Proof of The Bump Construction for Plan-Jet Viscosity Subsolutions on Square-Integrable Noncommutative Laws

Each result cited is universally quantified over the data in its own statement.

Conventions. For $L^{2}$ tuples of one tracial W*-probability space, sums, real multiples, the pairing and $\lVert\cdot\rVert_{2}$ are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing, the operations, inner product and norm of $H^{d}$ (Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §operations, Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §pairing); so the triangle and Cauchy--Schwarz inequalities hold. Law invariance: if $\mathrm{law}(Z)=\mathrm{law}(Z')$ and $T$ is an affine datum, then $\mathrm{law}(TZ)=\mathrm{law}(TZ')$, $\lVert TZ\rVert_{2}=\lVert TZ'\rVert_{2}$, and pairings of blocks of $TZ$ equal those of $TZ'$ (Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §moments); lifts take equal values on tuples of equal law (Lifts of Functions on Square-Integrable Noncommutative Laws to Square-Integrable Tuples §lift). Laws are realised by Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §law. For $\gamma\in\Sigma^{2}_{3d}$ with a realisation $(X,P,X')$ put $s(\gamma)=\lVert X'-X\rVert_{2}$ and $p(\gamma)=\langle P,X'-X\rangle_{2}$, which do not depend on the realisation; $B$ and $C$ are the affine data from $3d$ variables selecting $(x,p)$ and $x'$.

Step 0 (the failure). By Test Functions and Plan Jets: Touching Transfers Plan Jets, and the Jet Form of Plan-Jet Viscosity Solutions §super, since $w_{*}$ is not a supersolution there are real $\delta\ge0$, $\mu\in\Sigma^{2}_{d}$, $\pi\in J^{-}_{\delta}w_{*}(\mu)$ and real $\theta>0$ such that

for every tracial W*-probability space $(H,M,\Omega)$ and all $L^{2}$ $d$-tuples $X,P,Q$ of it with $\mathrm{law}(X,P)=\pi$ and $\lVert Q\rVert_{2}\le\delta$.

Step 1 ($w_{*}(\mu)<g(\mu)$). By Properties of the Lower Semicontinuous Envelope, by Duality §bounds, $w_{*}\le w\le g$. If $w_{*}(\mu)=g(\mu)$, then $g-w_{*}\ge0=(g-w_{*})(\mu)$, so $g-w_{*}$ has a local minimum at $\mu$ and $\pi\in J^{-}_{\delta}g(\mu)$ by Test Functions and Plan Jets: Touching Transfers Plan Jets, and the Jet Form of Plan-Jet Viscosity Solutions §below. Since $g$ is a supersolution, Test Functions and Plan Jets: Touching Transfers Plan Jets, and the Jet Form of Plan-Jet Viscosity Solutions §super (tolerance $\theta$) gives a realisation with $\mathrm{law}(X,P)=\pi$, $\lVert Q\rVert_{2}\le\delta$ and $\rho g(\mu)+\mathcal{H}_{M}(X,P+Q)\ge-\theta$, contradicting (V). Hence $\beta=\tfrac12(g(\mu)-w_{*}(\mu))>0$, and since $g$ is lower semicontinuous there is $r_{g}>0$ with $g(\nu)>w_{*}(\mu)+\beta$ whenever $\widehat{W}_{2}(\nu,\mu)<r_{g}$.

