Proof of The Wall-Confined Free Energy Converges along Wasserstein-Convergent Sequences with Bounded Scores

Each result cited is universally quantified over the data in its own statement. Recall that $\mathcal{D}_{\Xi}\subseteq\mathcal{D}\subseteq\Sigma_{d,R}$ by The Wall-Confined Free Energy and Its Score §score and The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §metric, and that $W_{2}$ is symmetric on $\Sigma_{d,R}$ by The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §symmetry. Since $C\ge\lVert\Xi(\mu_{n})\rVert_{2}\ge0$ for any $n\in\mathbb{N}$, we have $C\ge0$.

Step 1 (a pairing bound). Fix $n\in\mathbb{N}$. Since $\mu_{n},\mu\in\Sigma_{d,R}$, The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §attained (with the laws $\mu_{n}$ and $\mu$) gives an optimal coupling $\gamma_{n}\in\Pi(\mu_{n},\mu)$; it lies in $\Sigma_{2d}$ by Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §bounded-couplings, and its first marginal is $\gamma_{n}\circ\iota^{1}=\mu_{n}$ by Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling. Optimality means $I(\gamma_{n})=W_{2}(\mu_{n},\mu)^{2}$ by The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §optimal. Write $\zeta=\Xi(\mu_{n})\in\mathcal{H}_{\mu_{n}}^{d}$ and $V=V^{1}_{\gamma_{n}}$, the isometry from $\mathcal{H}_{\gamma_{n}\circ\iota^{1}}=\mathcal{H}_{\mu_{n}}$ to $\mathcal{H}_{\gamma_{n}}$ of Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §isometries; by Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §isometry (with the law $\gamma_{n}$ and the tuple $(x_{1},\dots,x_{d})$) it satisfies $V^{*}V=I$, so $\lVert V\zeta_{j}\rVert^{2}=\langle\zeta_{j},V^{*}V\zeta_{j}\rangle=\lVert\zeta_{j}\rVert^{2}$ for $j\in[d]$. By Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §coupling-pairing, $\mathcal{J}^{1}_{\gamma_{n}}(\zeta)=\operatorname{Re}\langle U,G\rangle_{\mathcal{H}_{\gamma_{n}}^{d}}$, where $U=(V\zeta_{1},\dots,V\zeta_{d})$ and $G=(\widehat{h_{1}},\dots,\widehat{h_{d}})$ lie in the complex Hilbert space $\mathcal{H}_{\gamma_{n}}^{d}$, whose inner product is the sum of the entrywise inner products, by Finite Direct Sums of a Complex Hilbert Space: the Hilbert Structure, Coordinate Inclusions, Block Entries of Bounded Operators and Commutation with Diagonal Operators §hilbert. By Cauchy-Schwarz Inequality in a Complex Inner Product Space in $\mathcal{H}_{\gamma_{n}}^{d}$ (with the vectors $U$ and $G$), and $|\operatorname{Re}z|\le|z|$,

$$\bigl|\mathcal{J}^{1}_{\gamma_{n}}(\zeta)\bigr|\le\lVert U\rVert\,\lVert G\rVert.$$

By Finite Direct Sums of a Complex Hilbert Space: the Hilbert Structure, Coordinate Inclusions, Block Entries of Bounded Operators and Commutation with Diagonal Operators §hilbert, $\lVert U\rVert^{2}=\sum_{j=1}^{d}\lVert V\zeta_{j}\rVert^{2}=\sum_{j=1}^{d}\lVert\zeta_{j}\rVert^{2}=\lVert\Xi(\mu_{n})\rVert_{2}^{2}$, the last equality being the definition of the $L^{2}$ norm (Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples). By Finite Direct Sums of a Complex Hilbert Space: the Hilbert Structure, Coordinate Inclusions, Block Entries of Bounded Operators and Commutation with Diagonal Operators §hilbert again ($\lVert G\rVert^{2}=\sum_{j=1}^{d}\lVert\widehat{h_{j}}\rVert^{2}$), The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §isometry (for the form $h_{\gamma_{n}}(p,q)=\gamma_{n}(p^{*}q)$ of The Complex GNS Space of a Tracial State on Noncommutative Polynomials §gns, whose canonical map gives the classes by The Complex GNS Space of a Tracial State on Noncommutative Polynomials §classes), the self-adjointness $h_{j}^{*}=h_{j}$ (the variables being self-adjoint and the adjoint conjugate-linear, Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint) and the linearity of $\gamma_{n}$,

