The Gibbs Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded $C^2$ Function of Finitely Many Coordinates

In the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, let $V=v\circ p_{d}$ be an admissible cylindrical potential with the functions $\partial_{k}V$ ($k\in\mathbb{N}$) of Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §gradient, let $\beta\in\mathbb{R}$ be positive, 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 $V$ is integrable with respect to $\mu$, let $n\in\mathbb{N}$, and let $g\in C^{2}_{b}(\mathbb{R}^{n})$, the set of Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded with $q=n$, with partial derivatives $\partial_{k}g$ and $\partial_{j}\partial_{k}g$ as in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives. The temperature $\beta$ is fixed together with $V$ throughout this item and is not displayed in the notation below. The positive numbers $c_{k}$ and $a_{k}$ are those of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §gaussian and Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §weights.

For $k\in[n]$, $\partial_{k}g$ is of class $C^{1}$ on $\mathbb{R}^{n}$ by clause 2 of C^k Maps on a Euclidean Open Set, and it is bounded with bounded partial derivatives $\partial_{j}\partial_{k}g$ by Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded; so $\partial_{k}g$ belongs to the set $C^{1}_{b}(\mathbb{R}^{n})$ of Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded, and $(\partial_{k}g)\circ p_{n}$ is a bounded $C^{1}$ cylindrical function. Hence the function $x\mapsto x_{k}\,\partial_{k}g(p_{n}(x))$ 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. The function $\partial_{k}\partial_{k}g$ is bounded by Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded, and it is continuous at every point in the sense of clause 1 of C^k Maps on a Euclidean Open Set, applied to the function $\partial_{k}g$ of class $C^{1}$; since $d_{E}(y,y')=\lVert y-y'\rVert\ge0$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §distance and $\lVert y-y'\rVert^{2}=\sum_{j=1}^{n}(y_{j}-y'_{j})^{2}$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square, the condition $\sum_{j=1}^{n}(y_{j}-y'_{j})^{2}<\delta^{2}$ is equivalent to $d_{E}(y,y')<\delta$, and for real numbers $(s-s')^{2}<\varepsilon^{2}$ is equivalent to $|s-s'|<\varepsilon$, so $\partial_{k}\partial_{k}g$ is continuous from $(\mathbb{R}^{n},d_{E})$ to $\mathbb{R}$ with the absolute-value metric. Hence it is Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space, the Borel $\sigma$-algebra of the real line with that metric being $\mathcal{B}(\mathbb{R})$ by claim 2 of that lemma. The map $p_{n}$ is Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, so the function $x\mapsto\partial_{k}\partial_{k}g(p_{n}(x))$ is Borel by claim 4 of Borel Measurability and Bounded Integration on a Metric Space, and bounded; so it is integrable with respect to the probability measure $\mu$ by claim 6(b) of that lemma.

For the potential term, let $k\in[n]$. The function $\partial_{k}V$ is integrable with respect to $\mu$ by Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §integrable (for $k>d$ it is the zero function), and so measurable by Integrable Function and the Lebesgue Integral. The bounded $C^{1}$ cylindrical function $(\partial_{k}g)\circ p_{n}$ 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; let $M_{k}\in\mathbb{R}$ satisfy $|\partial_{k}g(p_{n}(x))|\le M_{k}$ for every $x\in X$, so that $0\le M_{k}$. The product $x\mapsto\partial_{k}V(x)\,\partial_{k}g(p_{n}(x))$ is measurable by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, its absolute value is measurable by claim 4 of that lemma, and that absolute value is at most $M_{k}\,|\partial_{k}V|$ at every point of $X$; so its integral is at most $M_{k}\int_{X}|\partial_{k}V|\,d\mu<\infty$ by Linearity and Monotonicity of the Lebesgue Integral §nonnegative, and the product is integrable with respect to $\mu$ by the criterion of Integrable Function and the Lebesgue Integral.

(The Gibbs Ornstein-Uhlenbeck functional)¶ The Gibbs Ornstein-Uhlenbeck functional of $\mu$ at $g$, with potential $V$ and temperature $\beta$, is the real number

$$L^{a,V}_{\mu}(g)=\sum_{k=1}^{n}a_{k}\int_{X}\Bigl(\Bigl(\frac{x_{k}}{c_{k}}+\frac{\partial_{k}V(x)}{\beta}\Bigr)\partial_{k}g(p_{n}(x))-\partial_{k}\partial_{k}g(p_{n}(x))\Bigr)\mu(dx),$$

each integrand being integrable with respect to $\mu$ as the linear combination, with coefficients $c_{k}^{-1}$, $\beta^{-1}$ and $-1$, of the three integrable functions above, by Linearity and Monotonicity of the Lebesgue Integral §integrable.