Step 2 (constants). Let $K\ge0$ bound $|w|$ and $|g|$; then $-K\le w_{*}\le K$ by Properties of the Lower Semicontinuous Envelope, by Duality §bounds and Properties of the Lower Semicontinuous Envelope, by Duality §greatest. Let $(X^{0},P^{0})$ realise $\pi$, $a_{X}=\lVert X^{0}\rVert_{2}$, $a_{P}=\lVert P^{0}\rVert_{2}$ (independent of the realisation), and $R=a_{X}+a_{P}+\delta+4$. By Hamiltonians on Phase-Space Noncommutative Laws that are Uniformly Continuous on Bounded Sets §uniform with $R$ and $\theta/2$ there is $r_{H}>0$; put $\eta=\min\{1,r_{H}/16\}$. By Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §sub there is $r_{0}>0$ such that

for every realisation $(X,P,X')$ with $\mathrm{law}(X,P)=\pi$ and $\lVert X'-X\rVert_{2}<r_{0}$. Put $r=\min\{r_{0},1\}$, $\kappa=1$ and $M=2a_{P}+(4K+2)/r$.

Step 3 (the test function). For $\nu\in\Sigma^{2}_{d}$ let $\mathcal{C}(\nu)=\{\gamma\in\Sigma^{2}_{3d}:B_{\#}\gamma=\pi,\ C_{\#}\gamma=\nu\}$, and for $\gamma\in\Sigma^{2}_{3d}$ put

(a) Couplings. For every $\nu$ and $\varepsilon>0$ there is $\gamma\in\mathcal{C}(\nu)$ with $s(\gamma)\le\widehat{W}_{2}(\mu,\nu)+\varepsilon$: take $Y,Y'$ with $\mathrm{law}(Y)=\mu$, $\mathrm{law}(Y')=\nu$ and $\lVert Y'-Y\rVert_{2}^{2}\le\widehat{W}_{2}(\mu,\nu)^{2}+\varepsilon^{2}$ (Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §distance), and glue $\pi$ with $\mathrm{law}(Y,Y')$ over $\mu=\mathrm{pr}^{1}_{\#}\pi$ by Gluing Two Square-Integrable Noncommutative Laws along a Common Marginal §glue ($k=m=n=d$). (b) Boundedness. For $\gamma\in\mathcal{C}(\nu)$ with realisation $(X,P,X')$, $\lVert P\rVert_{2}=a_{P}$, so by Cauchy--Schwarz $\Psi(\gamma)\le a_{P}s-\kappa s^{2}\le a_{P}^{2}/(4\kappa)$ with $s=s(\gamma)$. Hence $\psi(\nu)=w_{*}(\mu)+\sup\{\Psi(\gamma):\gamma\in\mathcal{C}(\nu)\}$ is a real number (The Real Numbers: Standing Notation and Background §bounds). (c) Upper bound. Let $\gamma\in\mathcal{C}(\nu)$, $s=s(\gamma)$; then $s\ge\widehat{W}_{2}(\mu,\nu)$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §lipschitz. If $s<r_{0}$, (J) gives $w_{*}(\mu)+\Psi(\gamma)\le w_{*}(\nu)-\kappa s^{2}$. If $s\ge r_{0}$, then $s\ge r$, $s-r/2\ge s/2$, and $\Psi(\gamma)\le a_{P}s-Ms/2\le-(M/2-a_{P})r=-(2K+1)$, so $w_{*}(\mu)+\Psi(\gamma)\le-K-1\le w_{*}(\nu)-1$. Hence

