nu_{n})$ (The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §bounded) gives $\mathcal{E}(\nu_{n})<\delta^{-1}(|b|+B+2)$; so $|\mathcal{E}(\nu_{n})|<\delta^{-1}(|b|+B+2)+|e_{0}|+1$.

Step 3 (Viscosity data on the supersolution side). In the same way, Intrinsic Test Functions at a Maximiser of the Wasserstein-Doubled Difference §supersolution with $\varepsilon=\epsilon_{n}$, the supersolution property Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §supersolution of $v$ for $F_{2}$, a gluing and A Triangle Inequality for Discrepancies Along a Composite Coupling §triangle give $\nu'_{n}\in\mathcal{D}_{\Sigma}$, $\kappa'_{n}\in\Pi(\nu'_{n},\sigma^{*})$, $t_{n}\in\mathbb{R}$, $q'_{n}\in L^{2}(\nu'_{n};\mathbb{R}^{d})$ and $Y_{n}\in\mathcal{S}(d)$ with $-\epsilon_{n}\le F^{+}_{2,\delta}(\nu'_{n},t_{n},q'_{n},Y_{n})$, hence, by (T) and $h_{2}(\nu'_{n})\le H$, $-\epsilon_{n}-H\le-\epsilon_{n}-h_{2}(\nu'_{n})\le F^{+}_{\delta}(\nu'_{n},t_{n},q'_{n},Y_{n})$, $\sqrt{I(\kappa'_{n})}<2\epsilon_{n}$, the discrepancy of $q'_{n}$ and $V'_{*}$ along $\kappa'_{n}$ below $4\epsilon_{n}^{2}$, $|v^{+}_{\delta}(\nu'_{n})-t_{*}|<2\epsilon_{n}$, $|t_{n}-t_{*}|<2\epsilon_{n}$ and $\lVert Y_{n}-\mathbb{Y}\rVert<2\epsilon_{n}$. So $(\kappa'_{n})_{n}$ has vanishing cost, $(q'_{n})_{n}$ converges strongly to $V'_{*}$ along it, $t_{n}\to t_{*}$ and $Y_{n}\to\mathbb{Y}$; and $\sqrt{M_{2}(\nu'_{n})}<\sqrt{M_{2}(\sigma^{*})}+2$, $\lVert q'_{n}\rVert_{\nu'_{n}}<\lVert V'_{*}\rVert_{\sigma^{*}}+2$, $|t_{n}|<B+2$, $\lVert Y_{n}\rVert<6\alpha+2$, and, from $b'+\delta\mathcal{E}(\nu'_{n})\le v^{+}_{\delta}(\nu'_{n})<B+2$ (The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §bounded) and $e_{0}\le\mathcal{E}(\nu'_{n})$, $|\mathcal{E}(\nu'_{n})|<\delta^{-1}(|b'|+B+2)+|e_{0}|+1$.

Step 4 (Shift-coercivity of the reference operator and closed score). Let $R_{0}$ be the number of the statement; every summand in it is nonnegative ($0\le m$, $1<\alpha$, $0\le B$ since $B\ge|b|+|b'|+|e_{0}|$, and $0<\delta^{-1}$ by claim 7 of Elementary Order Arithmetic in an Ordered Field), so each of $m+3$, $4\alpha B+3$, $B+2$, $6\alpha+2$, $\delta^{-1}(|b|+B+2)+|e_{0}|+1$, $\delta^{-1}(|b'|+B+2)+|e_{0}|+1$ and $2$ is less than $R_{0}$. By Step 1, $\sqrt{M_{2}(\rho^{*})}+2\le m+3$, $\sqrt{M_{2}(\sigma^{*})}+2\le m+3$, $\lVert V_{*}\rVert_{\rho^{*}}+2\le4\alpha B+3$ and $\lVert V'_{*}\rVert_{\sigma^{*}}+2\le4\alpha B+3$; so every bound listed at the end of Steps 2 and 3 is at most $R_{0}$. For $n\in\mathbb{N}$ put $\xi_{n}=(\nu_{n},s_{n},q_{n},X_{n})$ and $\eta_{n}=(\nu'_{n},t_{n},q'_{n},Y_{n})$, test data for $F$; they are $R_{0}$-bounded, hence $(R_{0}+2H)$-bounded as $0\le2H$, and by Steps 2 and 3,

