The Relative Score with Respect to the Gibbs Measure of an Admissible Cylindrical Potential

In the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, let $V$ be an admissible cylindrical potential with head dimension $d$, with the functions $\partial_{k}V$ of Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §gradient, let $\beta\in\mathbb{R}$ be positive, and let $\gamma^{V}_{\beta}$ be the Gibbs measure of $V$ at temperature $\beta$. $\mathcal{F}C^{1}_{b}(X)$ is the set of bounded $C^{1}$ cylindrical functions and $\partial_{k}\varphi$ are their partial derivatives, which exist by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §gradient. Let $\mu\in\mathcal{P}_{2}(X)$, the set of The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space, be such that $\partial_{k}V$ is integrable with respect to $\mu$ for every $k\in[d]$; the notion below is defined only for such $\mu$. Let $L^{2}(\mu)=L^{2}(X,\mathcal{B}(X),\mu)$ be the real Hilbert space of The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §hilbert, with the inner product $\langle\cdot,\cdot\rangle_{L^{2}(\mu)}$ and norm $\lVert\cdot\rVert_{L^{2}(\mu)}$ of The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product. For $\varphi\in\mathcal{F}C^{1}_{b}(X)$ and $k\in\mathbb{N}$, the class of $\varphi$ belongs to $L^{2}(\mu)$ and is again written $\varphi$, and $\partial_{k}\varphi$ is integrable with respect to $\mu$, by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §square-integrable; the function $x\mapsto x_{k}\varphi(x)$ is integrable with respect to $\mu$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §coordinate-integrable; hence, $c_{k}$ being positive by Variance Sequences and Their Truncations §variances, the function $x\mapsto x_{k}\varphi(x)/c_{k}-\partial_{k}\varphi(x)$ is integrable with respect to $\mu$ by Linearity and Monotonicity of the Lebesgue Integral §integrable. Next, $\varphi$ is Borel and bounded by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §bounded-borel, so there is $M\in\mathbb{R}$ with $|\varphi(x)|\le M$ for every $x\in X$; and $\partial_{k}V$ is Borel by Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity and integrable with respect to $\mu$, by hypothesis for $k\le d$ and being $0$ for $k>d$. So $\partial_{k}V\,\varphi$ is Borel with $|\partial_{k}V\,\varphi|\le M|\partial_{k}V|$ on $X$, and is integrable with respect to $\mu$ by the monotonicity of Linearity and Monotonicity of the Lebesgue Integral §nonnegative, applied with $f=|\partial_{k}V\,\varphi|$ and $g=M|\partial_{k}V|$. Hence the function

$$x\mapsto\Bigl(\frac{x_{k}}{c_{k}}+\frac{\partial_{k}V(x)}{\beta}\Bigr)\varphi(x)-\partial_{k}\varphi(x)$$

is integrable with respect to $\mu$ by Linearity and Monotonicity of the Lebesgue Integral §integrable.

(Relative score)¶ The measure $\mu$ has a relative score with respect to $\gamma^{V}_{\beta}$ if for every $k\in\mathbb{N}$ there is $\zeta^{V}_{k}\in L^{2}(\mu)$ with

$$\langle\zeta^{V}_{k},\varphi\rangle_{L^{2}(\mu)}=\int_{X}\Bigl(\Bigl(\frac{x_{k}}{c_{k}}+\frac{\partial_{k}V(x)}{\beta}\Bigr)\varphi(x)-\partial_{k}\varphi(x)\Bigr)\,\mu(dx)\qquad\text{for every }\varphi\in\mathcal{F}C^{1}_{b}(X).$$

For each $k$ there is at most one such $\zeta^{V}_{k}$: the difference of two of them is orthogonal to every $\varphi\in\mathcal{F}C^{1}_{b}(X)$, by linearity of the inner product, and so is the zero vector by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §density. The relative score of $\mu$ with respect to $\gamma^{V}_{\beta}$ is then the sequence $(\zeta^{V}_{k})_{k\in\mathbb{N}}$, and $\zeta^{V}_{k}$ is its $k$-th component.