$$\lVert G\rVert^{2}=\sum_{j=1}^{d}\gamma_{n}(h_{j}^{*}h_{j})=\gamma_{n}\Bigl(\sum_{j=1}^{d}(x_{j}-x_{d+j})^{2}\Bigr)=\gamma_{n}(\Delta_{d})=I(\gamma_{n})=W_{2}(\mu_{n},\mu)^{2},$$

using Couplings of Two Noncommutative Laws and Their Quadratic Cost §cost. Taking nonnegative square roots of these two identities gives $\lVert U\rVert=\lVert\Xi(\mu_{n})\rVert_{2}$ and $\lVert G\rVert=W_{2}(\mu_{n},\mu)$ (all four numbers being nonnegative); inserting them into the Cauchy--Schwarz bound above, $|\mathcal{J}^{1}_{\gamma_{n}}(\Xi(\mu_{n}))|\le\lVert\Xi(\mu_{n})\rVert_{2}\,W_{2}(\mu_{n},\mu)\le C\,W_{2}(\mu_{n},\mu)$.

Step 2 (upper bound). For each $n$, Tangent Inequalities for the Wall Energy along Couplings and for the Wall-Confined Free Energy along Optimal Couplings §tangent, applied with its $\mu:=\mu_{n}\in\mathcal{D}_{\Xi}$, its $\nu:=\mu\in\mathcal{D}$ and the optimal coupling $\gamma_{n}\in\Pi(\mu_{n},\mu)$ of Step 1, gives $\mathcal{E}(\mu)\ge\mathcal{E}(\mu_{n})-\mathcal{J}^{1}_{\gamma_{n}}(\Xi(\mu_{n}))$. With Step 1,

$$\mathcal{E}(\mu_{n})\le\mathcal{E}(\mu)+C\,W_{2}(\mu_{n},\mu)\qquad(n\in\mathbb{N}).$$

Step 3 (conclusion). Let $\varepsilon>0$. By Sublevel Sets of the Wall-Confined Free Energy are Closed for the Wasserstein Distance §lsc, $\mathcal{E}$ is lower semicontinuous at $\mu\in\mathcal{D}$ relative to $\mathcal{D}$ in $(\Sigma_{d,R},W_{2})$; by Lower Semicontinuous Function on a Subset of a Metric Space there is a real $\delta>0$ such that every $\lambda\in\mathcal{D}$ with $W_{2}(\mu,\lambda)<\delta$ satisfies $\mathcal{E}(\mu)-\varepsilon<\mathcal{E}(\lambda)$. Put $\varepsilon'=\min\{\delta,\varepsilon/(C+1)\}>0$. Since $(W_{2}(\mu_{n},\mu))_{n}$ converges to $0$ (Limit of a Sequence of Real Numbers, as fixed in The Real Numbers: Standing Notation and Background §sequences), there is $N\in\mathbb{N}$ with $W_{2}(\mu_{n},\mu)<\varepsilon'$ for all $n\ge N$. For such $n$, $W_{2}(\mu,\mu_{n})=W_{2}(\mu_{n},\mu)<\delta$ and $\mu_{n}\in\mathcal{D}$, so $\mathcal{E}(\mu)-\varepsilon<\mathcal{E}(\mu_{n})$; and by Step 2, $\mathcal{E}(\mu_{n})\le\mathcal{E}(\mu)+C\varepsilon/(C+1)<\mathcal{E}(\mu)+\varepsilon$. Hence $|\mathcal{E}(\mu_{n})-\mathcal{E}(\mu)|<\varepsilon$ for all $n\ge N$. As $\varepsilon>0$ was arbitrary, $(\mathcal{E}(\mu_{n}))_{n\in\mathbb{N}}$ converges to $\mathcal{E}(\mu)$ by Limit of a Sequence of Real Numbers.