The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $r>0$ be real, let $\Sigma_{d,r}$ be the set of laws with norm bound $r$ of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws, let $\kappa_{d}$ and $\kappa_{2d}$ be the canonical maps of Square-Integrable Noncommutative Laws: Standing Notation §laws, and let $(\Sigma_{d,r},W_{2})$ be the metric space of The Noncommutative Laws with a Norm Bound Form a Complete Bounded Metric Space with Interpolation Points. Let $\mu\in\Sigma_{d,r}$, let $\pi$ be a bounded plan at $\mu$ with momentum norm $|\pi|_{\mathrm{mom}}$, and let $\theta>0$ and $L\ge0$ be real. Let $h:[0,\infty)\to[0,\infty)$ be nondecreasing with $h(0)=0$, and suppose that for every real $s\ge0$ there is a real number $h'(s)$ with $|h'(s)|\le L$ such that for every real $\eta>0$ there is a real $r_{\eta}>0$ with

$$\bigl|h(t)-h(s)-h'(s)(t-s)\bigr|\le\eta\,|t-s|\qquad\text{for every real }t\ge0\text{ with }|t-s|<r_{\eta},$$

that $h'(0)=0$, and that $h'(s)\to0$ as $s\to0$: for every real $\eta>0$ there is a real $r_{h}>0$ with $|h'(s)|<\eta$ whenever $0\le s<r_{h}$. Write $\Phi=\Phi^{h,\theta}_{\kappa_{2d}(\pi)}:\Sigma^{2}_{d}\to\mathbb{R}$ for the coupling test function of $\kappa_{2d}(\pi)$ with gauge $h$ and weight $\theta$, and, for $\lambda\in\Sigma^{2}_{d}$, $\mathcal{C}(\lambda)=\mathcal{C}_{\kappa_{2d}(\pi)}(\lambda)$ for the set of couplings of the plan $\kappa_{2d}(\pi)$ with $\lambda$, with the displacement and momentum pairing $s(\gamma)$ and $p(\gamma)$ of $\gamma\in\mathcal{C}(\lambda)$. For $\nu\in\Sigma_{d,r}$, a maximising coupling for $\nu$ is a $\gamma\in\mathcal{C}(\kappa_{d}(\nu))$ with $\Phi(\kappa_{d}(\nu))=p(\gamma)-h(s(\gamma))-\theta\,s(\gamma)^{2}$.

1. (Maximising couplings)¶ For every $\nu\in\Sigma_{d,r}$ there is a maximising coupling for $\nu$.

2. (Continuity)¶ The function $\nu\mapsto\Phi(\kappa_{d}(\nu))$ on $\Sigma_{d,r}$ is continuous from $(\Sigma_{d,r},W_{2})$ to $\mathbb{R}$ with the metric of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §metrics.

3. (Plan subjets)¶ Let $\nu\in\Sigma_{d,r}$, let $\gamma$ be a maximising coupling for $\nu$, and let $X,P,X'$ be $L^{2}$ $d$-tuples of a tracial W*-probability space $(H,M,\Omega)$ with $\mathrm{law}(X,P,X')=\gamma$, laws, differences and the $L^{2}$ norm being those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples and sums and real multiples those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing. Put $a=X'-X$ and $s=\lVert a\rVert_{2}$, so that $s=s(\gamma)$, and let $S=P-\bigl(\tfrac{h'(s)}{s}+2\theta\bigr)a$ if $s>0$ and $S=P$ if $s=0$. Then $\mathrm{law}(X',S)\in J^{-}\Phi(\kappa_{d}(\nu))$, the plan subjet of $\Phi$ at $\kappa_{d}(\nu)$ with slack $0$, and $\lVert S-P\rVert_{2}\le h'(s)+2\theta s$.

4. (Localisation)¶ Let $r_{0}>0$ be real with $\theta r_{0}\ge2|\pi|_{\mathrm{mom}}$, and suppose that $p(\gamma)\le\frac{1}{2}h(s(\gamma))$ for every $\gamma\in\mathcal{C}(\kappa_{d}(\mu))$ with $s(\gamma)<r_{0}$. Then $\Phi(\kappa_{d}(\mu))=0$.

5. (Convergence)¶ Let $r_{0}>0$ be real with $\theta r_{0}\ge2|\pi|_{\mathrm{mom}}$, suppose that $p(\gamma)\le\frac{1}{2}h(s(\gamma))$ for every $\gamma\in\mathcal{C}(\kappa_{d}(\mu))$ with $s(\gamma)<r_{0}$, let $(\nu_{n})_{n\in\mathbb{N}}$ be a sequence in $\Sigma_{d,r}$ such that the real sequence $(W_{2}(\nu_{n},\mu))_{n\in\mathbb{N}}$ converges to $0$, and for every $n\in\mathbb{N}$ let $\gamma_{n}$ be a maximising coupling for $\nu_{n}$. Then the real sequences $(s(\gamma_{n}))_{n\in\mathbb{N}}$ and $(\Phi(\kappa_{d}(\nu_{n})))_{n\in\mathbb{N}}$ converge to $0$. Moreover, if for every $n\in\mathbb{N}$, $(X_{n},P_{n},X'_{n})$ is a triple of $L^{2}$ $d$-tuples of a tracial W*-probability space $(H_{n},M_{n},\Omega_{n})$ with $\mathrm{law}(X_{n},P_{n},X'_{n})=\gamma_{n}$ and $S_{n}$ is defined from $X_{n},P_{n},X'_{n}$ and $s(\gamma_{n})=\lVert X'_{n}-X_{n}\rVert_{2}$ as $S$ is defined from $X,P,X'$ and $s$ in clause 3, then the real sequence $(\lVert S_{n}-P_{n}\rVert_{2})_{n\in\mathbb{N}}$ converges to $0$.