Proof of The Gaussian Entropy Pair is Uniformly Displacement Convex, with Modulus the Temperature over the Variance-to-Noise Bound

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

Let $\mu\in\mathcal{D}_{\Sigma}$, let $\nu\in\mathcal{D}$, and let $\pi\in\Pi^{a}(\mu,\nu)$ be a noise-optimal coupling. Write $\langle\cdot,\cdot\rangle_{\mu}$ and $\lVert\cdot\rVert_{\mu}$ for the inner product and norm of $L^{2}(\mu;X^{a})$, a real Hilbert space by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, in particular a real inner product space. By The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §penalty-domain and The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §score-domain, $\mu$ and $\nu$ have finite relative entropy with respect to $\gamma_{c}$ and lie in $\mathcal{P}^{a}_{\rho}$ and in $\mathcal{P}_{2}(X)$, and $\mu$ has a relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$; by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §pair, $\mathcal{E}(\mu)=\beta H(\mu\,|\,\gamma_{c})$, $\mathcal{E}(\nu)=\beta H(\nu\,|\,\gamma_{c})$ and $\Sigma(\mu)=\beta Z^{a}_{\mu}$, with $Z^{a}_{\mu}$ the noise score field of The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency §field.

Step 1 (the noise-optimal map). Since $c_{k}\le\kappa a_{k}$ for every $k\in\mathbb{N}$, the set $\mathcal{D}$ is the set $\mathcal{D}_{H}$ of The Entropy Domain of a Diagonal Gaussian Reference Measure Has the Noise Map Property, so it has the noise map property by The Entropy Domain of a Diagonal Gaussian Reference Measure Has the Noise Map Property §map-property. As $\mu\in\mathcal{D}$ and $\nu\in\mathcal{P}^{a}_{\rho}$, the ordered pair $(\mu,\nu)$ is uniquely noise-mapped by The Noise Map Property of a Set of Probability Measures §map-property, so by Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §uniquely-mapped there is a noise-optimal map $T$ from $\mu$ to $\nu$ such that every noise-optimal coupling of $\mu$ and $\nu$ equals $(\mathrm{id},T)_{\#}\mu$; hence $\pi=(\mathrm{id},T)_{\#}\mu$. By Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §map, $T_{\#}\mu=\nu$ and the map $x\mapsto T(x)-x$ is measurable into $X^{a}$ and square-integrable with respect to $\mu$; let $h=T-\mathrm{id}\in L^{2}(\mu;X^{a})$ be its class. Apply Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §displacement to $h$, with its representative $x\mapsto T(x)-x$ and with $t=1$: the map $S_{t}$ there at $t=1$ is $x\mapsto x+(T(x)-x)=T(x)$, so it equals $T$, it pushes $\mu$ forward to $\nu$, and the coupling $(\mathrm{id},S_{t})_{\#}\mu$ at $t=1$ is $(\mathrm{id},T)_{\#}\mu=\pi$, and that clause gives

$$I^{a}(\pi)=\lVert h\rVert_{\mu}^{2},\qquad\mathcal{J}^{a}(\eta,\pi)=\langle\eta,h\rangle_{\mu}\quad\text{for every }\eta\in L^{2}(\mu;X^{a}).\tag{1}$$

Step 2 (the inequality for the rescaled heads). Fix $n\in\mathbb{N}$; the Euclidean items cited below are read with $d=n$ (each is stated for every dimension $d$), as A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §euclidean does for the items listed there. Since $\mu,\nu\in\mathcal{P}_{2}(X)$, the rescaled heads $\tilde{\mu}_{n}$ and $\tilde{\nu}_{n}$ of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §heads lie in $\mathcal{P}_{2}(\mathbb{R}^{n})$ by Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §head. By Gaussian-Tail Extensions: the Gaussian Reference Measure, Marginals, Densities, Relative Entropy, Relative Score and the Noise Wasserstein Distance §heads, applied to $\mu$ and to $\nu$, both $\tilde{\mu}_{n}$ and $\tilde{\nu}_{n}$ have finite relative entropy with respect to $\tilde{\gamma}_{n}$, the sequence $\bigl(H(\tilde{\mu}_{m}\,|\,\tilde{\gamma}_{m})\bigr)_{m\in\mathbb{N}}$ converges to $H(\mu\,|\,\gamma_{c})$, and $H(\tilde{\nu}_{n}\,|\,\tilde{\gamma}_{n})=H((p_{n})_{\#}\nu\,|\,\gamma_{c^{(n)}})$. As $\gamma_{c^{(n)}}=(p_{n})_{\#}\gamma_{c}$ by Diagonal Gaussian Measures on a Hilbert Space §measure, Relative Entropy of the Finite-Dimensional Projections of Borel Probability Measures on a Hilbert Space: Monotonicity and Approximation §projections gives

