A Log-Sobolev Inequality for the Galerkin Wick-Ordered $Phi^4$ Gibbs Measure on the Torus at Temperatures above the Semiconvexity Constant

In the setting of The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation, so that $X=H^{-m}(\mathbb{T}^{n})$ with orthonormal basis $(e_{j})_{j\in\mathbb{N}}$, noise weights $a_{j}$, variances $c_{j}$, reference measure $\gamma_{c}$, noise space $X^{a}$ with norm $|\cdot|_{a}$. Relative entropy and $H(\cdot\,|\,\cdot)$ are those of Relative Entropy of Probability Measures §relative-entropy. By White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space §ratio, $c_{j}\le1\cdot a_{j}$ for every $j\in\mathbb{N}$. Let $N\in\mathbb{N}$, let $\varkappa,\mathfrak{m}\in\mathbb{R}$ with $0<\varkappa$, let $V_{N}$ be the Galerkin Wick-ordered $\phi^{4}$ potential with cutoff $N$, coupling $\varkappa$ and mass $\mathfrak{m}$, let $K_{N}=\max\{3\varkappa C_{N}-\mathfrak{m},0\}$ with $C_{N}$ the Wick constant, as in The Galerkin Wick-Ordered Phi^4 Potential on the Torus is an Admissible Cylindrical Potential, with Semiconvexity Constant the Positive Part of Three Times the Coupling Times the Wick Constant Minus the Mass, and let $\beta\in\mathbb{R}$ satisfy $K_{N}<\beta$. $\gamma^{V_{N}}_{\beta}$ is the Gibbs measure of $V_{N}$ at temperature $\beta$, defined since $V_{N}$ is an admissible cylindrical potential by The Galerkin Wick-Ordered Phi^4 Potential on the Torus is an Admissible Cylindrical Potential, with Semiconvexity Constant the Positive Part of Three Times the Coupling Times the Wick Constant Minus the Mass §admissible and $\beta>0$ since $K_{N}\ge0$ by that claim; $\mathcal{I}_{a}(\cdot\,|\,\gamma^{V_{N}}_{\beta})$ is the Fisher information relative to $\gamma^{V_{N}}_{\beta}$ with weights $a$, $\mathcal{F}C^{1}_{b}(X)$ and $\nabla_{a}F$ are the bounded $C^{1}$ cylindrical functions and their noise gradients, and $\operatorname{Ent}_{\gamma^{V_{N}}_{\beta}}$ is the entropy with respect to $\gamma^{V_{N}}_{\beta}$.

1. (Log-Sobolev inequality, entropy form)¶ For every $\mu\in\mathcal{P}(X)$ that has finite relative entropy with respect to $\gamma^{V_{N}}_{\beta}$, a relative score with respect to $\gamma^{V_{N}}_{\beta}$ and finite Fisher information relative to $\gamma^{V_{N}}_{\beta}$ with weights $a$,

$$H(\mu\,|\,\gamma^{V_{N}}_{\beta})\le\frac{\beta}{2(\beta-K_{N})}\,\mathcal{I}_{a}(\mu\,|\,\gamma^{V_{N}}_{\beta}).$$

2. (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 $\gamma^{V_{N}}_{\beta}$, and

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