Proof of A Comparison Principle for Viscosity Solutions on 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 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 $\alpha$, $W=W_{2}$, and $I=\{\delta\in\mathbb{R}:0<\delta<1\}$.

Step 1 (The corrected maximum and its monotonicity). 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 $\sigma\in\mathcal{D}$; the set $\mathcal{D}$ contains the nonempty $\mathcal{D}_{\Sigma}$ (Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty), so fix $\mu_{0}\in\mathcal{D}$. For positive $\delta,\alpha$ let $\Psi_{\delta,\alpha}$ and $M(\delta,\alpha)$ be as in Existence, Penalty Bounds and Monotonicity of the Maximum of the Wasserstein-Doubled Difference of Bounded Functions, read with the present $e_{0}$, $u$, $v$, $b$, $b'$; by its clause Existence, Penalty Bounds and Monotonicity of the Maximum of the Wasserstein-Doubled Difference of Bounded Functions §maximiser, $M(\delta,\alpha)$ is a real number at most $b-b'-2\delta e_{0}$ and $\Psi_{\delta,\alpha}$ has a maximising pair. For $\alpha>0$ and $\delta\in I$ put

Lower bound. By Basic Properties of the Delta-Envelopes on the Wasserstein Space §semicontinuity, $u(\sigma)-\delta\mathcal{E}(\sigma)\le u^{-}_{\delta}(\sigma)$ and $v^{+}_{\delta}(\sigma)\le v(\sigma)+\delta\mathcal{E}(\sigma)$ for $\sigma\in\mathcal{D}$. As $W(\mu_{0},\mu_{0})=0$ (The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §separation), $M(\delta,\alpha)\ge\Psi_{\delta,\alpha}(\mu_{0},\mu_{0})\ge u(\mu_{0})-v(\mu_{0})-2\delta\mathcal{E}(\mu_{0})$; for $\delta\in I$, $2\delta\mathcal{E}(\mu_{0})\le2|\mathcal{E}(\mu_{0})|$ and $-2|e_{0}|\le2\delta e_{0}$ (claims 3, 4 of Properties of the Absolute Value in an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field), so

Halving the weight. Let $\alpha>0$, $\delta,\delta'\in I$ with $\delta'<\delta$, and let $(\hat{\mu},\hat{\nu})$ be a maximising pair of $\Psi_{\delta,\alpha}$. By Existence, Penalty Bounds and Monotonicity of the Maximum of the Wasserstein-Doubled Difference of Bounded Functions §weight, $M(\delta',\alpha)-M(\delta,\alpha)\ge(\delta-\delta')(\mathcal{E}(\hat{\mu})+\mathcal{E}(\hat{\nu}))$; 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, $G(\alpha,\cdot)$ 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 $\Psi_{\delta,\alpha}$. By Existence, Penalty Bounds and Monotonicity of the Maximum of the Wasserstein-Doubled Difference of Bounded Functions §strength,

so $G(\cdot,\delta)$ is nonincreasing on the positive reals.

Step 2 (Claim 1: constants). Let $\theta$ be positive. Put $B=|b|+|b'|+|e_{0}|+1$ and $R=2B$; then $|b|+|b'|+|e_{0}|\le B$ and $0<2B\le R$. Since $F$ is locally strictly proper, fix a properness constant $\lambda>0$ 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 $\kappa=\lambda\theta/4$, 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 $\eta_{1}=\tau_{1}/16$.

