Changing the Variances of a Diagonal Gaussian Reference Measure: Equal Domains, the Shift of the Relative Entropy and the Shift of the Relative Score

In the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $c$ and $c'$ be variance vectors, with scaling maps $S_{c}$ and $S_{c'}$, weighted squares $|x|_{c}^{2}$ and $|x|_{c'}^{2}$ ($x\in\mathbb{R}^{d}$) and normalizing constants $Z_{c}$ and $Z_{c'}$, and let $\gamma_{c}$ and $\gamma_{c'}$ be the diagonal Gaussian measures with variances $c$ and $c'$. Finite relative entropy and $H(\cdot\,|\,\cdot)$ are those of that definition on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$; finite Fisher information relative to a diagonal Gaussian measure and the relative scores $\zeta^{c}_{\mu},\zeta^{c'}_{\mu}\in T_{\mu}$ are those of that definition, and for $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ the classes $S_{c},S_{c'}\in L^{2}(\mu;\mathbb{R}^{d})$ are as there; integrable is as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures. Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$.

1. (Equal domains)¶ $\mu$ has finite relative entropy with respect to $\gamma_{c}$ if and only if it has finite relative entropy with respect to $\gamma_{c'}$, and $\mu$ has finite Fisher information relative to $\gamma_{c}$ if and only if it has finite Fisher information relative to $\gamma_{c'}$.

2. (The shift of the relative entropy)¶ The function $x\mapsto|x|_{c'}^{2}-|x|_{c}^{2}$ is integrable with respect to $\mu$, and if $\mu$ has finite relative entropy with respect to $\gamma_{c}$, then

$$H(\mu\,|\,\gamma_{c'})=H(\mu\,|\,\gamma_{c})+\tfrac12\int_{\mathbb{R}^{d}}\bigl(|x|_{c'}^{2}-|x|_{c}^{2}\bigr)\,\mu(dx)+Z_{c'}-Z_{c}.$$

3. (The shift of the relative score)¶ If $\mu$ has finite Fisher information relative to $\gamma_{c}$, so that $\zeta^{c'}_{\mu}$ is defined by clause 1, then $\zeta^{c'}_{\mu}=\zeta^{c}_{\mu}+S_{c'}-S_{c}$ in $L^{2}(\mu;\mathbb{R}^{d})$.