A Log-Sobolev Inequality Survives a Bounded Perturbation of the Measure (Holley-Stroock)

In the setting of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, with $\mathcal{F}C^{1}_{b}(X)$ and the noise gradients $\nabla_{a}F$ of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §cylindrical, for which $|\nabla_{a}F|_{a}^{2}$ is a bounded nonnegative Borel function by The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient; for $\nu\in\mathcal{P}(X)$, $\operatorname{Ent}_{\nu}$ is the entropy with respect to $\nu$, and $\phi$ 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. $\exp$ is the exponential function. For every $F\in\mathcal{F}C^{1}_{b}(X)$ and every $\theta\in\mathcal{P}(X)$, the functions $F^{2}$ and $\phi\circ F^{2}$ are bounded and Borel, hence integrable with respect to $\theta$, so that $\operatorname{Ent}_{\theta}(F^{2})$ is defined: $F$ is bounded and Borel by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §bounded-borel, so $F^{2}$ is Borel by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and its values lie in an interval $[0,b]$; $\phi$ is continuous by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §continuous, so $\phi\circ F^{2}$ is Borel by that clause and bounded on $[0,b]$ by Extreme Value Theorem on a Closed Real Interval; and bounded Borel functions are integrable with respect to a probability measure by claim 6 of Borel Measurability and Bounded Integration on a Metric Space.

Let $\mu\in\mathcal{P}(X)$ and let $C\in\mathbb{R}$ with $0\le C$ be such that

$$\operatorname{Ent}_{\mu}(F^{2})\le2C\int_{X}|\nabla_{a}F|_{a}^{2}\,d\mu\qquad\text{for every }F\in\mathcal{F}C^{1}_{b}(X).$$

Let $U:X\to\mathbb{R}$ be Borel and bounded, and let $u_{-},u_{+}\in\mathbb{R}$ satisfy $u_{-}\le U(x)\le u_{+}$ for every $x\in X$.

1. (The perturbed measure)¶ The function $\exp(-U)$ is Borel and bounded, hence integrable with respect to $\mu$; the number $Z=\int_{X}\exp(-U)\,d\mu$ satisfies $\exp(-u_{+})\le Z\le\exp(-u_{-})$, so $Z>0$; the function $\nu$ on $\mathcal{B}(X)$ given by $\nu(A)=Z^{-1}\int_{X}\mathbf{1}_{A}\exp(-U)\,d\mu$ belongs to $\mathcal{P}(X)$; and for every bounded Borel $f:X\to\mathbb{R}$, $\int_{X}f\,d\nu=Z^{-1}\int_{X}f\exp(-U)\,d\mu$.

2. (Perturbed log-Sobolev inequality)¶ For every $F\in\mathcal{F}C^{1}_{b}(X)$,

$$\operatorname{Ent}_{\nu}(F^{2})\le2C\exp(u_{+}-u_{-})\int_{X}|\nabla_{a}F|_{a}^{2}\,d\nu .$$