Gaussian Smoothing of a Measure with Finite Second Moment: Positive Density, Finite Entropy, a Tangent Score, Non-Increasing Fisher Information and Exponential Integrability

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 spaces $L^{2}(\nu;\mathbb{R}^{d})$ and the tangent spaces $T_{\nu}$; finite entropy and the set $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ are those of that definition, and finite Fisher information, the set $\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ and the score $\xi_{\nu}$ are those of that definition. Lebesgue measure $\lambda_{d}$ is that of Euclidean Space and Lebesgue Measure: Standing Notation §measure, $\bar{B}(x,R)$ is the closed ball of Euclidean Space and Lebesgue Measure: Standing Notation §space, $\log$ is the natural logarithm and $\exp$ the exponential function. For $0<s\le1$ let $g_{s}$ be the Gaussian weight of The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails, read with $q=d$, and for $\nu\in\mathcal{P}(\mathbb{R}^{d})$ let $g_{s}*\nu$ be the Gaussian smoothing of $\nu$ at scale $s$, that lemma also being read with $q=d$. The natural number $d$ is read in $\mathbb{R}$ where a real number is required.

Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $s$ be a real number with $0<s\le\tfrac12$. Write $\rho_{s}=g_{s}*\mu$, which is nonnegative, Borel and of integral $1$ with respect to $\lambda_{d}$ by Gaussian Smoothing of a Probability Measure on Euclidean Space: Regularity, Mass, Duality, Approximation of a Bounded Function with a Modulus of Continuity, and the Pairing Identity §duality, and let $\mu_{s}\in\mathcal{P}(\mathbb{R}^{d})$ be the measure with density $\rho_{s}$ with respect to $\lambda_{d}$.

1. (Density and distance)¶ $\rho_{s}$ is of class $C^{2}$ on $\mathbb{R}^{d}$ and $0<\rho_{s}(y)$ for every $y\in\mathbb{R}^{d}$; moreover $\mu_{s}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $W_{2}(\mu_{s},\mu)^{2}\le d\,s$.

2. (Entropy)¶ $\mu_{s}\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$.

3. (Score)¶ Let $r_{s}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be the map $y\mapsto\rho_{s}(y)^{-1}\nabla\rho_{s}(y)$, with $\nabla\rho_{s}$ the gradient map of $\rho_{s}$. Then $r_{s}$ is Borel with $\int_{\mathbb{R}^{d}}\lVert r_{s}\rVert^{2}\,d\mu_{s}\le d\,s^{-1}$; the function $y\mapsto\tfrac12\lVert y\rVert^{2}+s\log\rho_{s}(y)$ is of class $C^{2}$ and convex on $\mathbb{R}^{d}$; and $\mu_{s}\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$, the class of $r_{s}$ belonging to $T_{\mu_{s}}$ and being equal to $\xi_{\mu_{s}}$.

4. (Fisher information does not increase)¶ If $\mu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$, then $\lVert\xi_{\mu_{s}}\rVert_{\mu_{s}}\le\lVert\xi_{\mu}\rVert_{\mu}$.

5. (Exponential integrability)¶ Suppose that $\mu(\bar{B}(0_{\mathbb{R}^{d}},R))=1$ for some nonnegative real number $R$, and let $h:\mathbb{R}^{d}\to\mathbb{R}$ be Borel with $|h(y)|\le A\exp(B\lVert y\rVert)$ for every $y\in\mathbb{R}^{d}$, where $A$ and $B$ are nonnegative real numbers. Then $h$ is integrable with respect to $\mu_{s}$.