$$H(\tilde{\nu}_{n}\,|\,\tilde{\gamma}_{n})\le H(\nu\,|\,\gamma_{c}).\tag{2}$$

Since $\mu$ has a relative score with respect to $\gamma_{c}$, $\tilde{\mu}_{n}$ has finite Fisher information relative to $\tilde{\gamma}_{n}$ by The Relative Score of the Rescaled Head: Projections of the Score, the Ornstein-Uhlenbeck Functional of the Head, and the Converse under Bounded Head Fisher Information §head-score; write $\zeta_{n}=\zeta^{\tilde{c}^{(n)}}_{\tilde{\mu}_{n}}\in L^{2}(\tilde{\mu}_{n};\mathbb{R}^{n})$ for its relative score. Let $(\mathcal{D}_{n},\mathcal{D}_{\Sigma,n},\mathcal{E}_{n},\Sigma_{n})$ be the Gaussian free-energy pair with variances $\tilde{c}^{(n)}$, a variance vector by A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §heads, and temperature $\beta$. By that definition and what was just shown, $\tilde{\nu}_{n}\in\mathcal{D}_{n}$, $\tilde{\mu}_{n}\in\mathcal{D}_{\Sigma,n}$, $\mathcal{E}_{n}(\tilde{\mu}_{n})=\beta H(\tilde{\mu}_{n}\,|\,\tilde{\gamma}_{n})$, $\mathcal{E}_{n}(\tilde{\nu}_{n})=\beta H(\tilde{\nu}_{n}\,|\,\tilde{\gamma}_{n})$ and $\Sigma_{n}(\tilde{\mu}_{n})=\beta\zeta_{n}$.

The set $\mathcal{D}_{n}$ has the map property by The Gaussian Free-Energy Pair is a Penalty Pair: Translations, the First Variation, Lower Semicontinuity, the Moment Bound and the Map Property §maps, so by The Map Property of a Set of Probability Measures §map-property the ordered pair $(\tilde{\mu}_{n},\tilde{\nu}_{n})$ is uniquely mapped, and by Optimal Transport Maps and Uniquely Mapped Pairs of Probability Measures §uniquely-mapped and Optimal Transport Maps and Uniquely Mapped Pairs of Probability Measures §map there is a Borel map $S_{n}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ with $(S_{n})_{\#}\tilde{\mu}_{n}=\tilde{\nu}_{n}$ such that $\pi_{n}=(\mathrm{id},S_{n})_{\#}\tilde{\mu}_{n}$ is an optimal coupling of $\tilde{\mu}_{n}$ and $\tilde{\nu}_{n}$. The maps $S_{n}$ ($n\in\mathbb{N}$) satisfy the hypothesis of Lifted Optimal Maps between Rescaled Heads Converge Strongly to the Noise-Optimal Displacement, which applies to $\mu,\nu\in\mathcal{P}^{a}_{\rho}$, the pair $(\mu,\nu)$ being uniquely noise-mapped with noise-optimal map $T$ by Step 1. By Lifted Optimal Maps between Rescaled Heads Converge Strongly to the Noise-Optimal Displacement §cost, $\int_{\mathbb{R}^{n}}\lVert S_{n}(u)-u\rVert^{2}\,\tilde{\mu}_{n}(du)$ is finite, so The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §displacement, applied with $S=S_{n}$ (for which $\pi_{S}=\pi_{n}$), gives that the class $S_{n}-\mathrm{id}$ lies in $L^{2}(\tilde{\mu}_{n};\mathbb{R}^{n})$ and, with $\mathcal{J}$ and $I$ the displacement pairing and cost of The Intrinsic Calculus on the Wasserstein Space: Standing Notation §couplings,