the equality because $\gamma=\mathrm{law}(X^{0},P^{0},X^{0})\in\mathcal{C}(\mu)$ has $\Psi(\gamma)=0$. (d) Lower bound. With $b(t)=(a_{P}+\delta+\eta+M)t+\kappa t^{2}$, (a) gives $\Psi(\gamma)\ge-b(s(\gamma))$ for such $\gamma$, so $\psi(\nu)\ge w_{*}(\mu)-b(\widehat{W}_{2}(\mu,\nu)+\varepsilon)$ for every $\varepsilon>0$ and hence $\psi(\nu)\ge w_{*}(\mu)-b(\widehat{W}_{2}(\mu,\nu))$, $b$ being continuous and nondecreasing on $[0,\infty)$. (e) Continuity. Fix $\nu$ and put $t_{\nu}=\widehat{W}_{2}(\mu,\nu)+1$ and $s_{\nu}=(a_{P}+(a_{P}^{2}+4\kappa(b(t_{\nu})+1))^{1/2})/(2\kappa)$. If $\widehat{W}_{2}(\nu,\nu')\le1$, $\nu''\in\{\nu,\nu'\}$ and $\gamma\in\mathcal{C}(\nu'')$ has $\Psi(\gamma)\ge\psi(\nu'')-w_{*}(\mu)-1$, then by (d) and $\widehat{W}_{2}(\mu,\nu'')\le t_{\nu}$, $a_{P}s-\kappa s^{2}\ge\Psi(\gamma)\ge-b(t_{\nu})-1$, so $s(\gamma)\le s_{\nu}$. Take such $\gamma\in\mathcal{C}(\nu)$ with $\Psi(\gamma)\ge\psi(\nu)-w_{*}(\mu)-\varepsilon$ ($0<\varepsilon\le1$), realised by $(X,P,X')$, and $Z,Z'$ with $\mathrm{law}(Z)=\nu$, $\mathrm{law}(Z')=\nu'$, $\lVert Z'-Z\rVert_{2}\le\widehat{W}_{2}(\nu,\nu')+\varepsilon$. Gluing $\mathrm{law}(X',X,P)$ with $\mathrm{law}(Z,Z')$ over $\nu$ (Gluing Two Square-Integrable Noncommutative Laws along a Common Marginal §glue, $k=d$, $m=2d$, $n=d$) gives $\hat X',\hat X,\hat P,\hat Z'$ in one space with $\mathrm{law}(\hat X,\hat P,\hat X')=\gamma$ and $\tau=\lVert\hat Z'-\hat X'\rVert_{2}\le\widehat{W}_{2}(\nu,\nu')+\varepsilon\le2$. Then $\gamma'=\mathrm{law}(\hat X,\hat P,\hat Z')\in\mathcal{C}(\nu')$, $|s(\gamma')-s(\gamma)|\le\tau$, and termwise $\Psi(\gamma')\ge\Psi(\gamma)-L_{\nu}\tau$ with $L_{\nu}=a_{P}+\delta+\eta+M+\kappa(2s_{\nu}+2)$. Hence $\psi(\nu')\ge\psi(\nu)-\varepsilon-L_{\nu}(\widehat{W}_{2}(\nu,\nu')+\varepsilon)$ for every $\varepsilon$, and $\psi(\nu')\ge\psi(\nu)-L_{\nu}\widehat{W}_{2}(\nu,\nu')$. Exchanging $\nu$ and $\nu'$ (the same bound $s_{\nu}$ applies to $\nu'$) gives $|\psi(\nu)-\psi(\nu')|\le L_{\nu}\widehat{W}_{2}(\nu,\nu')$ whenever $\widehat{W}_{2}(\nu,\nu')\le1$, so $\psi$ is continuous.

