Proof of A Comparison Principle for Viscosity Solutions on the Lift of the Wasserstein Space

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 quotient of $\alpha$ by $2=1+1$, $\alpha^{-1}$ is the multiplicative inverse of a positive $\alpha$, $2^{k}$ and $(\tfrac{1}{2})^{k}$ are natural powers, and $W_{2}(\mu,\nu)^{2}=W_{2}(\mu,\nu)W_{2}(\mu,\nu)$.

Suppose the conclusion fails. Since the order of $\mathbb{R}$ is total (an axiom of Ordered Field), there is then $\mu_{0}\in\mathcal{D}$ with $v(\mu_{0})<u(\mu_{0})$; put $\theta_{0}=u(\mu_{0})-v(\mu_{0})$, a positive real number by claim 1 of Elementary Order Arithmetic in an Ordered Field applied to $v(\mu_{0})<u(\mu_{0})$ with the summand $-v(\mu_{0})$.

Step 1. The least penalty and the penalised suprema. Claim 1 of The Doubled Difference on the Lift of a Wasserstein-Coercive Penalty Pair: Bounds, Closed Superlevel Sets and the Least Penalty, read with $\delta=\tfrac{1}{2}$, with the present $u$, $v$, $b$, $b'$ and with any lower bound for $\mathcal{E}$ on $\mathcal{D}$ as provided by Basic Properties of a Wasserstein-Coercive Penalty Pair (ii), gives $\mu_{\min}\in\mathcal{D}$ with $\mathcal{E}(\mu_{\min})\le\mathcal{E}(\sigma)$ for every $\sigma\in\mathcal{D}$. Put $e_{0}=\mathcal{E}(\mu_{\min})$; this is a real number satisfying $e_{0}\le\mathcal{E}(\sigma)$ for every $\sigma\in\mathcal{D}$, and it is with this choice that The Structure Estimate at a Maximiser of the Wasserstein-Doubled Difference on the Lift and Existence, Penalty Bounds and Optimal Realisation at a Maximiser of the Wasserstein-Doubled Difference on the Lift are read below.

Let $Z_{1}=\mathcal{D}\times\mathcal{D}$, a nonempty set because $\mathcal{D}$ contains the nonempty $\mathcal{D}_{\Sigma}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (i) and Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (ii), and let $D:Z_{1}\to\mathbb{R}$ have the value $D(\mu,\nu)=\mathcal{E}(\mu)+\mathcal{E}(\nu)-2e_{0}$; then $0\le D(\mu,\nu)$ for every $(\mu,\nu)\in Z_{1}$, and $D(\mu_{\min},\mu_{\min})=0$. For positive $\alpha\in\mathbb{R}$ let $\psi_{\alpha}:Z_{1}\to\mathbb{R}$ have the value $u(\mu)-v(\nu)-\tfrac{\alpha}{2}W_{2}(\mu,\nu)^{2}$; it is bounded above by $b-b'$.

For positive $\delta$ and $\alpha$ let $\Psi_{\delta,\alpha}:Z_{1}\to\mathbb{R}$ and $M(\delta,\alpha)$ be as in The Structure Estimate at a Maximiser of the Wasserstein-Doubled Difference on the Lift. Since $u^{-}_{\delta}=u-\delta\mathcal{E}$ and $v^{+}_{\delta}=v+\delta\mathcal{E}$ on $\mathcal{D}$ by Basic Properties of a Wasserstein-Coercive Penalty Pair (iv),

$$\Psi_{\delta,\alpha}(\mu,\nu)=\psi_{\alpha}(\mu,\nu)-\delta\,D(\mu,\nu)-2\delta e_{0}\qquad\bigl((\mu,\nu)\in Z_{1}\bigr).$$

