Proof of The Upper Semicontinuous Envelope of a Supremum of Plan-Jet Viscosity Subsolutions is a Plan-Jet Viscosity Subsolution

Each result cited is universally quantified over the data in its own statement. By Test Functions and Plan Jets: Touching Transfers Plan Jets, and the Jet Form of Plan-Jet Viscosity Solutions §sub it suffices to show: for every real $\delta\ge0$, every $\mu\in\Sigma^{2}_{d}$, every $\pi\in J^{+}_{\delta}w^{*}(\mu)$ and every real $\eta_{f}>0$ there are a tracial W*-probability space and $L^{2}$ $d$-tuples $X,P,Q$ of it with $\mathrm{law}(X,P)=\pi$, $\lVert Q\rVert_{2}\le\delta$ and $\rho\,w^{*}(\mu)+\mathcal{H}_{M}(X,P+Q)\le\eta_{f}$. Fix such $\delta,\mu,\pi,\eta_{f}$.

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; they are 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 $\mathrm{law}(Z)=\mathrm{law}(Z')$ for $L^{2}$ $k$-tuples $Z,Z'$ of possibly different spaces and $T$ is an affine datum from $k$ to $l$ variables, then $\mathrm{law}(TZ)=\mathrm{law}(TZ')$ and $\lVert TZ\rVert_{2}=\lVert TZ'\rVert_{2}$, and every pairing $\langle(TZ)^{1},(TZ)^{2}\rangle_{2}$ of two blocks of $TZ$ equals the corresponding one for $Z'$, by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward and Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §moments. In particular a block of a tuple has the law obtained by the coordinate push-forward, and the lift of a function on laws takes the same value on tuples of equal law (Lifts of Functions on Square-Integrable Noncommutative Laws to Square-Integrable Tuples §lift). Every law in $\Sigma^{2}_{k}$ has a realisation by Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §law.

For $\gamma\in\Sigma^{2}_{3d}$ and a realisation $(X,P,X')$ of $\gamma$ (three $L^{2}$ $d$-tuples of one space with $\mathrm{law}(X,P,X')=\gamma$), put $s(\gamma)=\lVert X'-X\rVert_{2}$ and $p(\gamma)=\langle P,X'-X\rangle_{2}$; by law invariance these do not depend on the realisation. Let $B$, $C$ be the affine data from $3d$ variables selecting $(x,p)$ and $x'$. Since $s(\gamma)^{2}$ is the second moment of the push-forward of $\gamma$ under the datum $(x,p,x')\mapsto x'-x$ and $p(\gamma)$ is a sum of quadratic moments of the push-forward under $(x,p,x')\mapsto(p,x'-x)$, the maps $s$, $p$, $B_{\#}$ and $C_{\#}$ are continuous by Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §lipschitz.

Step 1 (constants). Let $(X^{0},P^{0})$ realise $\pi$ and put $a_{X}=\lVert X^{0}\rVert_{2}$, $a_{P}=\lVert P^{0}\rVert_{2}$, which do not depend on the realisation. Put $R=a_{X}+a_{P}+\delta+4$. By Hamiltonians on Phase-Space Noncommutative Laws that are Uniformly Continuous on Bounded Sets §uniform with $R$ and $\eta_{f}/3$ there is $r_{H}>0$; put $\eta=\min\{1,r_{H}/8\}$. By Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §super (with this $\eta$) there is $r_{0}>0$ such that

for every realisation $(X,P,X')$ with $\mathrm{law}(X,P)=\pi$ and $\lVert X'-X\rVert_{2}<r_{0}$. Since the constant $K$ is an upper semicontinuous majorant of $w$, $w\le w^{*}\le K$ by Properties of the Upper Semicontinuous Envelope §bounds and Properties of the Upper Semicontinuous Envelope §least. Put $\kappa=(K-w^{*}(\mu)+1)/r_{0}^{2}+a_{P}/r_{0}>0$.