$$I(\pi_{n})=\lVert S_{n}-\mathrm{id}\rVert_{\tilde{\mu}_{n}}^{2},\qquad\mathcal{J}(\beta\zeta_{n},\pi_{n})=\langle\beta\zeta_{n},S_{n}-\mathrm{id}\rangle_{\tilde{\mu}_{n}}=\beta\,\langle\zeta_{n},S_{n}-\mathrm{id}\rangle_{\tilde{\mu}_{n}} .$$

By The Gaussian Free-Energy Pair is Uniformly Displacement Convex, with Closed Score Along Couplings and Regular Penalised Maxima §convex the pair $(\mathcal{D}_{n},\mathcal{D}_{\Sigma,n},\mathcal{E}_{n},\Sigma_{n})$ is $\frac{\beta}{\tilde{c}^{(n)}_{\max}}$-displacement convex, with $\tilde{c}^{(n)}_{\max}$ the greatest variance of The Diagonal Gaussian Density on Euclidean Space and Its Notation §variances; applying $\lambda$-Displacement Convexity of a Penalty Pair on the Wasserstein Space §convex to $\tilde{\mu}_{n}$, $\tilde{\nu}_{n}$ and $\pi_{n}$,

$$\beta H(\tilde{\mu}_{n}\,|\,\tilde{\gamma}_{n})+\beta\,\langle\zeta_{n},S_{n}-\mathrm{id}\rangle_{\tilde{\mu}_{n}}+\frac{\beta}{2\tilde{c}^{(n)}_{\max}}\,\lVert S_{n}-\mathrm{id}\rVert_{\tilde{\mu}_{n}}^{2}\le\beta H(\tilde{\nu}_{n}\,|\,\tilde{\gamma}_{n}).$$

By The Diagonal Gaussian Density on Euclidean Space and Its Notation §variances there is $j\in[n]$ with $\tilde{c}^{(n)}_{\max}=c_{j}/a_{j}$, and $c_{j}/a_{j}\le\kappa$ because $c_{j}\le\kappa a_{j}$ and $a_{j}>0$; since $\tilde{c}^{(n)}_{\max}>0$, $\beta>0$ and $\lVert S_{n}-\mathrm{id}\rVert_{\tilde{\mu}_{n}}^{2}\ge0$, the coefficient $\frac{\beta}{2\tilde{c}^{(n)}_{\max}}$ may be replaced by the smaller $\frac{\beta}{2\kappa}$ without affecting the inequality.

Step 3 (lifting). Let $u_{n}=\Lambda_{n}\zeta_{n}$ and $v_{n}=\Lambda_{n}(S_{n}-\mathrm{id})$, elements of $L^{2}(\mu;X^{a})$ by Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §lift. That clause gives $\langle\Lambda_{n}\xi,\Lambda_{n}\eta\rangle_{\mu}=\langle\xi,\eta\rangle_{\tilde{\mu}_{n}}$ for all $\xi,\eta\in L^{2}(\tilde{\mu}_{n};\mathbb{R}^{n})$; with $(\xi,\eta)=(\zeta_{n},S_{n}-\mathrm{id})$ and with $\xi=\eta=S_{n}-\mathrm{id}$ it yields $\langle\zeta_{n},S_{n}-\mathrm{id}\rangle_{\tilde{\mu}_{n}}=\langle u_{n},v_{n}\rangle_{\mu}$ and $\lVert S_{n}-\mathrm{id}\rVert_{\tilde{\mu}_{n}}^{2}=\lVert v_{n}\rVert_{\mu}^{2}$. Together with Step 2 and (2), for every $n\in\mathbb{N}$,

$$\beta H(\tilde{\mu}_{n}\,|\,\tilde{\gamma}_{n})+\beta\,\langle u_{n},v_{n}\rangle_{\mu}+\frac{\beta}{2\kappa}\,\lVert v_{n}\rVert_{\mu}^{2}\le\beta H(\nu\,|\,\gamma_{c}).\tag{3}$$