Let $g(\alpha,\delta)$ be the supremum of the values of $\psi_{\alpha}-\delta D$ on $Z_{1}$, a real number by claim 1 of Penalised Suprema: Monotonicity, Near-Maximisers, and Vanishing Penalty along a Doubling Sequence read with $Z=Z_{1}$, with $\psi=\psi_{\alpha}$, with this $D$ and with $z_{0}=(\mu_{\min},\mu_{\min})$. If $(\hat{\mu},\hat{\nu})\in Z_{1}$ satisfies $\Psi_{\delta,\alpha}(\hat{\mu},\hat{\nu})=M(\delta,\alpha)$, then adding the constant $2\delta e_{0}$ to the inequalities $\Psi_{\delta,\alpha}(\mu,\nu)\le\Psi_{\delta,\alpha}(\hat{\mu},\hat{\nu})$ shows that $(\psi_{\alpha}-\delta D)(\hat{\mu},\hat{\nu})$ is an upper bound for the values of $\psi_{\alpha}-\delta D$ and is one of them, so that

$$g(\alpha,\delta)=(\psi_{\alpha}-\delta D)(\hat{\mu},\hat{\nu}),\qquad M(\delta,\alpha)=g(\alpha,\delta)-2\delta e_{0}.$$

Let $I=\{\delta\in\mathbb{R}:0<\delta<1\}$, let $Z_{2}=Z_{1}\times I$, and let $\psi',D':Z_{2}\to\mathbb{R}$ have the values $\psi'\bigl((\mu,\nu),\delta\bigr)=u(\mu)-v(\nu)-\delta D(\mu,\nu)$ and $D'\bigl((\mu,\nu),\delta\bigr)=\tfrac{1}{2}W_{2}(\mu,\nu)^{2}$. Then $\psi'$ is bounded above by $b-b'$, the values of $D'$ are nonnegative, $D'(z'_{0})=0$ for $z'_{0}=\bigl((\mu_{\min},\mu_{\min}),\tfrac{1}{2}\bigr)\in Z_{2}$, and

$$(\psi'-\alpha D')\bigl((\mu,\nu),\delta\bigr)=(\psi_{\alpha}-\delta D)(\mu,\nu)$$

for positive $\alpha$. Let $N(\alpha)$ be the supremum of the values of $\psi'-\alpha D'$ on $Z_{2}$, a real number by claim 1 of Penalised Suprema: Monotonicity, Near-Maximisers, and Vanishing Penalty along a Doubling Sequence read with $Z=Z_{2}$, $\psi=\psi'$, $D=D'$ and $z_{0}=z'_{0}$. Two consequences will be used. First, $g(\alpha,\delta)\le N(\alpha)$ for every $\delta\in I$ and every positive $\alpha$: by the last display the values of $\psi_{\alpha}-\delta D$ on $Z_{1}$ are among the values of $\psi'-\alpha D'$ on $Z_{2}$, for which $N(\alpha)$ is an upper bound. Secondly, by claim 2 of Penalised Suprema: Monotonicity, Near-Maximisers, and Vanishing Penalty along a Doubling Sequence, read once with $Z_{1}$ and once with $Z_{2}$, the function $g(\alpha,\cdot)$ is nonincreasing on the positive reals for each fixed positive $\alpha$, and $N$ is nonincreasing on the positive reals.

Step 2. The level and the constants. Put

$$B=2\bigl(|b|+|b'|+|e_{0}|+|\mathcal{E}(\mu_{0})|+\theta_{0}\bigr),\qquad R=2B,$$

positive because $\theta_{0}$ is. Let $\delta\in I$ and let $\alpha\in\mathbb{R}$ satisfy $1<\alpha$, and let $(\hat{\mu},\hat{\nu})\in Z_{1}$ satisfy $\Psi_{\delta,\alpha}(\hat{\mu},\hat{\nu})=M(\delta,\alpha)$, as Existence, Penalty Bounds and Optimal Realisation at a Maximiser of the Wasserstein-Doubled Difference on the Lift (i) provides. Since $W_{2}(\mu_{0},\mu_{0})=0$,

$$\theta_{0}-2\delta\mathcal{E}(\mu_{0})=\Psi_{\delta,\alpha}(\mu_{0},\mu_{0})\ \le\ M(\delta,\alpha)=\Psi_{\delta,\alpha}(\hat{\mu},\hat{\nu}),$$

