Proof of Stability of Comparison on the Wasserstein Space under Locally Vanishing Cost Defects

Each result cited is universally quantified over the data in its own statement, and is applied to the data named here. Throughout, $|s|$ is the absolute value of $s\in\mathbb{R}$, $\tfrac{\alpha}{2}$ is the product of $\alpha$ with the multiplicative inverse of $2$ (claim 8 of Elementary Order Arithmetic in an Ordered Field), $\alpha^{-1}$ is the multiplicative inverse of a positive $\alpha$, $W=W_{2}$, $I=\{\delta\in\mathbb{R}:0<\delta<1\}$, and $(u_{n})^{-}_{\delta}$, $(v_{n})^{+}_{\delta}$ are the $\delta$-envelopes of $u_{n}$ and $v_{n}$. The rules for adding inequalities and multiplying them by positive reals are claims 1, 3 and 10 of Elementary Order Arithmetic in an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field; the least of two reals is provided by claim 9 of Elementary Order Arithmetic in an Ordered Field.

Order of choice. Let $R$ and $\theta$ be positive. The proof argues by contradiction; the objects are chosen in the following order, each depending only on the data of the statement, on $R$, $\theta$ and on those chosen before it: the index set $J$ and $e_{0}$ (Steps 0 and 1); $\delta_{*}$ (Step 1); $B$, $R_{\mathrm{s}}$, $\lambda$, $(\omega_{1},\omega_{2})$, $\kappa$, $\tau_{1}$, $\beta_{0}$, $\eta_{1}$ (Step 3); $\alpha_{1}$, $\alpha$, $\tau_{2}$ (Step 4); $\eta_{2}$, $\delta_{1}$, $\delta$ (Step 5); $m$, $R_{0}$, $C$, $n_{0}$, $N_{1}$, $N_{2}$ and finally the index $n$ (Step 6). In particular $\alpha$, $\delta$, and with them the set $K$ of Step 6, are fixed before the index $n$ is chosen.

Step 0 (The contradiction hypothesis). Suppose that no $n_{1}$ as in the claim exists. Then, the order of $\mathbb{R}$ being total, for every $k\in\mathbb{N}$ there are $n\in\mathbb{N}$ with $k\le n$ and $\mu\in\mathcal{D}$ with $|\mathcal{E}(\mu)|\le R$ and $\theta<u_{n}(\mu)-v_{n}(\mu)$. Let $J$ be the set of those $n\in\mathbb{N}$ for which some $\mu\in\mathcal{D}$ satisfies $|\mathcal{E}(\mu)|\le R$ and $\theta<u_{n}(\mu)-v_{n}(\mu)$. Then

Step 1 (The corrected maxima). By Basic Properties of a Wasserstein-Coercive Penalty Pair §bounded-below fix $e_{0}\in\mathbb{R}$ with $e_{0}\le\mathcal{E}(\sigma)$ for every $\sigma\in\mathcal{D}$. For $n\in\mathbb{N}$ and positive $\delta,\alpha$ let $\Psi^{n}_{\delta,\alpha}$ and $M_{n}(\delta,\alpha)$ be the function $\Psi_{\delta,\alpha}$ and the number $M(\delta,\alpha)$ of Existence, Penalty Bounds and Monotonicity of the Maximum of the Wasserstein-Doubled Difference of Bounded Functions, read with the present $e_{0}$, with $u_{n}$, $v_{n}$ in place of $u$, $v$, and with $b$, $b'$; by its clause Existence, Penalty Bounds and Monotonicity of the Maximum of the Wasserstein-Doubled Difference of Bounded Functions §maximiser, $M_{n}(\delta,\alpha)$ is a real number at most $b-b'-2\delta e_{0}$ and $\Psi^{n}_{\delta,\alpha}$ has a maximising pair. For $\alpha>0$ and $\delta\in I$ put

Halving the weight. Let $\alpha>0$, $\delta,\delta'\in I$ with $\delta'<\delta$, and let $(\hat{\mu},\hat{\nu})$ be a maximising pair of $\Psi^{n}_{\delta,\alpha}$. By Existence, Penalty Bounds and Monotonicity of the Maximum of the Wasserstein-Doubled Difference of Bounded Functions §weight, adding $2(\delta'-\delta)e_{0}$,

the last because $\mathcal{E}(\hat{\mu})-e_{0}$ and $\mathcal{E}(\hat{\nu})-e_{0}$ are nonnegative (claim 3 of Elementary Arithmetic in an Ordered Field). Since a maximising pair exists, $G_{n}(\alpha,\cdot)$ is nonincreasing on $I$.

