Proof of Exact Inequalities at a Touching Point of a Penalised Viscosity Subsolution or Supersolution on the Wasserstein Space

Each result cited is universally quantified over the data in its own statement. We write $W$ for $W_{2}$. For $n\in\mathbb{N}$ let $\epsilon_{n}=\tfrac{1}{n}$ (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, and a real sequence $(t_{n})_{n}$ with $0\le t_{n}<\epsilon_{n}^{2}$, or with $0\le t_{n}<\epsilon_{n}$, for every $n$ converges to $0$ by Order Properties 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), so that $\lVert P\rVert=d_{\mathcal{S}(d)}(P,0)\le\lVert P-P'\rVert+\lVert P'\rVert$. Elementary real arithmetic and order, including square roots of nonnegative reals, are used through The Real Numbers: Standing Notation and Background §background. Throughout, $\delta$ with $0<\delta<1$ and the intrinsic test function $\varphi$ on $\mathcal{D}$ are those of the statement.

Step 0 (Common data). (a) By Basic Properties of a Wasserstein-Closed Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets, Bounded Distances, and Coercive Pairs §bounded-below fix $e_{0}\in\mathbb{R}$ with $e_{0}\le\mathcal{E}(\sigma)$ for every $\sigma\in\mathcal{D}$; by Basic Properties of a Wasserstein-Closed Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets, Bounded Distances, and Coercive Pairs §lsc, $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W)$.

(b) By Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty fix $\nu_{0}\in\mathcal{D}_{\Sigma}$, and let $\eta_{0}=(\nu_{0},0,0,0)$, the second entry the real $0$, the third the zero field of $L^{2}(\nu_{0};\mathbb{R}^{d})$ and the fourth the zero matrix of $\mathcal{S}(d)$. Then $(\nu_{0},0)\in\mathcal{V}(\mathcal{D}_{\Sigma})$ (The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts §bundle), so $\eta_{0}$ is a test datum for $F$ and $F^{-}_{\delta}(\eta_{0})$ and $F^{+}_{\delta}(\eta_{0})$ are real numbers. As the zero field and the zero matrix have norm $0$, $\eta_{0}$ is $R$-bounded for every real $R$ with $R>\sqrt{M_{2}(\nu_{0})}+|\mathcal{E}(\nu_{0})|$.

(c) (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

$$\lVert q\rVert_{\nu}\le\Bigl(\int_{\mathbb{R}^{d+d}}\lVert q(x)-\eta(y)\rVert^{2}\,\pi(dz)\Bigr)^{1/2}+\lVert\eta\rVert_{\mu}.$$

Likewise, by The Coordinate Fields of a Coupling, and the Second Moment as a Lipschitz Function of the Wasserstein Distance §fields (with $m=d$) the classes of $\mathrm{pr}_{1}$, $\mathrm{pr}_{2}$ and $\mathrm{pr}_{1}-\mathrm{pr}_{2}$ in $L^{2}(\pi;\mathbb{R}^{d})$ have squared norms $M_{2}(\nu)$, $M_{2}(\mu)$ and $I(\pi)$, so The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle gives $\sqrt{M_{2}(\nu)}\le\sqrt{I(\pi)}+\sqrt{M_{2}(\mu)}$.

Step 1 (Clause subsolution: viscosity data). Let $u$ and $\hat{\mu}$ be as in clause subsolution, and write $a=u^{-}_{\delta}(\hat{\mu})$, $g=\nabla\varphi(\hat{\mu})$, which lies in $T_{\hat{\mu}}\subseteq L^{2}(\hat{\mu};\mathbb{R}^{d})$ by property (b) of intrinsic test functions, and $H=H_{\varphi}(\hat{\mu})$. By Penalty-Subordinate Growth of a Function on the Penalty Domain §above with the positive weight $\tfrac{\delta}{2}$, fix $C_{1}\in\mathbb{R}$ with $u(\sigma)\le C_{1}+\tfrac{\delta}{2}\,\mathcal{E}(\sigma)$ for every $\sigma\in\mathcal{D}$. Since $\mathcal{E}$ is lower semicontinuous (Step 0(a)) and $0\le\tfrac{\delta}{2}\le\delta$, Basic Properties of the Delta-Envelopes on the Wasserstein Space §bound with $C=C_{1}$ and $\eta=\tfrac{\delta}{2}$ gives

$$u^{-}_{\delta}(\sigma)\le C_{1}-\tfrac{\delta}{2}\,\mathcal{E}(\sigma)\qquad(\sigma\in\mathcal{D}).\qquad(1.1)$$