so claim 2 of Existence, Penalty Bounds and Optimal Realisation at a Maximiser of the Wasserstein-Doubled Difference on the Lift, read with this $\mu_{0}$, this $e_{0}$ and with the nonnegative number there taken to be $0$, gives $\mathcal{E}(\hat{\mu})\le c_{0}$ and $\mathcal{E}(\hat{\nu})\le c_{0}$, where $c_{0}=\delta^{-1}(b-b'-\theta_{0}+2\delta\mathcal{E}(\mu_{0}))-e_{0}$. Multiplying by the positive $\delta$ (claim 5 of Elementary Arithmetic in an Ordered Field),

$$\delta\,\mathcal{E}(\hat{\mu})\ \le\ b-b'-\theta_{0}+2\delta\mathcal{E}(\mu_{0})-\delta e_{0}\ \le\ |b|+|b'|+\theta_{0}+2|\mathcal{E}(\mu_{0})|+|e_{0}|\ \le\ B,$$

the middle step by claim 3 of Properties of the Absolute Value in an Ordered Field, which bounds each of $b$, $-b'$, $-\theta_{0}$, $2\delta\mathcal{E}(\mu_{0})$ and $-\delta e_{0}$ by the corresponding absolute value, combined with $0<\delta<1$ and claim 5 of Elementary Arithmetic in an Ordered Field. Since $e_{0}\le\mathcal{E}(\hat{\mu})$ we also have $-|e_{0}|\le\delta e_{0}\le\delta\mathcal{E}(\hat{\mu})$ and $|e_{0}|\le B$, so $-B\le\delta\mathcal{E}(\hat{\mu})\le B$ and hence $|\delta\mathcal{E}(\hat{\mu})|\le B$ by claim 6 of Properties of the Absolute Value in an Ordered Field; as $\delta$ is positive, $\delta|\mathcal{E}(\hat{\mu})|=|\delta\mathcal{E}(\hat{\mu})|\le B$ by claim 4 of that lemma; the same argument gives $\delta|\mathcal{E}(\hat{\nu})|\le B$. Also $|b|+|b'|+|e_{0}|\le B$ and $0<2B\le R$. These bounds hold for every $\delta\in I$, every $\alpha>1$ and every maximiser.

Since $F$ is locally strictly proper, fix a properness constant $\lambda$ for $F$ at $R$; since $F$ satisfies the second-order structure condition, fix a second-order structure pair $(\omega_{1},\omega_{2})$ for $F$ at $R$. Put $\varsigma=\theta_{0}/4$ and $\kappa=\lambda\theta_{0}/8$, both positive.

Step 3. The choice of the doubling strength. As $\omega_{1}$ is a modulus of continuity, its clause 2 provides a positive $\tau_{1}$ with $\omega_{1}(t)\le\kappa$ whenever $0\le t\le\tau_{1}$. Put $\eta_{1}=\tau_{1}/16$ and $\beta_{0}=1+2\tau_{1}^{-1}$, both positive, and $\beta_{k}=2^{k}\beta_{0}$ for $k\in\mathbb{N}$.

First, $1\le2^{k}$ for every $k\in\mathbb{N}$. The set $S=\{k\in\mathbb{N}:1\le2^{k}\}$ contains $1$, since $2^{1}=2$ and $1\le2$, the latter because $0<1$ by claim 6 of Elementary Order Arithmetic in an Ordered Field and claim 1 of that lemma then gives $1<1+1=2$; and if $k\in S$ then $2^{k+1}=2\cdot2^{k}$ and $1\le2=2\cdot1\le2\cdot2^{k}$ by claim 5 of Elementary Arithmetic in an Ordered Field with the nonnegative multiplier $2$, so $k+1\in S$. By Principle of Induction for the Natural Numbers, $S=\mathbb{N}$. Multiplying $1\le2^{k}$ by the nonnegative $\beta_{0}$ gives $\beta_{0}\le\beta_{k}$ for every $k\in\mathbb{N}$.