Step 4 (the bump). Choose $c\in(0,\tfrac12)$ so small that, with $\zeta=(2c/\kappa)^{1/2}$: $\zeta+c^{1/2}<r/2$; $(c/\kappa)^{1/2}<r_{g}$; $a_{P}\zeta+2c<\beta$; $\rho(a_{P}\zeta+2c)<\theta/2$; $(1+2\kappa)\zeta<r_{H}/4$; and $2\kappa\zeta\le1$. Define $u=\max\{w,\psi+c\}$. Let $A=\{\nu:\psi(\nu)+c\ge w(\nu)\}$. For $\nu\in A$: (i) by (U) and $w_{*}\le w$, $\min\{\kappa\widehat{W}_{2}(\mu,\nu)^{2},1\}\le c<1$, so $\widehat{W}_{2}(\mu,\nu)\le(c/\kappa)^{1/2}$; (ii) every $\gamma\in\mathcal{C}(\nu)$ with $\Psi(\gamma)\ge\psi(\nu)-w_{*}(\mu)-\varepsilon$, $0<\varepsilon\le c$, has $s(\gamma)\le\zeta$: if $s(\gamma)\ge r_{0}$, Step 3(c) gives $\psi(\nu)-\varepsilon\le w_{*}(\nu)-1\le\psi(\nu)+c-1$, impossible as $c+\varepsilon<1$; otherwise $\psi(\nu)-\varepsilon\le w_{*}(\nu)-\kappa s(\gamma)^{2}\le\psi(\nu)+c-\kappa s(\gamma)^{2}$, so $\kappa s(\gamma)^{2}\le2c$; (iii) taking such $\gamma$ with $\varepsilon=c$ in (b): $\psi(\nu)\le w_{*}(\mu)+a_{P}\zeta+c$. Properties of $u$. $u$ is upper semicontinuous by The Maximum of Two Upper Semicontinuous Functions §max-usc ($\psi+c$ being continuous) and $u\ge w$. If $u(\nu)>w(\nu)$ then $\nu\in A$, and by (i), (iii) and Step 1, $u(\nu)=\psi(\nu)+c\le w_{*}(\mu)+a_{P}\zeta+2c<w_{*}(\mu)+\beta<g(\nu)$; so $u\le g$, and $u$ is bounded. By Properties of the Lower Semicontinuous Envelope, by Duality §approximation there are $\nu_{k}\to\mu$ with $w(\nu_{k})\to w_{*}(\mu)$, and $\psi(\nu_{k})\to\psi(\mu)=w_{*}(\mu)$ by (e) and (U); so $\psi(\nu_{k})+c>w(\nu_{k})$, i.e. $u(\nu_{k})>w(\nu_{k})$, for some $k$.

Step 5 ($u$ is a subsolution). We verify Test Functions and Plan Jets: Touching Transfers Plan Jets, and the Jet Form of Plan-Jet Viscosity Solutions §sub. Let $\nu\in\Sigma^{2}_{d}$, $\delta'\ge0$, $\sigma\in J^{+}_{\delta'}u(\nu)$ and $\eta''>0$.

Case $u(\nu)=w(\nu)$. Then $w-u\le0=(w-u)(\nu)$, so $\sigma\in J^{+}_{\delta'}w(\nu)$ by Test Functions and Plan Jets: Touching Transfers Plan Jets, and the Jet Form of Plan-Jet Viscosity Solutions §above, and the subsolution property of $w$ (Test Functions and Plan Jets: Touching Transfers Plan Jets, and the Jet Form of Plan-Jet Viscosity Solutions §sub) gives the required realisation, as $\rho u(\nu)=\rho w(\nu)$.

Case $u(\nu)>w(\nu)$. Then $\nu\in A$ and $u(\nu)=\psi(\nu)+c$; as $(\psi+c)-u\le0=((\psi+c)-u)(\nu)$, Test Functions and Plan Jets: Touching Transfers Plan Jets, and the Jet Form of Plan-Jet Viscosity Solutions §above gives $\sigma\in J^{+}_{\delta'}(\psi+c)(\nu)$, which equals $J^{+}_{\delta'}\psi(\nu)$ because the defining inequality of Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §super involves only differences of values. Let $r_{1}>0$ be as in Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §super for $\sigma$, $\psi$ and $\eta$. Fix $\varepsilon\in(0,c]$ with $\varepsilon^{1/2}<r_{1}$ and $(1+\kappa)\varepsilon^{1/2}\le\eta$, and $\gamma\in\mathcal{C}(\nu)$ with $\Psi(\gamma)\ge\psi(\nu)-w_{*}(\mu)-\varepsilon$; by (ii), $s(\gamma)\le\zeta$. Realise $\gamma$ by $(X_{1},P_{1},X'_{1})$ and $\sigma$ by $(Y,S_{1})$, and glue $\mathrm{law}(X'_{1},X_{1},P_{1})$ with $\sigma$ over $\nu$ (Gluing Two Square-Integrable Noncommutative Laws along a Common Marginal §glue, $k=d$, $m=2d$, $n=d$): this gives $X',X,P,S$ in one space $(H,M,\Omega)$ with $\mathrm{law}(X,P,X')=\gamma$ and $\mathrm{law}(X',S)=\sigma$. Put $s=\lVert X'-X\rVert_{2}\le\zeta$, $G=P-2\kappa(X'-X)$ and $D=G-S$.