Step 2 (the functions on plans). Let $\Gamma=\{\gamma\in\Sigma^{2}_{3d}:B_{\#}\gamma=\pi\}$ with the metric $\widehat{W}_{2}$. It is complete: a Cauchy sequence in $\Gamma$ converges in $\Sigma^{2}_{3d}$ (The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It §complete), and its limit lies in $\Gamma$ because $B_{\#}$ is continuous and limits are unique (Uniqueness of Limits in a Metric Space). For $v\in\mathcal{F}$ define $F_{v}:\Gamma\to\mathbb{R}$,

It is upper semicontinuous on $\Gamma$: $v\circ C_{\#}$ is upper semicontinuous as the composite of an upper semicontinuous function with a continuous map, and so is its restriction to $\Gamma$ (Semicontinuity and Continuity Under Composition with a Continuous Map, claims 1 and 4); subtracting the continuous function $p+(\delta+2\eta)s+\kappa s^{2}$ preserves this (Sums and Nonnegative Multiples of Semicontinuous Functions). Let $\gamma\in\Gamma$ with realisation $(X,P,X')$ and $s=s(\gamma)$; then $\mathrm{law}(X,P)=\pi$, $\lVert P\rVert_{2}=a_{P}$ and $v\le w\le w^{*}$. If $s<r_{0}$, (J) gives $F_{v}(\gamma)\le w^{*}(\mu)-\eta s-\kappa s^{2}$. If $s\ge r_{0}$, then $-p(\gamma)\le a_{P}s$ and $\kappa s^{2}\ge(K-w^{*}(\mu)+1)+a_{P}s$ (as $s/r_{0}\ge1$), so $F_{v}(\gamma)\le K+a_{P}s-\kappa s^{2}\le w^{*}(\mu)-1$. Hence

Step 3 (almost maximisers and Ekeland points). Let $j\in\mathbb{N}$, $j\ge2$, and choose $\varepsilon_{j}\in(0,1]$ with $\varepsilon_{j}\bigl(6+2a_{P}+2\delta+4\kappa\bigr)\le j^{-2}$. By Properties of the Upper Semicontinuous Envelope §approximation there is $\nu\in\Sigma^{2}_{d}$ with $\widehat{W}_{2}(\nu,\mu)\le\varepsilon_{j}$ and $w(\nu)>w^{*}(\mu)-\varepsilon_{j}$, and by the definition of $w$ there is $v_{j}\in\mathcal{F}$ with $v_{j}(\nu)>w^{*}(\mu)-2\varepsilon_{j}$. By Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §distance (with its $\varepsilon$ equal to $\varepsilon_{j}^{2}$) there are $L^{2}$ $d$-tuples $Y,Y'$ of one space with $\mathrm{law}(Y)=\mu$, $\mathrm{law}(Y')=\nu$ and $\lVert Y'-Y\rVert_{2}^{2}\le\widehat{W}_{2}(\mu,\nu)^{2}+\varepsilon_{j}^{2}\le2\varepsilon_{j}^{2}$, so $\lVert Y'-Y\rVert_{2}\le2\varepsilon_{j}$. Since $\mathrm{pr}^{1}_{\#}\pi=\mu=\mathrm{pr}^{1}_{\#}\mathrm{law}(Y,Y')$ (Plans at a Square-Integrable Noncommutative Law §plan, Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §coupling), Gluing Two Square-Integrable Noncommutative Laws along a Common Marginal §glue with $k=m=n=d$ gives $L^{2}$ $d$-tuples $X,P,X'$ of one space with $\mathrm{law}(X,P)=\pi$ and $\mathrm{law}(X,X')=\mathrm{law}(Y,Y')$. Then $\gamma^{0}_{j}=\mathrm{law}(X,P,X')\in\Gamma$, $C_{\#}\gamma^{0}_{j}=\nu$, $s(\gamma^{0}_{j})=\lVert Y'-Y\rVert_{2}\le2\varepsilon_{j}$, and