Step 3 (Choice of the doubling strength). For $\alpha\ge\beta_{0}$ the set $\{G(\alpha,\delta):\delta\in I\}$ is nonempty and bounded above by $b-b'$ by (1a); let $N(\alpha)$ be its least upper bound (The Real Numbers: Standing Notation and Background §bounds). By (1b), $\ell\le N(\alpha)$; by (1d), $N$ is nonincreasing on $\{\alpha:\alpha\ge\beta_{0}\}$, since $G(\alpha,\delta)\le G(\alpha',\delta)\le N(\alpha')$ 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 $\ell$; 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 $\alpha^{-1}\tau_{1}/2$ (claim 10 of Elementary Order Arithmetic in an Ordered Field), $\alpha^{-1}<\tau_{1}/2$. 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 4 (Choice of the threshold). Let $\eta_{2}$ be the least of $\eta_{1}$ and $\tau_{2}/4$ (claim 9 of Elementary Order Arithmetic in an Ordered Field). By claim 3 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ fix $\delta_{1}\in I$ with $N(\alpha)-\eta_{2}<G(\alpha,\delta_{1})$, and let $\delta_{0}$ be the least of $\delta_{1}$ and $\tau_{2}\bigl(4|e_{0}|+2\bigr)^{-1}$ (claims 7 and 9 of Elementary Order Arithmetic in an Ordered Field), a positive number. Let $\delta\in\mathbb{R}$ satisfy $0<\delta<\delta_{0}$. Then $\delta\in I$ and $\delta<\delta_{1}$, so, $G(\alpha,\cdot)$ being nonincreasing,

Step 5 (Claim 1: the bound). With $\delta$ as in Step 4 we show $M(\delta,\alpha)\le\theta$. Suppose instead $\theta<M(\delta,\alpha)$; then $0\le M(\delta,\alpha)$. The hypotheses of The Structure Estimate at a Maximiser of the Wasserstein-Doubled Difference hold for the present pair, $F$, $u$, $v$, $b$, $b'$, $e_{0}$, $\delta$, $\alpha$, $B$, $R$, $\lambda$ and $(\omega_{1},\omega_{2})$: the pair is Wasserstein-coercive with closed score along couplings and $\mathcal{D}$ has the map property, $F$ satisfies the shift-coercivity and shift-semicontinuity conditions, $u$ is a viscosity subsolution bounded above by $b$ and $v$ a viscosity supersolution bounded below by $b'$, $0<\delta<1$, $1<\alpha$, $0\le M(\delta,\alpha)$, and $B$, $R$, $\lambda$, $(\omega_{1},\omega_{2})$ are as there. Its clause The Structure Estimate at a Maximiser of the Wasserstein-Doubled Difference §estimate provides a maximising pair $(\rho^{*},\sigma^{*})$ of $\Psi_{\delta,\alpha}$ with

The first modulus. By (1d) with $\alpha'=\tfrac{\alpha}{2}$, whose coefficient is $\tfrac{\alpha-\alpha/2}{2}=\tfrac{\alpha}{4}$, then $G(\tfrac{\alpha}{2},\delta)\le N(\tfrac{\alpha}{2})$, (4a) and (3a),

so $\alpha W(\rho^{*},\sigma^{*})^{2}<8\eta_{1}=\tau_{1}/2$, 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 $G(\alpha,\tfrac{\delta}{2})\le N(\alpha)$ and (4a),

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 $\lambda M(\delta,\alpha)\le2\kappa=\lambda\theta/2$. But $\theta<M(\delta,\alpha)$ and $0<\lambda$ give $\lambda\theta<\lambda M(\delta,\alpha)$ (claim 10 of Elementary Order Arithmetic in an Ordered Field), so $\lambda\theta<\lambda\theta/2$, that is $\lambda\theta/2<0$ (claim 1 of that lemma), contradicting $0<\lambda\theta/2$ (claims 5 and 8 of that lemma). Therefore $M(\delta,\alpha)\le\theta$. For $\mu\in\mathcal{D}$, since $W(\mu,\mu)=0$,

As $\delta$ with $0<\delta<\delta_{0}$ was arbitrary, this is claim 1.

Step 6 (Claim 2). Let $\mu\in\mathcal{D}$ and let $\theta$ be positive; let $\delta_{0}$ be as in claim 1 for $\theta$, and let $\delta$ be half the least of $\delta_{0}$ and $\theta\bigl(2|\mathcal{E}(\mu)|+1\bigr)^{-1}$, so that $0<\delta<\delta_{0}$ and $2\delta|\mathcal{E}(\mu)|\le\theta$. By Basic Properties of the Delta-Envelopes on the Wasserstein Space §semicontinuity and claim 1,

As $\theta$ was arbitrary, $u(\mu)-v(\mu)\le0$ by Comparison of Real Numbers with Arbitrary Positive Slack §slack-above, that is $u(\mu)\le v(\mu)$ (claim 3 of Elementary Arithmetic in an Ordered Field).