Basic Properties of the Entropy on the Wasserstein Space: Comparison with the Gaussian Relative Entropy, Lower Bound, Translation Invariance, Absolute Continuity, Closed Sublevel Sets and Lower Semicontinuity

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, with the Wasserstein space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$, the second moment $M_{2}$ and the translations $\tau_{a}$, $a\in\mathbb{R}^{d}$. Finite entropy, the entropy $\mathrm{Ent}$ and the set $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ are those of that definition; finite relative entropy and $H(\cdot\,|\,\cdot)$ are those of that definition on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$; absolute continuity is that of that definition; and lower semicontinuity on $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ is taken relative to that subset of $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$. Let $g_{1}$ and $c_{1}$ be the Gaussian smoothing weight and its normalising constant of The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails in dimension $q=d$ with $s=1$, so that $g_{1}(z)=c_{1}\exp(-\tfrac12\lVert z\rVert^{2})$ and $\int_{\mathbb{R}^{d}}g_{1}\,d\lambda_{d}=1$, with $\lambda_{d}$ Lebesgue measure, and let $\gamma_{d}$ be the measure with density $g_{1}$ with respect to $\lambda_{d}$, so that $\gamma_{d}(\mathbb{R}^{d})=\int_{\mathbb{R}^{d}}g_{1}\,d\lambda_{d}=1$ and $\gamma_{d}\in\mathcal{P}(\mathbb{R}^{d})$; $\log$ is the natural logarithm.

1. (Comparison with the Gaussian)¶ Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. Then $\mu$ has finite entropy if and only if $\mu$ has finite relative entropy with respect to $\gamma_{d}$, and in that case

2. (Lower bound)¶ $\log c_{1}-\tfrac12M_{2}(\mu)\le\mathrm{Ent}(\mu)$ for every $\mu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$.

3. (Translation invariance)¶ Let $\mu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ and $a\in\mathbb{R}^{d}$. Then $(\tau_{a})_{\#}\mu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ and $\mathrm{Ent}((\tau_{a})_{\#}\mu)=\mathrm{Ent}(\mu)$.

4. (Absolute continuity)¶ Every $\mu\in\mathcal{P}(\mathbb{R}^{d})$ with finite entropy is absolutely continuous.

5. (Closed sublevel sets)¶ Let $c\in\mathbb{R}$, let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, and let $(\mu_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ with $\mathrm{Ent}(\mu_{n})\le c$ for every $n\in\mathbb{N}$ and $\lim_{n\to\infty}W_{2}(\mu_{n},\mu)=0$. Then $\mu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ and $\mathrm{Ent}(\mu)\le c$.

6. (Lower semicontinuity)¶ $\mathrm{Ent}$ is lower semicontinuous on $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$.