Halving the strength. Let $0<\alpha'<\alpha$, $\delta\in I$, and let $(\hat{\mu},\hat{\nu})$ be a maximising pair of $\Psi^{n}_{\delta,\alpha}$. By Existence, Penalty Bounds and Monotonicity of the Maximum of the Wasserstein-Doubled Difference of Bounded Functions §strength,

so $G_{n}(\cdot,\delta)$ is nonincreasing on the positive reals.

Lower bound on $J$. Let $\delta_{*}$ be the least of $1$ and $\theta(4R)^{-1}$, positive by claims 5 and 7 of Elementary Order Arithmetic in an Ordered Field, and $I_{*}=\{\delta\in\mathbb{R}:0<\delta<\delta_{*}\}\subseteq I$. Let $n\in J$, $\alpha>0$ and $\delta\in I_{*}$, and let $\mu\in\mathcal{D}$ with $|\mathcal{E}(\mu)|\le R$ and $\theta<u_{n}(\mu)-v_{n}(\mu)$. By Basic Properties of the Delta-Envelopes on the Wasserstein Space §semicontinuity, $u_{n}(\mu)-\delta\mathcal{E}(\mu)\le(u_{n})^{-}_{\delta}(\mu)$ and $(v_{n})^{+}_{\delta}(\mu)\le v_{n}(\mu)+\delta\mathcal{E}(\mu)$, and $W(\mu,\mu)=0$ (The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §separation); as $2\delta\mathcal{E}(\mu)\le2\delta|\mathcal{E}(\mu)|\le2\delta R<2\theta(4R)^{-1}R=\tfrac{\theta}{2}$ (claim 3 of Properties of the Absolute Value in an Ordered Field),

and, since $2\delta e_{0}\ge-2\delta|e_{0}|\ge-2|e_{0}|$ for $\delta\in I$,

Step 2 (Limits superior over the indices). For $\alpha>0$, $\delta\in I_{*}$ and $n\in\mathbb{N}$ put $a_{n}(\alpha,\delta)=G_{n}(\alpha,\delta)$ if $n\in J$ and $a_{n}(\alpha,\delta)=-2|e_{0}|$ if $n\notin J$, and let $M_{0}=|b|+|b'|+2|e_{0}|+1$, positive. For $n\in J$, (1a) and (1b) give $-(|b|+|b'|+2|e_{0}|)\le-2|e_{0}|<a_{n}(\alpha,\delta)\le b-b'\le|b|+|b'|+2|e_{0}|$, so $|a_{n}(\alpha,\delta)|<M_{0}$ by claim 6 of Properties of the Absolute Value in an Ordered Field; for $n\notin J$, $|a_{n}(\alpha,\delta)|=2|e_{0}|<M_{0}$. So $(a_{n}(\alpha,\delta))_{n\in\mathbb{N}}$ is a bounded sequence of real numbers, and we let

be its limit superior, the infimum over $k\in\mathbb{N}$ of $\sup A_{k}$, where $A_{k}=\{a_{m}(\alpha,\delta):m\in\mathbb{N},\ k\le m\}$. It has the following properties.

(2a) $\hat{G}(\alpha,\delta)\le M_{0}$, by clause 2 of Basic Properties of the Limit Inferior and Limit Superior of a Bounded Real Sequence.

(2b) $\tfrac{\theta}{2}-2|e_{0}|\le\hat{G}(\alpha,\delta)$. Indeed, let $k\in\mathbb{N}$; by (0) there is $n\in J$ with $k\le n$, so $a_{n}(\alpha,\delta)\in A_{k}$ and, $\sup A_{k}$ being an upper bound of $A_{k}$, (1b) gives $\tfrac{\theta}{2}-2|e_{0}|<a_{n}(\alpha,\delta)\le\sup A_{k}$. So $\tfrac{\theta}{2}-2|e_{0}|$ is a lower bound of $\{\sup A_{k}:k\in\mathbb{N}\}$, and is at most its infimum, the greatest lower bound (Lower Bound and Greatest Lower Bound).

