Log-Sobolev and Talagrand Inequalities for the Gibbs Measure of a Semiconvex Cylindrical Potential below the Critical Curvature (Bakry-Emery)

In the setting of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, with the noise Wasserstein distance $W_{a}$ on $\mathcal{P}^{a}_{\rho}$ and relative entropy $H(\cdot\,|\,\cdot)$ of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §background, and $\mathcal{F}C^{1}_{b}(X)$ and the noise gradients $\nabla_{a}F$ of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §cylindrical. Let $\kappa,\beta\in\mathbb{R}$ be positive with $c_{k}\le\kappa\,a_{k}$ for every $k\in\mathbb{N}$, let $K\in\mathbb{R}$ satisfy $0\le K$ and $\kappa K<\beta$, and let $V$ be an admissible cylindrical potential with semiconvexity constant $K$. $\gamma^{V}_{\beta}$ is the Gibbs measure of $V$ at temperature $\beta$; the relative score with respect to $\gamma^{V}_{\beta}$ and finite Fisher information relative to $\gamma^{V}_{\beta}$ with weights $a$, with value $\mathcal{I}_{a}(\cdot\,|\,\gamma^{V}_{\beta})$, are those of the cited definitions, applicable to every $\mu$ of finite relative entropy with respect to $\gamma^{V}_{\beta}$ by Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §domain; and $\operatorname{Ent}_{\gamma^{V}_{\beta}}$ is the entropy with respect to $\gamma^{V}_{\beta}$.

1. (Log-Sobolev inequality, entropy form)¶ Let $\mu\in\mathcal{P}(X)$ have finite relative entropy with respect to $\gamma^{V}_{\beta}$, a relative score with respect to $\gamma^{V}_{\beta}$, and finite Fisher information relative to $\gamma^{V}_{\beta}$ with weights $a$. Then

$$H(\mu\,|\,\gamma^{V}_{\beta})\le\frac{\kappa\beta}{2(\beta-\kappa K)}\,\mathcal{I}_{a}(\mu\,|\,\gamma^{V}_{\beta}).$$

2. (Talagrand inequality)¶ Let $\mu\in\mathcal{P}(X)$ have finite relative entropy with respect to $\gamma^{V}_{\beta}$. Then $\mu$ and $\gamma^{V}_{\beta}$ belong to $\mathcal{P}^{a}_{\rho}$, and

$$W_{a}(\mu,\gamma^{V}_{\beta})^{2}\le\frac{2\kappa\beta}{\beta-\kappa K}\,H(\mu\,|\,\gamma^{V}_{\beta}).$$

3. (Log-Sobolev inequality, Gross's form)¶ For every $F\in\mathcal{F}C^{1}_{b}(X)$, the functions $F^{2}$ and $\phi\circ F^{2}$, with $\phi$ the function of The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm, are bounded and Borel, hence integrable with respect to $\gamma^{V}_{\beta}$, and

$$\operatorname{Ent}_{\gamma^{V}_{\beta}}(F^{2})\le\frac{2\kappa\beta}{\beta-\kappa K}\int_{X}|\nabla_{a}F|_{a}^{2}\,d\gamma^{V}_{\beta}.$$