Let $n\in\mathbb{N}$. Since $u$ is a viscosity subsolution and $u^{-}_{\delta}-\varphi$ has a local maximum relative to $\mathcal{D}$ at $\hat{\mu}$, Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution with this $\delta$, $\varphi$, $\hat{\mu}$ and $\varepsilon=\epsilon_{n}$ provides $\nu_{n}\in\mathcal{D}_{\Sigma}$, $\pi_{n}\in\Pi(\nu_{n},\hat{\mu})$, $s_{n}\in\mathbb{R}$, $q_{n}\in L^{2}(\nu_{n};\mathbb{R}^{d})$ and $Y_{n}\in\mathcal{S}(d)$ with

$$I(\pi_{n})<\epsilon_{n}^{2},\quad\bigl|u^{-}_{\delta}(\nu_{n})-a\bigr|<\epsilon_{n},\quad|s_{n}-a|<\epsilon_{n},\quad\int_{\mathbb{R}^{d+d}}\lVert q_{n}(x)-g(y)\rVert^{2}\,\pi_{n}(dz)<\epsilon_{n}^{2},\quad\lVert Y_{n}-H\rVert<\epsilon_{n},\quad F^{-}_{\delta}(\nu_{n},s_{n},q_{n},Y_{n})\le\epsilon_{n};$$

by Axiom of Countable Choice we choose such data for every $n\in\mathbb{N}$. Put $\xi_{n}=(\nu_{n},s_{n},q_{n},Y_{n})$, a test datum for $F$ since $(\nu_{n},q_{n})\in\mathcal{V}(\mathcal{D}_{\Sigma})$.

Step 2 (Uniform bounds). Let $n\in\mathbb{N}$. Taking square roots, $\sqrt{I(\pi_{n})}<\epsilon_{n}\le1$ and the discrepancy of $q_{n}$ and $g$ along $\pi_{n}$ has square root below $1$; so Step 0(c) with $\pi_{n}$ gives $\sqrt{M_{2}(\nu_{n})}<\sqrt{M_{2}(\hat{\mu})}+1$ and $\lVert q_{n}\rVert_{\nu_{n}}<\lVert g\rVert_{\hat{\mu}}+1$. Also $|s_{n}|<|a|+1$ and $\lVert Y_{n}\rVert\le\lVert Y_{n}-H\rVert+\lVert H\rVert<\lVert H\rVert+1$. For the penalty, $\nu_{n}\in\mathcal{D}_{\Sigma}\subseteq\mathcal{D}$, so $e_{0}\le\mathcal{E}(\nu_{n})$ by Step 0(a), and $a-1<u^{-}_{\delta}(\nu_{n})\le C_{1}-\tfrac{\delta}{2}\mathcal{E}(\nu_{n})$ by Step 1 and (1.1); multiplying by the positive $2\delta^{-1}$, $\mathcal{E}(\nu_{n})<2\delta^{-1}(C_{1}-a+1)$. With

$$A_{1}=|e_{0}|+2\delta^{-1}\bigl(|C_{1}|+|a|+1\bigr)+1$$

we get $-A_{1}<e_{0}\le\mathcal{E}(\nu_{n})<A_{1}$, that is, $|\mathcal{E}(\nu_{n})|<A_{1}$. None of these bounds depends on $n$.

Step 3 (Admissibility and the score bound). Let

$$R_{1}=\sqrt{M_{2}(\hat{\mu})}+\lVert g\rVert_{\hat{\mu}}+|a|+\lVert H\rVert+A_{1}+\sqrt{M_{2}(\nu_{0})}+|\mathcal{E}(\nu_{0})|+|F^{+}_{\delta}(\eta_{0})|+3,$$

a sum of nonnegative reals and $3$, hence positive, and exceeding each of the bounds of Step 2 as well as $\sqrt{M_{2}(\nu_{0})}+|\mathcal{E}(\nu_{0})|$ and $1+|F^{+}_{\delta}(\eta_{0})|$. By Step 2 every $\xi_{n}$ is $R_{1}$-bounded (Test Data for an Intrinsic Second-Order Equation Operator on the Wasserstein Space and the Admissible Sets §bounded), and $\eta_{0}$ is $R_{1}$-bounded by Step 0(b). Since $F^{-}_{\delta}(\xi_{n})\le\epsilon_{n}\le1$,

$$F^{-}_{\delta}(\xi_{n})-F^{+}_{\delta}(\eta_{0})\le1+|F^{+}_{\delta}(\eta_{0})|<R_{1},$$

so $\xi_{n}\in S^{-}_{\delta,R_{1}}$, with $\eta_{0}$ as the $R_{1}$-bounded datum required by Test Data for an Intrinsic Second-Order Equation Operator on the Wasserstein Space and the Admissible Sets §admissible. As $0<\delta<1$ and $0<R_{1}$, the shift-coercivity condition (The Shift-Coercivity Condition for an Equation Operator on the Wasserstein Space §coercivity) provides a score bound $C\ge0$ for $F$ at $(\delta,R_{1})$, so

