The Entropy Inequality for Nonnegative Functions, and Quadratic Moments Bounded by the Relative Entropy with Respect to a Diagonal Gaussian Measure

In the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation, finite relative entropy and $H(\cdot\,|\,\cdot)$ are those of that definition on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$; Borel and integrable are as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures, a nonnegative Borel function being integrated as a $[0,\infty]$-valued map; $\exp$ is the exponential function and $\log$ the natural logarithm.

1. (Entropy inequality)¶ Let $\gamma,\nu\in\mathcal{P}(\mathbb{R}^{d})$ with $\nu$ of finite relative entropy with respect to $\gamma$, and let $h:\mathbb{R}^{d}\to[0,\infty)$ be Borel with $\exp\circ h$ integrable with respect to $\gamma$. Then $h$ is integrable with respect to $\nu$, $\int_{\mathbb{R}^{d}}\exp\circ h\,d\gamma$ is a positive real number, and

$$\int_{\mathbb{R}^{d}}h\,d\nu\le H(\nu\,|\,\gamma)+\log\int_{\mathbb{R}^{d}}\exp\circ h\,d\gamma .$$

2. (Diagonal quadratic moments)¶ Let $c$ be a variance vector, with greatest variance $c_{\max}$, let $\gamma_{c}$ be the diagonal Gaussian measure with variances $c$, let $w=(w_{1},\dots,w_{d})\in\mathbb{R}^{d}$ with $w_{i}\ge0$ for every $i\in[d]$, and let $t\in\mathbb{R}$ be positive with $4tw_{i}c_{i}\le1$ for every $i\in[d]$. Then for every $\mu\in\mathcal{P}(\mathbb{R}^{d})$ of finite relative entropy with respect to $\gamma_{c}$ the function $x\mapsto\sum_{i=1}^{d}w_{i}x_{i}^{2}$ is integrable with respect to $\mu$ and

$$\int_{\mathbb{R}^{d}}\sum_{i=1}^{d}w_{i}x_{i}^{2}\,\mu(dx)\le\frac{1}{t}\,H(\mu\,|\,\gamma_{c})+2\sum_{i=1}^{d}w_{i}c_{i}.$$

3. (Second moment)¶ Let $c$ and $\gamma_{c}$ be as in clause 2. Every $\mu\in\mathcal{P}(\mathbb{R}^{d})$ of finite relative entropy with respect to $\gamma_{c}$ belongs to $\mathcal{P}_{2}(\mathbb{R}^{d})$, and its second moment satisfies

$$M_{2}(\mu)\le4c_{\max}\,H(\mu\,|\,\gamma_{c})+2\sum_{i=1}^{d}c_{i}.$$