the last step by (B). Ekeland's Variational Principle for Upper Semicontinuous Functions on a Complete Metric Space (with its $\eta=j^{-2}$ and $\kappa=j^{-1}$, applied on the nonempty complete metric space $\Gamma$ to the upper semicontinuous function $F_{v_{j}}$, bounded above by (B)) gives $\gamma_{j}\in\Gamma$ with $F_{v_{j}}(\gamma_{j})\ge F_{v_{j}}(\gamma^{0}_{j})\ge w^{*}(\mu)-j^{-2}$ by Ekeland's Variational Principle for Upper Semicontinuous Functions on a Complete Metric Space §value, and, by Ekeland's Variational Principle for Upper Semicontinuous Functions on a Complete Metric Space §perturbed,

Put $s_{j}=s(\gamma_{j})$. By (B), $\min\{\eta s_{j}+\kappa s_{j}^{2},1\}\le j^{-2}<1$, so $\eta s_{j}\le j^{-2}$ and $s_{j}\to0$. Realise $\gamma_{j}$ by $(X_{j},P_{j},X'_{j})$ and put $\nu_{j}=\mathrm{law}(X'_{j})=C_{\#}\gamma_{j}$; then $\widehat{W}_{2}(\nu_{j},\mu)\le s_{j}$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §lipschitz. Moreover $v_{j}(\nu_{j})=F_{v_{j}}(\gamma_{j})+p(\gamma_{j})+(\delta+2\eta)s_{j}+\kappa s_{j}^{2}\ge w^{*}(\mu)-j^{-2}-a_{P}s_{j}$, while $v_{j}(\nu_{j})\le w^{*}(\nu_{j})$ and $w^{*}$ is upper semicontinuous at $\mu$ (Properties of the Upper Semicontinuous Envelope §usc); hence $v_{j}(\nu_{j})\to w^{*}(\mu)$.

Step 4 (a plan superjet of $v_{j}$). Put $S_{j}=P_{j}+2\kappa(X'_{j}-X_{j})$, $\sigma_{j}=\mathrm{law}(X'_{j},S_{j})$, a plan at $\nu_{j}$, and $\delta_{j}=\delta+2\eta+j^{-1}$. We claim $\sigma_{j}\in J^{+}_{\delta_{j}}v_{j}(\nu_{j})$. Let $\eta'>0$ and $r'=\eta'/\kappa$, and let $Y,S,Y''$ be $L^{2}$ $d$-tuples of a tracial W*-probability space with $\mathrm{law}(Y,S)=\sigma_{j}$ and $\lVert Y''-Y\rVert_{2}<r'$. The laws $\mathrm{law}(X'_{j},S_{j},X_{j},P_{j})\in\Sigma^{2}_{4d}$ and $\mathrm{law}(Y,S,Y'')\in\Sigma^{2}_{3d}$ have the same first-$2d$ marginal $\sigma_{j}$, so Gluing Two Square-Integrable Noncommutative Laws along a Common Marginal §glue (with $k=2d$, $m=2d$, $n=d$) gives $L^{2}$ $d$-tuples $\hat Y,\hat S,\hat X,\hat P,\hat Y''$ of one space with $\mathrm{law}(\hat Y,\hat S,\hat X,\hat P)=\mathrm{law}(X'_{j},S_{j},X_{j},P_{j})$ and $\mathrm{law}(\hat Y,\hat S,\hat Y'')=\mathrm{law}(Y,S,Y'')$. By law invariance: $\lVert\hat S-\hat P-2\kappa(\hat Y-\hat X)\rVert_{2}=\lVert S_{j}-P_{j}-2\kappa(X'_{j}-X_{j})\rVert_{2}=0$, so $\hat S=\hat P+2\kappa(\hat Y-\hat X)$; $\mathrm{law}(\hat X,\hat P,\hat Y)=\gamma_{j}$; and $\gamma''=\mathrm{law}(\hat X,\hat P,\hat Y'')\in\Gamma$ with $\widehat{W}_{2}(\gamma'',\gamma_{j})\le\lVert\hat Y''-\hat Y\rVert_{2}$ (Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §lipschitz). Writing out (E) for $\gamma''$ with these realisations, and using $\lVert\hat Y''-\hat X\rVert_{2}-\lVert\hat Y-\hat X\rVert_{2}\le\lVert\hat Y''-\hat Y\rVert_{2}$ and $\lVert\hat Y''-\hat X\rVert_{2}^{2}-\lVert\hat Y-\hat X\rVert_{2}^{2}=2\langle\hat Y-\hat X,\hat Y''-\hat Y\rangle_{2}+\lVert\hat Y''-\hat Y\rVert_{2}^{2}$,