By claim 4 of Penalised Suprema: Monotonicity, Near-Maximisers, and Vanishing Penalty along a Doubling Sequence, read with $Z_{2}$, $\psi'$, $D'$, $z'_{0}$ and this $\beta_{0}$, the sequence whose $k$-th term is $N(\beta_{k})-N(\beta_{k+1})$ converges to $0$. Fix $k\in\mathbb{N}$ with $N(\beta_{k})-N(\beta_{k+1})\le\eta_{1}$ and put $\alpha=\beta_{k+1}$. Then $\alpha=2^{k+1}\beta_{0}=2\cdot2^{k}\beta_{0}=2\beta_{k}$, so $\beta_{k}=\alpha/2$; and $1<\beta_{0}\le\alpha$, so $1<\alpha$. Moreover $2\tau_{1}^{-1}\le\beta_{0}\le\alpha$, and multiplying first by the nonnegative $\alpha^{-1}$ and then by the nonnegative $\tau_{1}/2$ gives $\alpha^{-1}\le\tau_{1}/2$.

By clause 2 of The Second-Order Structure Condition at Optimally Coupled Pairs on the Lift of the Wasserstein Space the function on the nonnegative reals with value $\omega_{2}(t,\alpha)$ at $t$ is a modulus of continuity; fix a positive $\tau_{2}$ with $\omega_{2}(t,\alpha)\le\kappa$ whenever $0\le t\le\tau_{2}$.

Step 4. The choice of the penalty weight. Since $N(\alpha)$ is the supremum of the values of $\psi'-\alpha D'$ on $Z_{2}$, claim 3 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ provides $z=((\mu,\nu),\delta_{1})\in Z_{2}$ with $N(\alpha)-\eta_{1}<(\psi'-\alpha D')(z)$. By Step 1, $(\psi'-\alpha D')(z)=(\psi_{\alpha}-\delta_{1}D)(\mu,\nu)\le g(\alpha,\delta_{1})$, so $N(\alpha)-\eta_{1}\le g(\alpha,\delta_{1})$ with $\delta_{1}\in I$.

For $j\in\mathbb{N}$ put $\rho_{j}=(\tfrac{1}{2})^{j}\delta_{1}$; then $0<\rho_{j}\le\delta_{1}<1$, so $\rho_{j}\in I$, and $\rho_{j+1}=\rho_{j}/2$ because $(\tfrac{1}{2})^{j+1}=(\tfrac{1}{2})^{j}\cdot\tfrac{1}{2}$. By claim 4 of Series of Nonnegative Real Numbers, Comparison, and the Geometric Series, read with $r=\tfrac{1}{2}$, the sequence whose $j$-th term is $(\tfrac{1}{2})^{j}$ converges to $0$, so by claim 3 of Arithmetic of Limits of Real Sequences the sequence whose $j$-th term is $\rho_{j}$ converges to $0$.

Because $g(\alpha,\cdot)$ is nonincreasing and $\rho_{j+1}\le\rho_{j}$, the sequence whose $j$-th term is $g(\alpha,\rho_{j})$ is nondecreasing, and it is bounded above by $N(\alpha)$; by claim 1 of A Bounded Monotone Sequence of Real Numbers Converges it converges. The sequence whose $j$-th term is $g(\alpha,\rho_{j+1})$ is a subsequence of it, hence converges to the same limit by A Subsequence of a Convergent Sequence Has the Same Limit, so by claim 3 of Arithmetic of Limits of Real Sequences the sequence whose $j$-th term is $g(\alpha,\rho_{j+1})-g(\alpha,\rho_{j})$ converges to $0$.

