Proof of The Structure Estimate at a Maximiser of the Wasserstein-Doubled Difference

Each result cited is universally quantified over the data in its own statement. 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 1 (A maximising pair and admitted matrices). By Existence, Penalty Bounds and the Least Penalty at a Maximiser of the Wasserstein-Doubled Difference §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, $\Psi(\rho^{*},\sigma^{*})=M$, so by hypothesis $\delta|\mathcal{E}(\rho^{*})|\le B$ and $\delta|\mathcal{E}(\sigma^{*})|\le B$; 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 $u^{-}_{\delta}=u-\delta\mathcal{E}\le b-\delta e_{0}$ and $v^{+}_{\delta}=v+\delta\mathcal{E}\ge b'+\delta e_{0}$ on $\mathcal{D}$ (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).

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, Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution 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

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 $-B-2<u^{-}_{\delta}(\nu_{n})=u(\nu_{n})-\delta\mathcal{E}(\nu_{n})\le b-\delta\mathcal{E}(\nu_{n})$ 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 $v$, 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 $-\epsilon_{n}\le F^{+}_{\delta}(\nu'_{n},t_{n},q'_{n},Y_{n})$, $\sqrt{I(\kappa'_{n})}<2\epsilon_{n}$, 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 $b'+\delta\mathcal{E}(\nu'_{n})\le v^{+}_{\delta}(\nu'_{n})<B+2$ and $e_{0}\le\mathcal{E}(\nu'_{n})$, $|\mathcal{E}(\nu'_{n})|<\delta^{-1}(|b'|+B+2)+|e_{0}|+1$.

Step 4 (Shift-coercivity and closed score). Let

a sum of nonnegative reals and $10$, which exceeds every bound listed at the end of Steps 2 and 3 and exceeds $2$. 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 $R'$-bounded, and $F^{-}_{\delta}(\xi_{n})-F^{+}_{\delta}(\eta_{n})\le2\epsilon_{n}\le2<R'$, so $\xi_{n}\in S^{-}_{\delta,R'}$ and $\eta_{n}\in S^{+}_{\delta,R'}$ (Test Data for an Intrinsic Second-Order Equation Operator on the Wasserstein Space and the Admissible Sets §admissible), each witnessing the other. By the shift-coercivity condition there is a score bound $C\ge0$ for $F$ at $(\delta,R')$, so $\lVert\Sigma(\nu_{n})\rVert_{\nu_{n}}\le C$ 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 $F$.

Step 5 (Shift-semicontinuity). Put $R''=R'+C$, 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 $(\delta,R'')$ (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 $F^{-}_{\delta}(\xi_{n})\le0+\varepsilon$ and $0-\varepsilon\le F^{+}_{\delta}(\eta_{n})$ for $n\ge N$; the two implications of The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §level with $c=0$ give $F^{-}_{\delta}(\xi)\le0\le F^{+}_{\delta}(\eta)$.

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, $F^{-}_{\delta}(\rho^{*},r,V_{*},\mathbb{X})=F(\rho^{*},r+\delta\mathcal{E}(\rho^{*}),V_{*}+\delta\Sigma(\rho^{*}),\mathbb{X})$ for every $r\in\mathbb{R}$. 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 $F^{-}_{\delta}(\xi)-F^{+}_{\delta}(\eta)\le0$ 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.