By law invariance the three quantities $\mathrm{law}(\hat Y'')$, $\langle\hat S,\hat Y''-\hat Y\rangle_{2}$ and $\lVert\hat Y''-\hat Y\rVert_{2}$ equal those for $(Y,S,Y'')$, and $\kappa\lVert Y''-Y\rVert_{2}<\eta'$; so $v_{j}(\mathrm{law}(Y''))\le v_{j}(\nu_{j})+\langle S,Y''-Y\rangle_{2}+(\delta_{j}+\eta')\lVert Y''-Y\rVert_{2}$, which is Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §super.

Step 5 (conclusion). Choose $j\ge2$ with $j^{-1}\le\eta_{f}/3$, $(1+2\kappa)s_{j}+j^{-1}<r_{H}/2$, $s_{j}\le1$, $2\kappa s_{j}\le1$ and $\rho\,|v_{j}(\nu_{j})-w^{*}(\mu)|\le\eta_{f}/3$ (Step 3). Since $v_{j}$ is a subsolution, Test Functions and Plan Jets: Touching Transfers Plan Jets, and the Jet Form of Plan-Jet Viscosity Solutions §sub with $\delta'=\delta_{j}$, $\mu'=\nu_{j}$, the plan $\sigma_{j}$ and the tolerance of that clause equal to $j^{-1}$ gives $L^{2}$ $d$-tuples $Y,S,Q$ of a tracial W*-probability space with $\mathrm{law}(Y,S)=\sigma_{j}$, $\lVert Q\rVert_{2}\le\delta_{j}$ and $\rho\,v_{j}(\nu_{j})+\mathcal{H}(\mathrm{law}(Y,S+Q))\le j^{-1}$. Glue $\mathrm{law}(X'_{j},S_{j},X_{j},P_{j})$ and $\mathrm{law}(Y,S,Q)$ as in Step 4: this gives $\hat Y,\hat S,\hat X,\hat P,\hat Q$ in one space $(H,M,\Omega)$ with $\hat S=\hat P+2\kappa(\hat Y-\hat X)$, $\mathrm{law}(\hat X,\hat P)=\pi$, $\lVert\hat X-\hat Y\rVert_{2}=s_{j}$, $\lVert\hat Q\rVert_{2}\le\delta_{j}$ and $\mathrm{law}(\hat Y,\hat S+\hat Q)=\mathrm{law}(Y,S+Q)$ (law invariance). Let $\tilde Q=\hat Q$ if $\lVert\hat Q\rVert_{2}\le\delta$, and $\tilde Q=(\delta/\lVert\hat Q\rVert_{2})\hat Q$ otherwise; then $\lVert\tilde Q\rVert_{2}\le\delta$ and $\lVert\hat Q-\tilde Q\rVert_{2}\le\max\{0,\lVert\hat Q\rVert_{2}-\delta\}\le2\eta+j^{-1}$. Therefore

and all four tuples $\hat X,\hat P+\tilde Q,\hat Y,\hat S+\hat Q$ have $L^{2}$ norm at most $R$ (their norms are at most $a_{X}$, $a_{P}+\delta$, $a_{X}+1$ and $a_{P}+1+\delta+2+1$). By the choice of $r_{H}$,

using $\mathcal{H}_{M}(\hat Y,\hat S+\hat Q)=\mathcal{H}(\mathrm{law}(Y,S+Q))$. With $X=\hat X$, $P=\hat P$ and $Q=\tilde Q$ this is the required inequality.