Integration by Parts for the Gibbs Measure of a Gibbs-Sobolev Potential, and Closability of the Noise Gradient in Its Square-Integrable Space

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 $1<q$, and let $V$ be a Gibbs-Sobolev potential at temperature $\beta$ with exponent $q$, with partial derivatives $\partial_{k}V$ ($k\in\mathbb{N}$). $\mathcal{F}C^{1}_{b}(X)$, the partial derivatives $\partial_{k}\varphi$ and the noise gradients $\nabla_{a}\varphi$ of its members are those of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §cylindrical.

1. (The Gibbs measure)¶ The function $\exp(-V/\beta)$, with $\exp$ the exponential function, is Borel and integrable with respect to $\gamma_{c}$, so the Gibbs measure $\gamma^{V}_{\beta}$ of $V$ at temperature $\beta$ is defined; and $\gamma^{V}_{\beta}\in\mathcal{P}_{2}(X)$.

2. (Square-integrability)¶ For every $k\in\mathbb{N}$ the coordinate $x\mapsto x_{k}$ and $\partial_{k}V$ are square-integrable with respect to $\gamma^{V}_{\beta}$.

3. (Integration by parts)¶ For every $\varphi\in\mathcal{F}C^{1}_{b}(X)$ and $k\in\mathbb{N}$, the function $x\mapsto\varphi(x)\bigl(x_{k}/c_{k}+\partial_{k}V(x)/\beta\bigr)$ is integrable with respect to $\gamma^{V}_{\beta}$, and

$$\int_{X}\partial_{k}\varphi\,d\gamma^{V}_{\beta}=\int_{X}\varphi(x)\Bigl(\frac{x_{k}}{c_{k}}+\frac{\partial_{k}V(x)}{\beta}\Bigr)\gamma^{V}_{\beta}(dx).$$

4. (Closability)¶ Let $L^{2}(\gamma^{V}_{\beta};X^{a})$ be the real Hilbert space of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields for the measure $\gamma^{V}_{\beta}$, with norm $\lVert\cdot\rVert_{\gamma^{V}_{\beta}}$. If $\varphi_{j}\in\mathcal{F}C^{1}_{b}(X)$ ($j\in\mathbb{N}$) and $G\in L^{2}(\gamma^{V}_{\beta};X^{a})$ satisfy $\int_{X}\varphi_{j}^{2}\,d\gamma^{V}_{\beta}\to0$ and $\lVert\nabla_{a}\varphi_{j}-G\rVert_{\gamma^{V}_{\beta}}\to0$ as $j\to\infty$, then $G=0$.