so $\xi_{n}\in S^{-}_{\delta,R_{0}+2H}(F)$ and $\eta_{n}\in S^{+}_{\delta,R_{0}+2H}(F)$ (Test Data for an Intrinsic Second-Order Equation Operator on the Wasserstein Space and the Admissible Sets §admissible), each witnessing the other. This is where the reference operator enters: the viscosity inequalities for $F_{1}$ and $F_{2}$ have been transferred to $F$ at the cost $2H$. As $C$ is a score bound for $F$ at $(\delta,R_{0}+2H)$, a nonnegative real number by that clause, we get $\lVert\Sigma(\nu_{n})\rVert_{\nu_{n}}\le C$ and $\lVert\Sigma(\nu'_{n})\rVert_{\nu'_{n}}\le C$ for every $n$. By Penalty Pairs with Closed Score Along Couplings §closed at the level $C$, applied to $(\nu_{n})_{n}$, $\rho^{*}$ and $(\kappa_{n})_{n}$, we get $\rho^{*}\in\mathcal{D}_{\Sigma}$ and that $(\Sigma(\nu_{n}))_{n}$ converges weakly to $\Sigma(\rho^{*})$ along $(\kappa_{n})_{n}$; likewise $\sigma^{*}\in\mathcal{D}_{\Sigma}$, with $(\Sigma(\nu'_{n}))_{n}$ converging weakly to $\Sigma(\sigma^{*})$ along $(\kappa'_{n})_{n}$. So $(\rho^{*},\sigma^{*})\in\mathcal{D}_{\Sigma}\times\mathcal{D}_{\Sigma}$, and $\xi=(\rho^{*},s_{*},V_{*},\mathbb{X})$ and $\eta=(\sigma^{*},t_{*},V'_{*},\mathbb{Y})$ are test data for $F$. Finally, for every $n$ we have $\nu_{n},\nu'_{n}\in\mathcal{D}_{\Sigma}$, $|\mathcal{E}(\nu_{n})|<R_{0}$, $|\mathcal{E}(\nu'_{n})|<R_{0}$ and the two score bounds just obtained, so $\nu_{n},\nu'_{n}\in K$; hence $h_{1}(\nu_{n})\le H_{1}$ and $h_{2}(\nu'_{n})\le H_{2}$, and (T), with Steps 2 and 3, sharpens the bounds there to

Step 5 (Shift-semicontinuity of the reference operator at the defect levels). Put $R''=R_{0}+C$, positive. Every $\xi_{n}$ is $R''$-bounded and $\lVert\Sigma(\nu_{n})\rVert_{\nu_{n}}\le R''$; with Steps 2 and 4 this says that $(\xi_{n})_{n}$ converges to $\xi$ along $(\kappa_{n})_{n}$ with score bounded by $R''$, and likewise $(\eta_{n})_{n}$ converges to $\eta$ along $(\kappa'_{n})_{n}$ with score bounded by $R''$. $F$ is shift-semicontinuous at $(\delta,R'')$ by The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §semicontinuity. Given $\varepsilon>0$ there is $N$ with $\epsilon_{n}\le\varepsilon$ for $n\ge N$, so, by the last display of Step 4, $F^{-}_{\delta}(\xi_{n})\le H_{1}+\varepsilon$ and $(-H_{2})-\varepsilon\le F^{+}_{\delta}(\eta_{n})$ for $n\ge N$. The first implication of The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §level with $c=H_{1}$, and the second with $c=-H_{2}$, give

No regularity of $h_{1}$, $h_{2}$ enters: only the constant levels $H_{1}$ and $-H_{2}$ pass to the limit.

Step 6 (Properness and the structure condition). The measures $\rho^{*},\sigma^{*}$ lie in $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}$, which has the map property, so both ordered pairs $(\rho^{*},\sigma^{*})$ and $(\sigma^{*},\rho^{*})$ are uniquely mapped (The Map Property of a Set of Probability Measures §map-property); $\delta(|\mathcal{E}(\rho^{*})|+|\mathcal{E}(\sigma^{*})|)\le2B\le R$; $-R\le t_{*}\le R$ since $|t_{*}|\le B\le R$; and $(\mathbb{X},\mathbb{Y})$ is admitted at $\alpha$. So The Second-Order Structure Condition at Uniquely Mapped Pairs on the Wasserstein Space §pair, for the pair $(\omega_{1},\omega_{2})$ at $R$ with the value slot $t_{*}$, gives

By The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts §shifted, $F^{-}_{\delta}(\rho^{*},r,V_{*},\mathbb{X})=F(\rho^{*},r+\delta\mathcal{E}(\rho^{*}),V_{*}+\delta\Sigma(\rho^{*}),\mathbb{X}+\delta H_{\mathcal{E}}(\rho^{*}))$ for every $r\in\mathbb{R}$, the last three arguments not depending on $r$. We have $t_{*}+\delta\mathcal{E}(\rho^{*})\le s_{*}+\delta\mathcal{E}(\rho^{*})$, and both have absolute value at most $B+B\le R$ (claim 5 of Properties of the Absolute Value in an Ordered Field), so the properness constant $\lambda$ at $R$ (Locally Strictly Proper Second-Order Equation Operator on the Wasserstein Space §constant, with $Q=\mathcal{D}_{\Sigma}$) gives

Adding the two displays and using $F^{-}_{\delta}(\xi)-F^{+}_{\delta}(\eta)\le H_{1}+H_{2}$ from Step 5,

Finally $M\le s_{*}-t_{*}$ and $0<\lambda$ give $\lambda M\le\lambda(s_{*}-t_{*})$ by claim 5 of Elementary Arithmetic in an Ordered Field. With $\Psi(\rho^{*},\sigma^{*})=M$ and $(\rho^{*},\sigma^{*})\in\mathcal{D}_{\Sigma}\times\mathcal{D}_{\Sigma}$, this is the claim.