Proof of The Gibbs Entropy Pair of a K-Semiconvex Potential is Lambda-Displacement Convex with Lambda the Temperature over the Variance-to-Noise Bound Minus K

Each result cited is universally quantified over the data in its own statement. Elementary real-number order and arithmetic are carried by The Real Numbers: Standing Notation and Background §background and are not cited step by step.

Let $\gamma^{V}_{\beta}$ be the Gibbs measure of $V$ at temperature $\beta$, with normaliser $Z_{V,\beta}>0$ (The Gibbs Measure of an Admissible Cylindrical Potential Relative to a Diagonal Gaussian Measure §normaliser), and let $(\mathcal{D}^{G},\mathcal{D}^{G}_{\Sigma},\mathcal{E}^{G},\Sigma^{G})$ be the Gaussian entropy pair with temperature $\beta$; its hypothesis holds with the given $\kappa$, since $c_{k}\le\kappa a_{k}$ for every $k$. Fix $\mu\in\mathcal{D}_{\Sigma}$, $\nu\in\mathcal{D}$ and a noise-optimal coupling $\pi\in\Pi^{a}(\mu,\nu)$; write $x=\pi_{1}(z)$, $y=\pi_{2}(z)$ for $z\in X\times X$.

Step 1 (Splitting the penalty). By The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §penalty-domain and The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §score-domain, $\mu,\nu$ have finite relative entropy with respect to $\gamma^{V}_{\beta}$, so by Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §entropy they have finite relative entropy with respect to $\gamma_{c}$, $V$ is integrable with respect to both, and, multiplying the identity there by $\beta$ and using the definition of $\mathcal{E}$ in The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §pair and of $\mathcal{E}^{G}$ in The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §pair,

$$\mathcal{E}(\mu)=\mathcal{E}^{G}(\mu)+\int_{X}V\,d\mu+\beta\log Z_{V,\beta},\qquad\mathcal{E}(\nu)=\mathcal{E}^{G}(\nu)+\int_{X}V\,d\nu+\beta\log Z_{V,\beta}.\tag{1}$$

In particular $\mu,\nu\in\mathcal{D}^{G}$ by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §penalty-domain.

Step 2 (Splitting the score). By The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §score-domain, $\mu$ has a relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$, and $\int_{X}|\nabla_{a}V|_{a}^{2}\,d\mu<\infty$. Hence $\mu\in\mathcal{D}^{G}_{\Sigma}$ by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §score-domain, with $\Sigma^{G}(\mu)=\beta Z^{a}_{\mu}$, and the class of $\nabla_{a}V$ lies in $L^{2}(\mu;X^{a})$ by Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §tangent. Since $\Sigma(\mu)=\beta Z^{a}_{\mu}+\nabla_{a}V$ by The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §pair, linearity of the noise displacement pairing in the field, Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §linear (with $s=t=1$), gives

$$\mathcal{J}^{a}(\Sigma(\mu),\pi)=\mathcal{J}^{a}(\Sigma^{G}(\mu),\pi)+\mathcal{J}^{a}(\nabla_{a}V,\pi).\tag{2}$$

Step 3 (The Gaussian part). $\pi$ is a noise-optimal coupling in $\Pi^{a}(\mu,\nu)$ with $\mu\in\mathcal{D}^{G}_{\Sigma}$ and $\nu\in\mathcal{D}^{G}$, so The Gaussian Entropy Pair is Uniformly Displacement Convex, with Modulus the Temperature over the Variance-to-Noise Bound §convex gives

$$\mathcal{E}^{G}(\mu)+\mathcal{J}^{a}(\Sigma^{G}(\mu),\pi)+\frac{\beta}{2\kappa}I^{a}(\pi)\le\mathcal{E}^{G}(\nu).\tag{3}$$