(2c) Monotonicity. If $0<\alpha'<\alpha$ and $\delta\in I_{*}$, then $a_{n}(\alpha,\delta)\le a_{n}(\alpha',\delta)$ for every $n$, by (1d) when $n\in J$ and trivially otherwise; so $\hat{G}(\alpha,\delta)\le\hat{G}(\alpha',\delta)$ by clause 2 of Basic Properties of the Limit Inferior and Limit Superior of a Bounded Real Sequence. Likewise, if $\alpha>0$ and $\delta,\delta'\in I_{*}$ with $\delta'<\delta$, then (1c) gives $\hat{G}(\alpha,\delta)\le\hat{G}(\alpha,\delta')$.

(2d) Eventually below. For every positive $\varepsilon$ there is $N\in\mathbb{N}$ with $a_{m}(\alpha,\delta)<\hat{G}(\alpha,\delta)+\varepsilon$ for every $m\ge N$, by clause 3 of Basic Properties of the Limit Inferior and Limit Superior of a Bounded Real Sequence.

(2e) Frequently above. For every positive $\varepsilon$ and every $k\in\mathbb{N}$ there is $m\in\mathbb{N}$ with $k\le m$ and $\hat{G}(\alpha,\delta)-\varepsilon<a_{m}(\alpha,\delta)$. Indeed $\hat{G}(\alpha,\delta)\le\sup A_{k}$, the infimum being a lower bound, so $\hat{G}(\alpha,\delta)-\varepsilon<\sup A_{k}$, and claim 1 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$, applied to the nonempty set $A_{k}$, bounded above by $M_{0}$, provides an element $a_{m}(\alpha,\delta)$ of $A_{k}$ exceeding $\hat{G}(\alpha,\delta)-\varepsilon$.

Step 3 (Constants of the reference operator). Put $B=|b|+|b'|+|e_{0}|+1$ and $R_{\mathrm{s}}=2B$; then $|b|+|b'|+|e_{0}|\le B$ and $0<2B\le R_{\mathrm{s}}$. Since $F$ is locally strictly proper, fix a properness constant $\lambda>0$ for $F$ at $R_{\mathrm{s}}$; since $F$ satisfies the second-order structure condition, fix a second-order structure pair $(\omega_{1},\omega_{2})$ for $F$ at $R_{\mathrm{s}}$. Put $\kappa=\lambda\theta/16$, positive. By clause 2 of Modulus of Continuity fix a positive $\tau_{1}$ with $\omega_{1}(t)\le\kappa$ whenever $0\le t\le\tau_{1}$, and put $\beta_{0}=1+2\tau_{1}^{-1}$ and $\eta_{1}=\tau_{1}/32$.

Step 4 (Choice of the doubling strength). For $\alpha\ge\beta_{0}$ the set $\{\hat{G}(\alpha,\delta):\delta\in I_{*}\}$ is nonempty and bounded above by $M_{0}$ by (2a); let $N(\alpha)$ be its least upper bound (The Real Numbers: Standing Notation and Background §bounds). By (2b), $\tfrac{\theta}{2}-2|e_{0}|\le N(\alpha)$; by (2c), $N$ is nonincreasing on $\{\alpha:\alpha\ge\beta_{0}\}$, since $\hat{G}(\alpha,\delta)\le\hat{G}(\alpha',\delta)\le N(\alpha')$ for $\beta_{0}\le\alpha'<\alpha$ and every $\delta\in I_{*}$. The set $\{N(\alpha):\alpha\ge\beta_{0}\}$ is nonempty and bounded below by $\tfrac{\theta}{2}-2|e_{0}|$; let $L$ be its greatest lower bound (The Real Numbers: Standing Notation and Background §bounds). By claim 4 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ fix $\alpha_{1}\ge\beta_{0}$ with $N(\alpha_{1})<L+\eta_{1}$, and put $\alpha=2\alpha_{1}$, so that $\tfrac{\alpha}{2}=\alpha_{1}$. Then $\alpha\ge\alpha_{1}\ge\beta_{0}$, so $L\le N(\alpha)$ and

Moreover $1<\beta_{0}\le\alpha$, and from $2\tau_{1}^{-1}<\beta_{0}\le\alpha$, multiplying by the positive $\alpha^{-1}\tau_{1}/2$, $\alpha^{-1}<\tau_{1}/2$. By The Second-Order Structure Condition at Uniquely Mapped Pairs on the Wasserstein Space §pair the function with value $\omega_{2}(t,\alpha)$ at $t\ge0$ is a modulus of continuity; by clause 2 of Modulus of Continuity fix a positive $\tau_{2}$ with $\omega_{2}(t,\alpha)\le\kappa$ whenever $0\le t\le\tau_{2}$.

