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

Each result cited is universally quantified over the data in its own statement. We write $W$ for $W_{a}$, a metric on $\mathcal{P}^{a}_{\rho}$ by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §metric, and recall $\rho\in\mathcal{P}^{a}_{\rho}$ (The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §reference) and $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ (Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair). 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. Square roots are the nonnegative square roots of Existence and Uniqueness of the Nonnegative Square Root, and elementary real arithmetic and order are used through The Real Numbers: Standing Notation and Background §background. For $\sigma\in\mathcal{P}(X)$, $L^{2}(\sigma;X^{a})$ is a real Hilbert space (Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields), whose zero element $0_{\sigma}$ has norm $0$. Throughout, $\delta$ with $0<\delta<1$ and the noise intrinsic test function $\varphi$ on $\mathcal{D}$ are those of the statement.

Step 0 (Common data). (a) By Basic Properties of a Noise-Closed Noise Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets and Bounded Distances §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 Noise-Closed Noise Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets and Bounded Distances §lsc, $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$ in $(\mathcal{P}^{a}_{\rho},W)$.

(b) By Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty fix $\nu_{0}\in\mathcal{D}_{\Sigma}$, and let $\eta_{0}=(\nu_{0},0,0_{\nu_{0}})$, the second entry the real $0$ and the third the zero element of $L^{2}(\nu_{0};X^{a})$. Then $(\nu_{0},0_{\nu_{0}})\in\mathcal{V}^{a}(\mathcal{D}_{\Sigma})$ (The Bundle of Noise Fields over a Set of Measures, First-Order Equation Operators on the Noise 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 $\lVert0_{\nu_{0}}\rVert_{\nu_{0}}=0$, $\eta_{0}$ is $R$-bounded for every real $R$ with $R>W(\nu_{0},\rho)+|\mathcal{E}(\nu_{0})|$.

(c) (Norms along a coupling.) Let $\nu,\mu\in\mathcal{P}(X)$, $\pi\in\Pi(\nu,\mu)$, $q\in L^{2}(\nu;X^{a})$ and $\eta\in L^{2}(\mu;X^{a})$, and write $\Delta$ for the discrepancy $\int_{X\times X}|q(x)-\eta(y)|_{a}^{2}\,\pi(dz)$. By Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §discrepancy (with $\mu$ as its fixed measure, and $\nu$, $\pi$, $q$, $\eta$), $\Delta=\lVert q\rVert_{\nu}^{2}-2\,\mathcal{K}^{a}(q,\eta,\pi)+\lVert\eta\rVert_{\mu}^{2}$, and by Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §cross-bound, $\mathcal{K}^{a}(q,\eta,\pi)\le\lVert q\rVert_{\nu}\lVert\eta\rVert_{\mu}$. Hence, with $t=\lVert q\rVert_{\nu}-\lVert\eta\rVert_{\mu}$, $|t|^{2}=t^{2}=\lVert q\rVert_{\nu}^{2}-2\lVert q\rVert_{\nu}\lVert\eta\rVert_{\mu}+\lVert\eta\rVert_{\mu}^{2}\le\Delta=(\sqrt{\Delta})^{2}$, and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $|t|\le\sqrt{\Delta}$. Therefore

$$\lVert q\rVert_{\nu}\le\Bigl(\int_{X\times X}|q(x)-\eta(y)|_{a}^{2}\,\pi(dz)\Bigr)^{1/2}+\lVert\eta\rVert_{\mu}.\qquad(0.1)$$

(d) (Distances along a coupling of finite noise cost.) Let $\nu,\mu\in\mathcal{P}^{a}_{\rho}$ and $\pi\in\Pi^{a}(\nu,\mu)$. By The Noise Wasserstein Distance §distance, the pair being noise-connected (The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §connected), $W(\nu,\mu)^{2}\le I^{a}(\pi)=(\sqrt{I^{a}(\pi)})^{2}$, so $W(\nu,\mu)\le\sqrt{I^{a}(\pi)}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; and the triangle inequality The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §triangle gives

$$W(\nu,\rho)\le W(\nu,\mu)+W(\mu,\rho)\le\sqrt{I^{a}(\pi)}+W(\mu,\rho).\qquad(0.2)$$

Step 1 (Clause subsolution: viscosity data). Let $u$ and $\hat{\mu}$ be as in clause subsolution, and write $\hat{r}=u^{-}_{\delta}(\hat{\mu})$ and $g=\nabla\varphi(\hat{\mu})$, which lies in $T^{a}_{\hat{\mu}}\subseteq L^{2}(\hat{\mu};X^{a})$ by property (b) of noise intrinsic test functions. By Penalty-Subordinate Growth of a Function on the Domain of a Noise Penalty Pair §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 Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §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 First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §subsolution with this $\delta$, $\varphi$, $\hat{\mu}$ and $\varepsilon=\epsilon_{n}$ provides $\nu_{n}\in\mathcal{D}_{\Sigma}$, $\pi_{n}\in\Pi^{a}(\nu_{n},\hat{\mu})$, $s_{n}\in\mathbb{R}$ and $q_{n}\in L^{2}(\nu_{n};X^{a})$ with

$$I^{a}(\pi_{n})<\epsilon_{n}^{2},\quad\bigl|u^{-}_{\delta}(\nu_{n})-\hat{r}\bigr|<\epsilon_{n},\quad|s_{n}-\hat{r}|<\epsilon_{n},\quad\int_{X\times X}|q_{n}(x)-g(y)|_{a}^{2}\,\pi_{n}(dz)<\epsilon_{n}^{2},\quad F^{-}_{\delta}(\nu_{n},s_{n},q_{n})\le\epsilon_{n};$$