$$\lVert\Sigma(\nu_{n})\rVert_{\nu_{n}}\le C\qquad(n\in\mathbb{N}).$$

Step 4 (Closed score; $\hat{\mu}\in\mathcal{D}_{\Sigma}$). Since $0\le I(\pi_{n})<\epsilon_{n}^{2}$ for every $n$, $\lim_{n\to\infty}I(\pi_{n})=0$, so $(\pi_{n})_{n}$ is a sequence of couplings of vanishing cost from $(\nu_{n})_{n}$ to $\hat{\mu}$, where $\hat{\mu}\in\mathcal{D}$. By Penalty Pairs with Closed Score Along Couplings §closed at the nonnegative level $C$, applied to the sequence $(\nu_{n})_{n}$ in $\mathcal{D}_{\Sigma}$ (Step 3), to $\hat{\mu}$ and to $(\pi_{n})_{n}$, we get $\hat{\mu}\in\mathcal{D}_{\Sigma}$, the first assertion of clause subsolution, and that $(\Sigma(\nu_{n}))_{n}$ converges weakly to $\Sigma(\hat{\mu})$ along $(\pi_{n})_{n}$.

Step 5 (Shift-semicontinuity; the inequality). As $\hat{\mu}\in\mathcal{D}_{\Sigma}$ and $g\in L^{2}(\hat{\mu};\mathbb{R}^{d})$, $\xi=(\hat{\mu},a,g,H)$ is a test datum for $F$. Put $R_{2}=R_{1}+C$, positive. Every $\xi_{n}$ is $R_{2}$-bounded (as $R_{1}\le R_{2}$) and $\lVert\Sigma(\nu_{n})\rVert_{\nu_{n}}\le C\le R_{2}$. Further: $\pi_{n}\in\Pi(\nu_{n},\hat{\mu})$ and $(\pi_{n})_{n}$ has vanishing cost (Step 4); $(q_{n})_{n}$ converges strongly to $g$ along $(\pi_{n})_{n}$, the discrepancies lying in $[0,\epsilon_{n}^{2})$; $(\Sigma(\nu_{n}))_{n}$ converges weakly to $\Sigma(\hat{\mu})$ along $(\pi_{n})_{n}$ (Step 4); $(s_{n})_{n}$ converges to $a$ since $|s_{n}-a|<\epsilon_{n}$ and $(\epsilon_{n})_{n}$ converges to $0$, by claim 3 of Order Properties of Limits of Real Sequences; and $(Y_{n})_{n}$ converges to $H$ in $(\mathcal{S}(d),d_{\mathcal{S}(d)})$ since $d_{\mathcal{S}(d)}(Y_{n},H)<\epsilon_{n}$. So $(\xi_{n})_{n}$ converges to $\xi$ along $(\pi_{n})_{n}$ with score bounded by $R_{2}$. By the shift-semicontinuity condition (The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §semicontinuity), $F$ is shift-semicontinuous at $(\delta,R_{2})$. Let $\varepsilon$ be positive; by The Archimedean Property of the Real Numbers there is $N\in\mathbb{N}$ with $\epsilon_{N}<\varepsilon$, and for $n\ge N$, $\epsilon_{n}\le\epsilon_{N}$, so $F^{-}_{\delta}(\xi_{n})\le\epsilon_{n}\le0+\varepsilon$. The first implication of The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §level with $c=0$ gives $F^{-}_{\delta}(\xi)\le0$, that is,

$$F^{-}_{\delta}\bigl(\hat{\mu},\,u^{-}_{\delta}(\hat{\mu}),\,\nabla\varphi(\hat{\mu}),\,H_{\varphi}(\hat{\mu})\bigr)\le0 .$$

Step 6 (Clause supersolution). Let $v$ and $\hat{\nu}$ be as in clause supersolution, and write $a'=v^{+}_{\delta}(\hat{\nu})$, $g'=\nabla\varphi(\hat{\nu})\in L^{2}(\hat{\nu};\mathbb{R}^{d})$ (property (b) of intrinsic test functions) and $H'=H_{\varphi}(\hat{\nu})$. We repeat Steps 1 to 5 with the following changes.

