The Gibbs Sobolev Space: Linearity, Closedness, the Chain Rule, and the Score of a Measure of Finite Intrinsic Fisher Information

In the settings of The Real Numbers: Standing Notation and Background and Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, let $\beta,q\in\mathbb{R}$ satisfy $0<\beta$ and $1<q$, let $V$ be a Gibbs-Sobolev potential at temperature $\beta$ with exponent $q$, with partial derivatives $\partial_{k}V$, Gibbs measure $\gamma^{V}_{\beta}$, Gibbs Sobolev space $\mathbb{D}^{1,2}_{a}(\gamma^{V}_{\beta})$ and Gibbs noise gradient $\nabla^{V}_{a}$, and with the spaces $L^{2}(\gamma^{V}_{\beta})$ and $L^{2}(\gamma^{V}_{\beta};X^{a})$ and their norms as in The Gibbs Sobolev Space of the Noise Gradient. Finite intrinsic Fisher information relative to $\gamma^{V}_{\beta}$ with weights $a$ and its value $\mathcal{I}^{V}_{a}$ are those of The Intrinsic Fisher Information Relative to the Gibbs Measure of a Gibbs-Sobolev Potential. $\mathcal{F}C^{2}_{b}(X)$ and partial derivatives are those of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §cylindrical; for a measure $\mu\in\mathcal{P}(X)$, $L^{2}(\mu;X^{a})$ is the space of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, with inner product $\langle\cdot,\cdot\rangle_{\mu}$ and norm $\lVert\cdot\rVert_{\mu}$.

1. (Linearity)¶ $\mathbb{D}^{1,2}_{a}(\gamma^{V}_{\beta})$ is a linear subspace of $L^{2}(\gamma^{V}_{\beta})$ containing the constant functions, and $\nabla^{V}_{a}$ is linear on it and vanishes on the constant functions.

2. (Closedness)¶ Let $F_{j}\in\mathbb{D}^{1,2}_{a}(\gamma^{V}_{\beta})$ ($j\in\mathbb{N}$), $F\in L^{2}(\gamma^{V}_{\beta})$ and $G\in L^{2}(\gamma^{V}_{\beta};X^{a})$ satisfy $\lVert F_{j}-F\rVert_{L^{2}(\gamma^{V}_{\beta})}\to0$ and $\lVert\nabla^{V}_{a}F_{j}-G\rVert_{\gamma^{V}_{\beta}}\to0$. Then $F\in\mathbb{D}^{1,2}_{a}(\gamma^{V}_{\beta})$ and $\nabla^{V}_{a}F=G$.

3. (Chain rule)¶ Let $h:\mathbb{R}\to\mathbb{R}$ be of class $C^{1}$ on $\mathbb{R}=\mathbb{R}^{1}$ in the sense of C^k Maps on a Euclidean Open Set, with derivative $h'=\partial_{1}h$ bounded. Let $F\in\mathbb{D}^{1,2}_{a}(\gamma^{V}_{\beta})$, let $f$ be a Borel function in the class $F$ and $g$ a map in the class $\nabla^{V}_{a}F$. Then $h\circ f$ is Borel and square-integrable with respect to $\gamma^{V}_{\beta}$, its class lies in $\mathbb{D}^{1,2}_{a}(\gamma^{V}_{\beta})$, and $\nabla^{V}_{a}(h\circ f)$ is the class of $x\mapsto h'(f(x))\,g(x)$.

4. (The Gibbs measure)¶ $\gamma^{V}_{\beta}$ has finite intrinsic Fisher information relative to $\gamma^{V}_{\beta}$ with weights $a$, and $\mathcal{I}^{V}_{a}(\gamma^{V}_{\beta})=0$.

For claims 5 to 7, let $\mu\in\mathcal{P}(X)$ have finite intrinsic Fisher information relative to $\gamma^{V}_{\beta}$ with weights $a$, let $f$ be a density of $\mu$ relative to $\gamma^{V}_{\beta}$ such that the class of $\sqrt{f}$ lies in $\mathbb{D}^{1,2}_{a}(\gamma^{V}_{\beta})$, let $g$ be a map in the class $\nabla^{V}_{a}\sqrt{f}$, and let $\zeta:X\to X^{a}$ be given by $\zeta(x)=2g(x)/\sqrt{f(x)}$ if $f(x)>0$ and $\zeta(x)=0$ if $f(x)=0$.

5. (The score field)¶ The class of $\zeta$ lies in $L^{2}(\mu;X^{a})$ and does not depend on the choices of $f$ and $g$.

6. (Its norm)¶ $\lVert\zeta\rVert_{\mu}^{2}=\mathcal{I}^{V}_{a}(\mu)$.

7. (Integration by parts)¶ Suppose moreover that $\mu\in\mathcal{P}_{2}(X)$ and that $\partial_{k}V$ is integrable with respect to $\mu$ for every $k\in\mathbb{N}$. Then for every $\psi\in\mathcal{F}C^{2}_{b}(X)$ with a representation $\psi=\tilde\psi\circ p_{n}$ ($n\in\mathbb{N}$), the functions $x\mapsto\bigl(x_{k}/c_{k}+\partial_{k}V(x)/\beta\bigr)\partial_{k}\psi(x)-\partial_{k}\partial_{k}\psi(x)$ ($k\in[n]$) are integrable with respect to $\mu$, and

$$\langle\zeta,\nabla_{a}\psi\rangle_{\mu}=\sum_{k=1}^{n}a_{k}\int_{X}\Bigl(\Bigl(\frac{x_{k}}{c_{k}}+\frac{\partial_{k}V(x)}{\beta}\Bigr)\partial_{k}\psi(x)-\partial_{k}\partial_{k}\psi(x)\Bigr)\mu(dx).$$