The Gibbs Sobolev Space 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 $q$, and let $\gamma^{V}_{\beta}$ be its Gibbs measure, which is defined and lies in $\mathcal{P}_{2}(X)$ by Integration by Parts for the Gibbs Measure of a Gibbs-Sobolev Potential, and Closability of the Noise Gradient in Its Square-Integrable Space §measure. $L^{2}(\gamma^{V}_{\beta})=L^{2}(X,\mathcal{B}(X),\gamma^{V}_{\beta})$ is the real Hilbert space of The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §hilbert, with the norm $\lVert\cdot\rVert_{L^{2}(\gamma^{V}_{\beta})}$ of The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product, and $L^{2}(\gamma^{V}_{\beta};X^{a})$ is the real Hilbert space of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, with inner product $\langle\cdot,\cdot\rangle_{\gamma^{V}_{\beta}}$ and norm $\lVert\cdot\rVert_{\gamma^{V}_{\beta}}$. For $\varphi\in\mathcal{F}C^{1}_{b}(X)$, the set of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §cylindrical, the classes of $\varphi$ and of its noise gradient $\nabla_{a}\varphi$ lie in these spaces 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 and The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient, and are again written $\varphi$ and $\nabla_{a}\varphi$.

1. (The Gibbs Sobolev space)¶ $\mathbb{D}^{1,2}_{a}(\gamma^{V}_{\beta})$ is the set of those $F\in L^{2}(\gamma^{V}_{\beta})$ for which there is a sequence $(F_{j})_{j\in\mathbb{N}}$ in $\mathcal{F}C^{1}_{b}(X)$ such that $(\lVert F_{j}-F\rVert_{L^{2}(\gamma^{V}_{\beta})})_{j\in\mathbb{N}}$ converges to $0$ and $(\nabla_{a}F_{j})_{j\in\mathbb{N}}$ is a Cauchy sequence in $L^{2}(\gamma^{V}_{\beta};X^{a})$; such a sequence approximates $F$.

2. (The Gibbs noise gradient)¶ For $F\in\mathbb{D}^{1,2}_{a}(\gamma^{V}_{\beta})$ and an approximating sequence $(F_{j})$, the Cauchy sequence $(\nabla_{a}F_{j})$ converges in the complete space $L^{2}(\gamma^{V}_{\beta};X^{a})$, and its limit, the Gibbs noise gradient $\nabla^{V}_{a}F$ of $F$, does not depend on the approximating sequence: for a second approximating sequence $(F'_{j})$ with $\nabla_{a}F'_{j}\to G'$ and $\nabla_{a}F_{j}\to G$, the functions $F_{j}-F'_{j}$ lie in $\mathcal{F}C^{1}_{b}(X)$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §linear and satisfy $\lVert F_{j}-F'_{j}\rVert_{L^{2}(\gamma^{V}_{\beta})}\le\lVert F_{j}-F\rVert_{L^{2}(\gamma^{V}_{\beta})}+\lVert F-F'_{j}\rVert_{L^{2}(\gamma^{V}_{\beta})}\to0$, and $\nabla_{a}(F_{j}-F'_{j})=\nabla_{a}F_{j}-\nabla_{a}F'_{j}\to G-G'$, because the partial derivatives $\partial_{k}$ are linear on $\mathcal{F}C^{1}_{b}(X)$ (they are given by the gradient of Partial Derivatives along an Orthonormal Basis of a Function Differentiable on a Hilbert Space §partial, which is linear in the function) and the noise gradient is the finite sum displayed in The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient; so $G=G'$ by Integration by Parts for the Gibbs Measure of a Gibbs-Sobolev Potential, and Closability of the Noise Gradient in Its Square-Integrable Space §closable. For $\varphi\in\mathcal{F}C^{1}_{b}(X)$ the constant sequence approximates $\varphi$, so $\varphi\in\mathbb{D}^{1,2}_{a}(\gamma^{V}_{\beta})$ and $\nabla^{V}_{a}\varphi=\nabla_{a}\varphi$.