(a) By Penalty-Subordinate Growth of a Function on the Penalty Domain §below with the weight $\tfrac{\delta}{2}$, fix $C_{2}\in\mathbb{R}$ with $-C_{2}-\tfrac{\delta}{2}\mathcal{E}(\sigma)\le v(\sigma)$ for $\sigma\in\mathcal{D}$; the second part of Basic Properties of the Delta-Envelopes on the Wasserstein Space §bound, with $C=C_{2}$, $\eta=\tfrac{\delta}{2}$ and Step 0(a), gives $-C_{2}+\tfrac{\delta}{2}\mathcal{E}(\sigma)\le v^{+}_{\delta}(\sigma)$ for $\sigma\in\mathcal{D}$.

(b) As $v$ is a viscosity supersolution and $v^{+}_{\delta}-\varphi$ has a local minimum relative to $\mathcal{D}$ at $\hat{\nu}$, Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §supersolution with $\varepsilon=\epsilon_{n}$, and Axiom of Countable Choice, give for every $n\in\mathbb{N}$ data $\nu'_{n}\in\mathcal{D}_{\Sigma}$, $\pi'_{n}\in\Pi(\nu'_{n},\hat{\nu})$, $t_{n}\in\mathbb{R}$, $q'_{n}\in L^{2}(\nu'_{n};\mathbb{R}^{d})$ and $Y'_{n}\in\mathcal{S}(d)$ satisfying the bounds of Step 1 with $\nu_{n},\pi_{n},s_{n},q_{n},Y_{n},u^{-}_{\delta},a,g,H$ replaced by $\nu'_{n},\pi'_{n},t_{n},q'_{n},Y'_{n},v^{+}_{\delta},a',g',H'$, and $-\epsilon_{n}\le F^{+}_{\delta}(\zeta_{n})$ for the test datum $\zeta_{n}=(\nu'_{n},t_{n},q'_{n},Y'_{n})$.

(c) The bounds of Step 2 hold in the same way, except that the penalty bound now reads: $e_{0}\le\mathcal{E}(\nu'_{n})$, and $-C_{2}+\tfrac{\delta}{2}\mathcal{E}(\nu'_{n})\le v^{+}_{\delta}(\nu'_{n})<a'+1$ by (a), so $\mathcal{E}(\nu'_{n})<2\delta^{-1}(C_{2}+a'+1)$ and $|\mathcal{E}(\nu'_{n})|<A_{2}=|e_{0}|+2\delta^{-1}(|C_{2}|+|a'|+1)+1$.

(d) With $R'_{1}$ defined as $R_{1}$ but from $\hat{\nu}$, $g'$, $a'$, $H'$, $A_{2}$ and with $|F^{-}_{\delta}(\eta_{0})|$ in place of $|F^{+}_{\delta}(\eta_{0})|$, every $\zeta_{n}$ and $\eta_{0}$ are $R'_{1}$-bounded, and $F^{-}_{\delta}(\eta_{0})-F^{+}_{\delta}(\zeta_{n})\le|F^{-}_{\delta}(\eta_{0})|+\epsilon_{n}<R'_{1}$; so $\zeta_{n}\in S^{+}_{\delta,R'_{1}}$, with $\eta_{0}$ as the $R'_{1}$-bounded datum required by Test Data for an Intrinsic Second-Order Equation Operator on the Wasserstein Space and the Admissible Sets §admissible. A score bound $C'\ge0$ for $F$ at $(\delta,R'_{1})$ (The Shift-Coercivity Condition for an Equation Operator on the Wasserstein Space §coercivity) applies to the elements of $S^{+}_{\delta,R'_{1}}$ by The Shift-Coercivity Condition for an Equation Operator on the Wasserstein Space §bound, so $\lVert\Sigma(\nu'_{n})\rVert_{\nu'_{n}}\le C'$ for every $n$.

(e) As in Step 4, Penalty Pairs with Closed Score Along Couplings §closed at the level $C'$ gives $\hat{\nu}\in\mathcal{D}_{\Sigma}$ and the weak convergence of $(\Sigma(\nu'_{n}))_{n}$ to $\Sigma(\hat{\nu})$ along $(\pi'_{n})_{n}$; so $\zeta=(\hat{\nu},a',g',H')$ is a test datum and, as in Step 5, $(\zeta_{n})_{n}$ converges to $\zeta$ along $(\pi'_{n})_{n}$ with score bounded by $R'_{2}=R'_{1}+C'$. For positive $\varepsilon$ and $N$ as in Step 5, $0-\varepsilon\le-\epsilon_{n}\le F^{+}_{\delta}(\zeta_{n})$ for $n\ge N$, and the second implication of The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §level at $(\delta,R'_{2})$ with $c=0$ gives

$$0\le F^{+}_{\delta}\bigl(\hat{\nu},\,v^{+}_{\delta}(\hat{\nu}),\,\nabla\varphi(\hat{\nu}),\,H_{\varphi}(\hat{\nu})\bigr).\qquad\blacksquare$$