Proof of Test Functions and Plan Jets: Touching Transfers Plan Jets, and the Jet Form of Plan-Jet Viscosity Solutions

Each result cited is universally quantified over the data in its own statement. The metric $\widehat{W}_{2}$ is symmetric, since $(\Sigma^{2}_{d},\widehat{W}_{2})$ is a metric space by Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §metrics.

Clause 1. Let $u-\varphi$ have a local maximum at $\mu$ and let $\pi\in J^{+}_{\delta}\varphi(\mu)$. By Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §superjet, $\pi$ is a plan at $\mu$, so $\mathrm{pr}^{1}_{\#}\pi=\mu$, and $\pi$ is a plan superdifferential of $\varphi$ at $\mu$ with slack $\delta$ in the sense of Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §super. By Local Maximum of a Function Relative to a Subset of a Metric Space there is a real $r_{0}>0$ with $u(\nu)-\varphi(\nu)\le u(\mu)-\varphi(\mu)$ for every $\nu\in\Sigma^{2}_{d}$ with $\widehat{W}_{2}(\mu,\nu)<r_{0}$.

Let $\eta>0$ be real. By Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §super for $\varphi$ there is a real $r_{1}>0$ such that for every tracial W*-probability space $(H,M,\Omega)$ and all $L^{2}$ $d$-tuples $X,P,X'$ of it with $\mathrm{law}(X,P)=\pi$ and $\lVert X'-X\rVert_{2}<r_{1}$,

Let $r=\min\{r_{0},r_{1}\}>0$. Let $(H,M,\Omega)$ be a tracial W*-probability space and $X,P,X'$ be $L^{2}$ $d$-tuples of it with $\mathrm{law}(X,P)=\pi$ and $\lVert X'-X\rVert_{2}<r$. By Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §coupling, $\mathrm{law}(X)=\mathrm{pr}^{1}_{\#}\pi=\mu$. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §lipschitz, $\widehat{W}_{2}(\mu,\mathrm{law}(X'))\le\lVert X'-X\rVert_{2}<r_{0}$. Hence $u(\mathrm{law}(X'))-\varphi(\mathrm{law}(X'))\le u(\mu)-\varphi(\mu)$, which in the notation of the lifts reads $u_{M}(X')\le u(\mu)+\varphi_{M}(X')-\varphi(\mu)$. Since $\lVert X'-X\rVert_{2}<r_{1}$, the display gives

As $\eta$ was arbitrary, $\pi$ is a plan superdifferential of $u$ at $\mu$ with slack $\delta$ (Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §super); being a plan at $\mu$, it lies in $J^{+}_{\delta}u(\mu)$ by Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §superjet.

Clause 2. The argument is that of clause 1 with the inequalities reversed. Let $u-\varphi$ have a local minimum at $\mu$ and $\pi\in J^{-}_{\delta}\varphi(\mu)$. By Local Minimum of a Function Relative to a Subset of a Metric Space there is $r_{0}>0$ with $u(\nu)-\varphi(\nu)\ge u(\mu)-\varphi(\mu)$ whenever $\widehat{W}_{2}(\mu,\nu)<r_{0}$; given $\eta>0$, Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §sub for $\varphi$ gives $r_{1}>0$ with $\varphi_{M}(X')\ge\varphi(\mu)+\langle P,X'-X\rangle_{2}-(\delta+\eta)\lVert X'-X\rVert_{2}$ whenever $\mathrm{law}(X,P)=\pi$ and $\lVert X'-X\rVert_{2}<r_{1}$. For such $X,P,X'$ with $\lVert X'-X\rVert_{2}<\min\{r_{0},r_{1}\}$, the same two citations give $\mathrm{law}(X)=\mu$ and $\widehat{W}_{2}(\mu,\mathrm{law}(X'))<r_{0}$, so $u_{M}(X')\ge u(\mu)+\varphi_{M}(X')-\varphi(\mu)\ge u(\mu)+\langle P,X'-X\rangle_{2}-(\delta+\eta)\lVert X'-X\rVert_{2}$. Hence $\pi\in J^{-}_{\delta}u(\mu)$ by Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §sub and Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §subjet.

Clause 3. Suppose first that $u$ is a plan-jet viscosity subsolution of $(\mathrm{E})$, and let $\delta'\ge0$, $\mu'\in\Sigma^{2}_{d}$, $\pi\in J^{+}_{\delta'}u(\mu')$ and $\eta>0$ be given. The function $u-u$ is identically $0$, so it has a local maximum at $\mu'$ by Local Maximum of a Function Relative to a Subset of a Metric Space (with radius $1$). Applying Plan-Jet Viscosity Subsolutions, Supersolutions and Solutions of the Discounted Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws §sub with $\varphi=u$, $\delta'$, $\mu'$, $\pi$ and $\eta$ yields $(H,M,\Omega)$ and $X,P,Q$ with $\mathrm{law}(X,P)=\pi$, $\lVert Q\rVert_{2}\le\delta'$ and $\rho\,u(\mu')+\mathcal{H}_{M}(X,P+Q)\le\eta$.

Conversely, suppose the condition of clause 3 holds. Let $\delta'\ge0$, $\mu'\in\Sigma^{2}_{d}$, $\varphi:\Sigma^{2}_{d}\to\mathbb{R}$ such that $u-\varphi$ has a local maximum at $\mu'$, $\pi\in J^{+}_{\delta'}\varphi(\mu')$ and $\eta>0$ be given. By clause 1 (applied at $\mu'$ with slack $\delta'$), $\pi\in J^{+}_{\delta'}u(\mu')$, and the condition yields $(H,M,\Omega)$ and $X,P,Q$ with $\mathrm{law}(X,P)=\pi$, $\lVert Q\rVert_{2}\le\delta'$ and $\rho\,u(\mu')+\mathcal{H}_{M}(X,P+Q)\le\eta$. This is the requirement of Plan-Jet Viscosity Subsolutions, Supersolutions and Solutions of the Discounted Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws §sub.

Clause 4. Identical to clause 3, with local minima, $J^{-}$, clause 2 in place of clause 1, the inequality $\rho\,u(\mu')+\mathcal{H}_{M}(X,P+Q)\ge-\eta$, and Plan-Jet Viscosity Subsolutions, Supersolutions and Solutions of the Discounted Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws §super in place of Plan-Jet Viscosity Subsolutions, Supersolutions and Solutions of the Discounted Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws §sub; the function $u-u=0$ has a local minimum at $\mu'$ by Local Minimum of a Function Relative to a Subset of a Metric Space.