If $D\ne0$, let $t=\varepsilon^{1/2}/\lVert D\rVert_{2}$ and $X''=X'+tD$, so $\lVert X''-X'\rVert_{2}=\varepsilon^{1/2}<r_{1}$ and $s''=\lVert X''-X\rVert_{2}\le s+\varepsilon^{1/2}<r/2$. The superjet inequality gives $\psi(\mathrm{law}(X''))\le\psi(\nu)+t\langle S,D\rangle_{2}+(\delta'+\eta)\varepsilon^{1/2}$. On the other hand $\gamma''=\mathrm{law}(X,P,X'')\in\mathcal{C}(\mathrm{law}(X''))$, neither $\gamma$ nor $\gamma''$ activates the term with $M$, and $s''^{2}-s^{2}=2t\langle X'-X,D\rangle_{2}+\varepsilon$, $s''-s\le\varepsilon^{1/2}$; so

Comparing, $t\lVert D\rVert_{2}^{2}\le(1+\kappa)\varepsilon+(\delta+\delta'+2\eta)\varepsilon^{1/2}$, i.e. $\lVert D\rVert_{2}\le(1+\kappa)\varepsilon^{1/2}+\delta+\delta'+2\eta\le\delta+\delta'+3\eta$. This bound holds trivially if $D=0$.

Let $\lambda=1$ if $\lVert D\rVert_{2}\le\delta'$ and $\lambda=\delta'/\lVert D\rVert_{2}$ otherwise, $Q=\lambda D$ and $E=(1-\lambda)D$; then $\lVert Q\rVert_{2}\le\delta'$, $S+Q=G-E$ and $\lVert E\rVert_{2}=\max\{0,\lVert D\rVert_{2}-\delta'\}\le\delta+3\eta$. Let $E'=E$ if $\lVert E\rVert_{2}\le\delta$ and $E'=(\delta/\lVert E\rVert_{2})E$ otherwise, so $\lVert E'\rVert_{2}\le\delta$ and $\lVert E-E'\rVert_{2}\le3\eta$, and put $Q_{\pi}=(-1)E'$. Since $\mathrm{law}(X,P)=\pi$ and $\lVert Q_{\pi}\rVert_{2}\le\delta$, (V) gives $\rho w_{*}(\mu)+\mathcal{H}_{M}(X,P+Q_{\pi})<-\theta$. Now $\lVert X'-X\rVert_{2}+\lVert(S+Q)-(P+Q_{\pi})\rVert_{2}\le s+2\kappa s+3\eta<r_{H}$, and $\lVert X\rVert_{2}=a_{X}$, $\lVert X'\rVert_{2}\le a_{X}+1$, $\lVert P+Q_{\pi}\rVert_{2}\le a_{P}+\delta$, $\lVert S+Q\rVert_{2}\le a_{P}+2\kappa\zeta+\delta+3\eta\le R$; so the choice of $r_{H}$ gives $\mathcal{H}_{M}(X',S+Q)<\mathcal{H}_{M}(X,P+Q_{\pi})+\theta/2$. With (iii) and the choice of $c$, $\rho u(\nu)=\rho(\psi(\nu)+c)<\rho w_{*}(\mu)+\theta/2$, hence

with $\mathrm{law}(X',S)=\sigma$ and $\lVert Q\rVert_{2}\le\delta'$. Thus $u$ is a subsolution, and it has all the stated properties.