Step 4 (The potential part). Let $D_{a}$, $c_{a}$ be as in The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §pairs and let $\delta$ be the displacement field of Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §displacement-field for $\pi$, so $\delta(z)=y-x$ on $D_{a}$. Consider the functions on $X\times X$

$$f_{1}(z)=V(x),\qquad f_{2}(z)=V(y),\qquad P(z)=\langle\nabla_{a}V(x),\delta(z)\rangle_{a},\qquad c_{a}(z).$$

$V$ is Borel by Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity and $\pi_{1},\pi_{2}$ are Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §product-sigma; since $(\pi_{1})_{\#}\pi=\mu$ and $(\pi_{2})_{\#}\pi=\nu$ (Couplings of Two Borel Probability Measures on a Hilbert Space and Their Quadratic Cost §coupling), claim 2 of Image Measures, Measures with Densities, and Change of Variables shows that $f_{1},f_{2}$ are integrable with respect to $\pi$, with $\int f_{1}\,d\pi=\int_{X}V\,d\mu$ and $\int f_{2}\,d\pi=\int_{X}V\,d\nu$. By Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §pairing (with $\eta$ the class of $\nabla_{a}V$, represented by $\nabla_{a}V$ itself), $P$ is Borel and integrable with $\int P\,d\pi=\mathcal{J}^{a}(\nabla_{a}V,\pi)$. Since $\pi$ has finite noise cost, $c_{a}$ is Borel, nonnegative and integrable with $\int c_{a}\,d\pi=I^{a}(\pi)$, and $\pi(D_{a})=1$ (Couplings of Finite Noise Cost and Their Noise Cost §finite, Couplings of Finite Noise Cost and Their Noise Cost §cost). Hence, by Linearity and Monotonicity of the Lebesgue Integral §integrable, $A=f_{2}-f_{1}-P+\tfrac{K}{2}c_{a}$ is integrable with

$$\int_{X\times X}A\,d\pi=\int_{X}V\,d\nu-\int_{X}V\,d\mu-\mathcal{J}^{a}(\nabla_{a}V,\pi)+\frac{K}{2}I^{a}(\pi).$$

For $z\in D_{a}$ we have $y-x\in X^{a}$, $\delta(z)=y-x$ and $c_{a}(z)=|y-x|_{a}^{2}$ (Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §noise-space), so Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §tangent-inequality gives $A(z)\ge0$. Thus the negative part $A^{-}=\max(-A,0)$ of Integrable Function and the Lebesgue Integral, a nonnegative measurable function, vanishes off the set $(X\times X)\setminus D_{a}$, which is $\pi$-null as $\pi(D_{a})=1=\pi(X\times X)$; by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-integral, $\int A^{-}\,d\pi=0$, so $\int A\,d\pi=\int A^{+}\,d\pi-\int A^{-}\,d\pi=\int A^{+}\,d\pi\ge0$ (Integrable Function and the Lebesgue Integral). Therefore

$$\int_{X}V\,d\mu+\mathcal{J}^{a}(\nabla_{a}V,\pi)-\frac{K}{2}I^{a}(\pi)\le\int_{X}V\,d\nu.\tag{4}$$

Step 5 (Claim 1). Adding (3), (4) and $\beta\log Z_{V,\beta}$ to both sides, and using (1) and (2),

$$\mathcal{E}(\mu)+\mathcal{J}^{a}(\Sigma(\mu),\pi)+\frac{1}{2}\Bigl(\frac{\beta}{\kappa}-K\Bigr)I^{a}(\pi)\le\mathcal{E}(\nu).$$

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

Step 6 (Claim 2). If $K\le\beta/\kappa$, then $\tfrac12(\tfrac{\beta}{\kappa}-K)I^{a}(\pi)\ge0$, since $I^{a}(\pi)\ge0$ by Couplings of Finite Noise Cost and Their Noise Cost §cost. So the inequality of Step 5 gives $\mathcal{E}(\mu)+\mathcal{J}^{a}(\Sigma(\mu),\pi)+\tfrac{0}{2}I^{a}(\pi)\le\mathcal{E}(\nu)$ for all such $\mu,\nu,\pi$, i.e. the pair is $0$-displacement convex, which is displacement convexity.