Step 5 (Choice of the penalty weight). Let $\eta_{2}$ be the least of $\eta_{1}$, $\tau_{2}/12$ and $\theta/4$, positive. By claim 3 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ fix $\delta_{1}\in I_{*}$ with $N(\alpha)-\eta_{2}<\hat{G}(\alpha,\delta_{1})$, and let $\delta$ be half the least of $\delta_{1}$ and $\tau_{2}\bigl(4|e_{0}|+2\bigr)^{-1}$ (claims 7, 8 and 9 of Elementary Order Arithmetic in an Ordered Field). Then $0<\tfrac{\delta}{2}<\delta<\delta_{1}<\delta_{*}\le1$, so $\delta,\tfrac{\delta}{2}\in I_{*}$, and $\delta<\tau_{2}(4|e_{0}|+2)^{-1}$. By (2c),

and $\hat{G}(\alpha,\tfrac{\delta}{2})\le N(\alpha)$, $\hat{G}(\tfrac{\alpha}{2},\delta)\le N(\tfrac{\alpha}{2})$, as $\tfrac{\delta}{2}\in I_{*}$ and $\tfrac{\alpha}{2}=\alpha_{1}\ge\beta_{0}$.

Step 6 (The set $K$ and the index $n$). With $\delta$, $\alpha$ and $B$ now fixed, let $m\in\mathbb{R}$ be nonnegative with $M_{2}(\mu)\le m$ for every $\mu\in\mathcal{D}$ with $\mathcal{E}(\mu)\le\delta^{-1}B$, by Basic Properties of a Wasserstein-Coercive Penalty Pair §moment with $c=\delta^{-1}B$ (replacing the number provided there by its absolute value); put

positive; and, by The Shift-Coercivity Condition for an Equation Operator on the Wasserstein Space §coercivity, let $C$ be a score bound for $F$ at $(\delta,R_{0}+2H)$, a nonnegative real. Let $K=\{\nu\in\mathcal{D}_{\Sigma}:|\mathcal{E}(\nu)|\le R_{0},\ \lVert\Sigma(\nu)\rVert_{\nu}\le C\}$. By the hypothesis on the defects, with the positive numbers $R_{0}$, $C+1$ and $\kappa$, fix $n_{0}\in\mathbb{N}$ such that $h_{n}(\nu)+h'_{n}(\nu)\le\kappa$ for every $n\ge n_{0}$ and every $\nu\in\mathcal{D}_{\Sigma}$ with $|\mathcal{E}(\nu)|\le R_{0}$ and $\lVert\Sigma(\nu)\rVert_{\nu}\le C+1$. By (2d), at the points $(\tfrac{\alpha}{2},\delta)$ and $(\alpha,\tfrac{\delta}{2})$ with $\varepsilon=\eta_{2}$, fix $N_{1},N_{2}\in\mathbb{N}$ with

Let $k$ be the largest of $n_{0}$, $N_{1}$ and $N_{2}$, and by (2e) at $(\alpha,\delta)$ with $\varepsilon=\eta_{2}$ fix $n\in\mathbb{N}$ with $k\le n$ and $\hat{G}(\alpha,\delta)-\eta_{2}<a_{n}(\alpha,\delta)$. As $\eta_{2}\le\theta/4<\theta/2$, (2b) gives $-2|e_{0}|<\tfrac{\theta}{2}-2|e_{0}|-\eta_{2}\le\hat{G}(\alpha,\delta)-\eta_{2}<a_{n}(\alpha,\delta)$; were $n\notin J$, we would have $a_{n}(\alpha,\delta)=-2|e_{0}|$, which is impossible. So $n\in J$, $a_{n}=G_{n}$ at all three points, and

Moreover, for $\nu\in K$ we have $\lVert\Sigma(\nu)\rVert_{\nu}\le C\le C+1$ and $n\ge n_{0}$, so $h_{n}(\nu)+h'_{n}(\nu)\le\kappa$, and as both terms are nonnegative, $h_{n}(\nu)\le\kappa$ and $h'_{n}(\nu)\le\kappa$.