For each $j\in\mathbb{N}$ let $(\hat{\mu}_{j},\hat{\nu}_{j})\in Z_{1}$ satisfy $\Psi_{\rho_{j},\alpha}(\hat{\mu}_{j},\hat{\nu}_{j})=M(\rho_{j},\alpha)$, as Existence, Penalty Bounds and Optimal Realisation at a Maximiser of the Wasserstein-Doubled Difference on the Lift (i) provides. By Step 1 it satisfies $g(\alpha,\rho_{j})=(\psi_{\alpha}-\rho_{j}D)(\hat{\mu}_{j},\hat{\nu}_{j})$, so claim 3 of Penalised Suprema: Monotonicity, Near-Maximisers, and Vanishing Penalty along a Doubling Sequence, read with $Z_{1}$, $\psi_{\alpha}$, $D$, with $\rho_{j}$ in the role of the positive number written $\beta$ there and with an arbitrary positive $\eta$, gives

$$\tfrac{\rho_{j}}{2}\,D(\hat{\mu}_{j},\hat{\nu}_{j})\ \le\ g(\alpha,\rho_{j+1})-g(\alpha,\rho_{j})+\eta$$

for every positive $\eta$, whence $\rho_{j}D(\hat{\mu}_{j},\hat{\nu}_{j})\le2\bigl(g(\alpha,\rho_{j+1})-g(\alpha,\rho_{j})\bigr)$ by Comparison of Real Numbers with Arbitrary Positive Slack (i). The right-hand side converges to $0$ and the left-hand side is nonnegative, so the sequence whose $j$-th term is $\rho_{j}D(\hat{\mu}_{j},\hat{\nu}_{j})$ converges to $0$.

Since $e_{0}\le\mathcal{E}(\hat{\mu}_{j})$ the number $\mathcal{E}(\hat{\mu}_{j})-e_{0}$ is nonnegative, so $|\mathcal{E}(\hat{\mu}_{j})-e_{0}|=\mathcal{E}(\hat{\mu}_{j})-e_{0}$ by claim 1 of Properties of the Absolute Value in an Ordered Field, and the triangle inequality, claim 5 of that lemma, applied to $\mathcal{E}(\hat{\mu}_{j})=(\mathcal{E}(\hat{\mu}_{j})-e_{0})+e_{0}$ gives $|\mathcal{E}(\hat{\mu}_{j})|\le\mathcal{E}(\hat{\mu}_{j})-e_{0}+|e_{0}|$; likewise for $\hat{\nu}_{j}$. Adding the two and multiplying by the positive $\rho_{j}$,

$$\rho_{j}\bigl(|\mathcal{E}(\hat{\mu}_{j})|+|\mathcal{E}(\hat{\nu}_{j})|+1\bigr)\ \le\ \rho_{j}D(\hat{\mu}_{j},\hat{\nu}_{j})+\rho_{j}\bigl(2|e_{0}|+1\bigr).$$

Both terms on the right converge to $0$, and so does the sequence whose $j$-th term is $2\rho_{j}|\mathcal{E}(\mu_{0})|$, by claim 3 of Arithmetic of Limits of Real Sequences. Choosing $j$ beyond the three thresholds these convergences supply, fix $j\in\mathbb{N}$ with

$$\rho_{j}D(\hat{\mu}_{j},\hat{\nu}_{j})+\rho_{j}\bigl(2|e_{0}|+1\bigr)\le\tau_{2},\qquad 2\rho_{j}\,|\mathcal{E}(\mu_{0})|\le\varsigma,$$

and put $\delta=\rho_{j}$, $\hat{\mu}=\hat{\mu}_{j}$, $\hat{\nu}=\hat{\nu}_{j}$. Then $\delta\in I$ and $\delta\le\delta_{1}$, so $N(\alpha)-\eta_{1}\le g(\alpha,\delta_{1})\le g(\alpha,\delta)$ because $g(\alpha,\cdot)$ is nonincreasing; and

$$\delta\bigl(|\mathcal{E}(\hat{\mu})|+|\mathcal{E}(\hat{\nu})|+1\bigr)\le\tau_{2},\qquad\text{so}\qquad\omega_{2}\bigl(\delta(|\mathcal{E}(\hat{\mu})|+|\mathcal{E}(\hat{\nu})|+1),\alpha\bigr)\le\kappa .$$

Step 5. The contradiction. By Step 2 and the choice of $j$,

$$\theta_{0}-\varsigma\ \le\ \theta_{0}-2\delta\mathcal{E}(\mu_{0})\ \le\ M(\delta,\alpha),$$

using $2\delta\mathcal{E}(\mu_{0})\le2\delta|\mathcal{E}(\mu_{0})|\le\varsigma$; and $\theta_{0}-\varsigma=3\theta_{0}/4$ is positive, so $0\le M(\delta,\alpha)$.

