The Dirichlet Form of the Noise Gradient on the Gaussian Sobolev Space

In the setting of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, with the Gaussian Sobolev space $\mathbb{D}^{1,2}_{a}$ and the noise gradient $\nabla_{a}F\in L^{2}(\gamma_{c};X^{a})$ of $F\in\mathbb{D}^{1,2}_{a}$; $L^{2}(\gamma_{c};X^{a})$ is a real Hilbert space with inner product $\langle\cdot,\cdot\rangle_{\gamma_{c}}$ by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields.

(The Dirichlet form)¶ The Dirichlet form of the noise gradient is the map $\mathcal{E}_{a}:\mathbb{D}^{1,2}_{a}\times\mathbb{D}^{1,2}_{a}\to\mathbb{R}$,

$$\mathcal{E}_{a}(F,G)=\langle\nabla_{a}F,\nabla_{a}G\rangle_{\gamma_{c}} .$$