The Gibbs Dirichlet Form of the Noise Gradient

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 Gibbs measure $\gamma^{V}_{\beta}$, Gibbs Sobolev space $\mathbb{D}^{1,2}_{a}(\gamma^{V}_{\beta})$ and Gibbs noise gradient $\nabla^{V}_{a}$, and let $\langle\cdot,\cdot\rangle_{\gamma^{V}_{\beta}}$ be the inner product of $L^{2}(\gamma^{V}_{\beta};X^{a})$ as in The Gibbs Sobolev Space of the Noise Gradient.

(The Gibbs Dirichlet form)¶ The Gibbs Dirichlet form is the function $\mathcal{E}^{V}_{a}$ on pairs of elements of $\mathbb{D}^{1,2}_{a}(\gamma^{V}_{\beta})$ given by

$$\mathcal{E}^{V}_{a}(F,G)=\langle\nabla^{V}_{a}F,\nabla^{V}_{a}G\rangle_{\gamma^{V}_{\beta}}.$$