Apply claim 3 of Penalised Suprema: Monotonicity, Near-Maximisers, and Vanishing Penalty along a Doubling Sequence with $Z_{2}$, $\psi'$, $D'$, $z'_{0}$, with $\alpha$ in the role of $\beta$, with $\eta=\eta_{1}$ and with the point $z=\bigl((\hat{\mu},\hat{\nu}),\delta\bigr)$: its hypothesis $N(\alpha)-\eta_{1}\le(\psi'-\alpha D')(z)=g(\alpha,\delta)$ holds by Step 4, so

$$\tfrac{\alpha}{2}\cdot\tfrac{1}{2}W_{2}(\hat{\mu},\hat{\nu})^{2}\ \le\ N(\alpha/2)-N(\alpha)+\eta_{1}\ \le\ \eta_{1}+\eta_{1},$$

the last step because $\alpha/2=\beta_{k}$, $\alpha=\beta_{k+1}$ and $N(\beta_{k})-N(\beta_{k+1})\le\eta_{1}$. Hence $\alpha W_{2}(\hat{\mu},\hat{\nu})^{2}\le8\eta_{1}=\tau_{1}/2$, and with $\alpha^{-1}\le\tau_{1}/2$ from Step 3,

$$\alpha\,W_{2}(\hat{\mu},\hat{\nu})^{2}+\alpha^{-1}\ \le\ \tau_{1},\qquad\text{so}\qquad\omega_{1}\bigl(\alpha W_{2}(\hat{\mu},\hat{\nu})^{2}+\alpha^{-1}\bigr)\le\kappa .$$

All hypotheses of The Structure Estimate at a Maximiser of the Wasserstein-Doubled Difference on the Lift now hold: the probability space is rich, the penalty pair is Wasserstein-coercive with closed score, $F$ satisfies the shift-coercivity and shift-semicontinuity conditions, $u$ and $v$ are as required with the bounds $b$ and $b'$ and are respectively a viscosity subsolution and a viscosity supersolution on the lift, $e_{0}$ is a lower bound for $\mathcal{E}$ on $\mathcal{D}$, $0<\delta<1$ and $1<\alpha$, the pair $(\hat{\mu},\hat{\nu})$ maximises $\Psi_{\delta,\alpha}$ and $0\le M(\delta,\alpha)$, a pair $\hat{X},\hat{Y}$ as required exists by Existence, Penalty Bounds and Optimal Realisation at a Maximiser of the Wasserstein-Doubled Difference on the Lift (iv), the numbers $B$ and $R$ satisfy the four inequalities by Step 2, and $\lambda$ and $(\omega_{1},\omega_{2})$ are a properness constant and a structure pair at $R$. Its clause The Structure Estimate at a Maximiser of the Wasserstein-Doubled Difference on the Lift (i) therefore gives

$$\lambda\,(\theta_{0}-\varsigma)\ \le\ \lambda\,M(\delta,\alpha)\ \le\ \kappa+\kappa=\tfrac{\lambda\theta_{0}}{4},$$

the first inequality by claim 5 of Elementary Arithmetic in an Ordered Field with the nonnegative multiplier $\lambda$. As $\theta_{0}-\varsigma=3\theta_{0}/4$, this reads $\tfrac{3}{4}\lambda\theta_{0}\le\tfrac{1}{4}\lambda\theta_{0}$, that is $\tfrac{1}{2}\lambda\theta_{0}\le0$ by claim 3 of Elementary Arithmetic in an Ordered Field. But $\lambda$ and $\theta_{0}$ are positive, so $0<\tfrac{1}{2}\lambda\theta_{0}$ by claim 5 of Elementary Order Arithmetic in an Ordered Field applied twice, a contradiction.

Therefore no such $\mu_{0}$ exists, and $u(\mu)\le v(\mu)$ for every $\mu\in\mathcal{D}$.