Step 7 (The structure estimate with defects, and the contradiction). As $n\in J$ and $\delta\in I_{*}$, (1b) gives $\tfrac{\theta}{2}<M_{n}(\delta,\alpha)$, in particular $0\le M_{n}(\delta,\alpha)$. The hypotheses of The Structure Estimate at a Maximiser for Operators within a Cost Defect of a Reference Operator hold for the present pair and reference operator $F$ (which satisfies the shift-coercivity and shift-semicontinuity conditions), with $F_{n}$, $F'_{n}$ as the operators $F_{1}$, $F_{2}$ there, $H$, $h_{n}$, $h'_{n}$ as $h_{1}$, $h_{2}$, $u_{n}$, $v_{n}$ as $u$, $v$, and with $b$, $b'$, $e_{0}$, $\delta$, $\alpha$ (where $0<\delta<1<\alpha$), $B$, $R_{\mathrm{s}}$ in place of the number named $R$ there, $\lambda$, $(\omega_{1},\omega_{2})$, $m$ and $C$ as above; the number $R_{0}$ and the set $K$ of that lemma are then those of Step 6, and by Step 6 we may take $H_{1}=H_{2}=\kappa$. Its clause The Structure Estimate at a Maximiser for Operators within a Cost Defect of a Reference Operator §estimate provides a maximising pair $(\rho^{*},\sigma^{*})\in\mathcal{D}_{\Sigma}\times\mathcal{D}_{\Sigma}$ of $\Psi^{n}_{\delta,\alpha}$ with

The first modulus. By (1d) with $\alpha'=\tfrac{\alpha}{2}$, whose coefficient is $\tfrac{\alpha-\alpha/2}{2}=\tfrac{\alpha}{4}$, then (6a), $\hat{G}(\tfrac{\alpha}{2},\delta)\le N(\tfrac{\alpha}{2})$, (5a) and (4a),

so $\alpha W(\rho^{*},\sigma^{*})^{2}<\tau_{1}/2$, and with $\alpha^{-1}<\tau_{1}/2$ the first argument lies in $[0,\tau_{1}]$; hence the first modulus is at most $\kappa$.

The second modulus. By (1c) with $\delta'=\tfrac{\delta}{2}\in I$, then (6a), $\hat{G}(\alpha,\tfrac{\delta}{2})\le N(\alpha)$ and (5a),

As $\mathcal{E}(\rho^{*})-e_{0}\ge0$, the triangle inequality (claims 1 and 5 of Properties of the Absolute Value in an Ordered Field) applied to $\mathcal{E}(\rho^{*})=(\mathcal{E}(\rho^{*})-e_{0})+e_{0}$ gives $|\mathcal{E}(\rho^{*})|\le\mathcal{E}(\rho^{*})-e_{0}+|e_{0}|$, and likewise for $\sigma^{*}$. Multiplying by the positive $\delta$ and using $\delta<\tau_{2}(4|e_{0}|+2)^{-1}$,

and the argument is positive; hence the second modulus is at most $\kappa$.

So $\lambda M_{n}(\delta,\alpha)\le4\kappa=\lambda\theta/4$. But $\tfrac{\theta}{2}<M_{n}(\delta,\alpha)$ and $0<\lambda$ give $\lambda\theta/2<\lambda M_{n}(\delta,\alpha)$ (claim 10 of Elementary Order Arithmetic in an Ordered Field), so $\lambda\theta/2<\lambda\theta/4$, that is $\lambda\theta/4<0$ (claim 1 of that lemma), contradicting $0<\lambda\theta/4$ (claims 5 and 8 of that lemma). Hence the supposition of Step 0 is false: there is $n_{1}\in\mathbb{N}$ with $u_{n}(\mu)-v_{n}(\mu)\le\theta$ for every $n\ge n_{1}$ and every $\mu\in\mathcal{D}$ with $|\mathcal{E}(\mu)|\le R$. As $R$ and $\theta$ were arbitrary positive reals, this is the claim.