The Gaussian Log-Sobolev, HWI and Poincare Inequalities in the Noise Geometry, in Entropy Form and in Gross's Functional Form

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 $\mathcal{P}^{a}_{\rho}$, relative entropy, the relative score with respect to $\gamma_{c}$, finite Fisher information relative to $\gamma_{c}$ with weights $a$ and its value $\mathcal{I}_{a}(\cdot\,|\,\gamma_{c})$, as in Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §background; $\mathcal{F}C^{1}_{b}(X)$ and the noise gradient $\nabla_{a}F$ of $F\in\mathcal{F}C^{1}_{b}(X)$ are those of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §cylindrical; and $\operatorname{Ent}_{\gamma_{c}}$ is the entropy with respect to $\gamma_{c}$. Let $\kappa\in\mathbb{R}$ be positive with $c_{k}\le\kappa\,a_{k}$ for every $k\in\mathbb{N}$.

1. (Log-Sobolev inequality, entropy form)¶ Let $\mu\in\mathcal{P}(X)$ have finite relative entropy with respect to $\gamma_{c}$, so that $\mu\in\mathcal{P}_{2}(X)$ by Relative Entropy with Respect to a Diagonal Gaussian Measure on a Hilbert Space: the Moment Bound, the Cutoff Projections, and Bounded, Tight, Weakly Closed, Wasserstein-Closed and Weakly Sequentially Compact Sublevel Sets §moment, and suppose that $\mu$ has a relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$. Then

$$H(\mu\,|\,\gamma_{c})\le\frac{\kappa}{2}\,\mathcal{I}_{a}(\mu\,|\,\gamma_{c}).$$

2. (HWI inequality)¶ For $\mu$ as in claim 1, $\mu$ and $\gamma_{c}$ belong to $\mathcal{P}^{a}_{\rho}$, and, with $\mathcal{I}_{a}(\mu\,|\,\gamma_{c})^{1/2}$ the nonnegative square root,

$$H(\mu\,|\,\gamma_{c})\le\mathcal{I}_{a}(\mu\,|\,\gamma_{c})^{1/2}\,W_{a}(\mu,\gamma_{c})-\frac{1}{2\kappa}\,W_{a}(\mu,\gamma_{c})^{2}.$$

3. (Log-Sobolev inequality, Gross's form)¶ For every $F\in\mathcal{F}C^{1}_{b}(X)$, the function $F^{2}$ is Borel, nonnegative and bounded, $F^{2}$ and $\phi\circ F^{2}$ are integrable with respect to $\gamma_{c}$, where $\phi$ is 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, and

$$\operatorname{Ent}_{\gamma_{c}}(F^{2})\le2\kappa\int_{X}|\nabla_{a}F|_{a}^{2}\,d\gamma_{c}.$$

4. (Poincar'e inequality)¶ For every $F\in\mathcal{F}C^{1}_{b}(X)$,

$$\int_{X}F^{2}\,d\gamma_{c}-\Bigl(\int_{X}F\,d\gamma_{c}\Bigr)^{2}\le\kappa\int_{X}|\nabla_{a}F|_{a}^{2}\,d\gamma_{c}.$$