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

Step 2 (Uniform bounds). Let $n\in\mathbb{N}$. By claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, $\sqrt{I^{a}(\pi_{n})}<\epsilon_{n}\le1$, and the discrepancy of $q_{n}$ and $g$ along $\pi_{n}$ has square root below $\epsilon_{n}\le1$. Since $\nu_{n},\hat{\mu}\in\mathcal{P}^{a}_{\rho}$, (0.2) with $\pi_{n}$ gives $W(\nu_{n},\rho)<W(\hat{\mu},\rho)+1$; since $\pi_{n}\in\Pi^{a}(\nu_{n},\hat{\mu})\subseteq\Pi(\nu_{n},\hat{\mu})$ (Couplings of Finite Noise Cost and Their Noise Cost §couplings), (0.1) with $\pi_{n}$ gives $\lVert q_{n}\rVert_{\nu_{n}}<\lVert g\rVert_{\hat{\mu}}+1$. Also $|s_{n}|<|\hat{r}|+1$. 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 $\hat{r}-1<u^{-}_{\delta}(\nu_{n})\le C_{1}-\tfrac{\delta}{2}\mathcal{E}(\nu_{n})$ by Step 1 and (1.1); multiplying by the positive $2\delta^{-1}$, $\mathcal{E}(\nu_{n})<2\delta^{-1}(C_{1}-\hat{r}+1)$. With

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

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

$$R_{1}=W(\hat{\mu},\rho)+\lVert g\rVert_{\hat{\mu}}+|\hat{r}|+A_{1}+W(\nu_{0},\rho)+|\mathcal{E}(\nu_{0})|+|F^{+}_{\delta}(\eta_{0})|+3,$$

a sum of nonnegative reals and $3$, hence positive, and exceeding each of the bounds of Step 2 as well as $W(\nu_{0},\rho)+|\mathcal{E}(\nu_{0})|$ and $1+|F^{+}_{\delta}(\eta_{0})|$. By Step 2 every $\xi_{n}$ is $R_{1}$-bounded (Test Data for a First-Order Equation Operator on the Noise 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 a First-Order Equation Operator on the Noise Wasserstein Space and the Admissible Sets §admissible. As $0<\delta<1$ and $0<R_{1}$, the shift-coercivity condition (The Shift-Coercivity Condition for a First-Order Equation Operator on the Noise 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 $0\le I^{a}(\pi_{n})<\epsilon_{n}^{2}$ for every $n$, $\lim_{n\to\infty}I^{a}(\pi_{n})=0$, so $(\pi_{n})_{n}$ is a sequence of couplings of vanishing noise cost from the sequence $(\nu_{n})_{n}$ in $\mathcal{P}^{a}_{\rho}$ to $\hat{\mu}\in\mathcal{P}^{a}_{\rho}$, where $\hat{\mu}\in\mathcal{D}$. By Noise Penalty Pairs with Closed Score Along Noise 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 $g\in L^{2}(\hat{\mu};X^{a})$, $\xi=(\hat{\mu},\hat{r},g)$ 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: $\pi_{n}\in\Pi^{a}(\nu_{n},\hat{\mu})$ and $(\pi_{n})_{n}$ has vanishing noise 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); and $(s_{n})_{n}$ converges to $\hat{r}$ since $|s_{n}-\hat{r}|<\epsilon_{n}$ and $(\epsilon_{n})_{n}$ converges to $0$, by claim 3 of Order Properties of Limits of Real Sequences. 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 a First-Order Equation Operator on the Noise 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 a First-Order Equation Operator on the Noise Wasserstein Space §level with $c=0$ gives $F^{-}_{\delta}(\xi)\le0$, that is,

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

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

(a) By Penalty-Subordinate Growth of a Function on the Domain of a Noise Penalty Pair §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 Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §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 First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §supersolution with $\varepsilon=\epsilon_{n}$, and Axiom of Countable Choice, give for every $n\in\mathbb{N}$ data $\nu'_{n}\in\mathcal{D}_{\Sigma}$, $\pi'_{n}\in\Pi^{a}(\nu'_{n},\hat{\nu})$, $t_{n}\in\mathbb{R}$ and $q'_{n}\in L^{2}(\nu'_{n};X^{a})$ satisfying the bounds of Step 1 with $\nu_{n},\pi_{n},s_{n},q_{n},u^{-}_{\delta},\hat{r},g$ replaced by $\nu'_{n},\pi'_{n},t_{n},q'_{n},v^{+}_{\delta},\hat{r}',g'$, and $-\epsilon_{n}\le F^{+}_{\delta}(\zeta_{n})$ for the test datum $\zeta_{n}=(\nu'_{n},t_{n},q'_{n})$.

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

(d) With $R'_{1}$ defined as $R_{1}$ but from $\hat{\nu}$, $g'$, $\hat{r}'$, $A_{2}$ 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 a First-Order Equation Operator on the Noise Wasserstein Space and the Admissible Sets §admissible. A score bound $C'\ge0$ for $F$ at $(\delta,R'_{1})$ (The Shift-Coercivity Condition for a First-Order Equation Operator on the Noise Wasserstein Space §coercivity) applies to the elements of $S^{+}_{\delta,R'_{1}}$ by The Shift-Coercivity Condition for a First-Order Equation Operator on the Noise Wasserstein Space §bound, so $\lVert\Sigma(\nu'_{n})\rVert_{\nu'_{n}}\le C'$ for every $n$.

(e) As in Step 4, Noise Penalty Pairs with Closed Score Along Noise 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 $\zeta=(\hat{\nu},\hat{r}',g')$ 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 a First-Order Equation Operator on the Noise Wasserstein Space §level at $(\delta,R'_{2})$ with $c=0$ gives

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