Step 4 (the limit). By The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency §head, $\lVert Z^{a}_{\mu}-u_{n}\rVert_{\mu}^{2}\to0$, hence $\lVert u_{n}-Z^{a}_{\mu}\rVert_{\mu}\to0$ (a nonnegative real whose square is below $\varepsilon^{2}$ is below $\varepsilon$), that is, $(u_{n})$ converges to $Z^{a}_{\mu}$ in the metric of $L^{2}(\mu;X^{a})$; by Lifted Optimal Maps between Rescaled Heads Converge Strongly to the Noise-Optimal Displacement §convergence, $(v_{n})$ converges to $h$ likewise. By The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §continuity, $\langle u_{n},v_{n}\rangle_{\mu}\to\langle Z^{a}_{\mu},h\rangle_{\mu}$ and $\lVert v_{n}\rVert_{\mu}\to\lVert h\rVert_{\mu}$, so $\lVert v_{n}\rVert_{\mu}^{2}\to\lVert h\rVert_{\mu}^{2}$ by the product rule for limits; and $H(\tilde{\mu}_{n}\,|\,\tilde{\gamma}_{n})\to H(\mu\,|\,\gamma_{c})$ by Step 2. By the sum rule and the scalar rule for limits, the left side of (3) converges to $\beta H(\mu\,|\,\gamma_{c})+\beta\langle Z^{a}_{\mu},h\rangle_{\mu}+\frac{\beta}{2\kappa}\lVert h\rVert_{\mu}^{2}$, and the comparison claim (claim 1) of Order Properties of Limits of Real Sequences, applied with the constant right side of (3), gives

$$\beta H(\mu\,|\,\gamma_{c})+\beta\,\langle Z^{a}_{\mu},h\rangle_{\mu}+\frac{\beta}{2\kappa}\,\lVert h\rVert_{\mu}^{2}\le\beta H(\nu\,|\,\gamma_{c}).$$

By (1) with $\eta=\beta Z^{a}_{\mu}=\Sigma(\mu)$ and homogeneity of the inner product, $\mathcal{J}^{a}(\Sigma(\mu),\pi)=\beta\langle Z^{a}_{\mu},h\rangle_{\mu}$, and $I^{a}(\pi)=\lVert h\rVert_{\mu}^{2}$; with the values of $\mathcal{E}(\mu)$ and $\mathcal{E}(\nu)$ recalled above, this is

$$\mathcal{E}(\mu)+\mathcal{J}^{a}(\Sigma(\mu),\pi)+\frac{\beta}{2\kappa}\,I^{a}(\pi)\le\mathcal{E}(\nu).$$

As $\mu$, $\nu$ and $\pi$ were arbitrary, the pair is $\frac{\beta}{\kappa}$-displacement convex in the sense of Lambda-Displacement Convexity of a Noise Penalty Pair §convex (its being a noise penalty pair is The Gaussian Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain §pair). This proves claim 1.

Step 5 (claim 2). With $\mu$, $\nu$, $\pi$ as above, $I^{a}(\pi)\ge0$ by Couplings of Finite Noise Cost and Their Noise Cost §cost and $\frac{\beta}{2\kappa}>0$, so by claim 1

$$\mathcal{E}(\mu)+\mathcal{J}^{a}(\Sigma(\mu),\pi)+\frac{0}{2}\,I^{a}(\pi)=\mathcal{E}(\mu)+\mathcal{J}^{a}(\Sigma(\mu),\pi)\le\mathcal{E}(\mu)+\mathcal{J}^{a}(\Sigma(\mu),\pi)+\frac{\beta}{2\kappa}\,I^{a}(\pi)\le\mathcal{E}(\nu).$$

Hence the pair is $0$-displacement convex, that is, displacement convex in the sense of Lambda-Displacement Convexity of a Noise Penalty Pair §displacement-convex.