The Gaussian Sobolev Space of the Noise Gradient

In the setting of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, with $\mathcal{F}C^{1}_{b}(X)$ and the noise gradients $\nabla_{a}F$ of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §cylindrical, whose classes lie in $L^{2}(\gamma_{c})$ 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 in $L^{2}(\gamma_{c};X^{a})$ by The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient respectively, and are again written $F$ and $\nabla_{a}F$; $L^{2}(\gamma_{c};X^{a})$ is a real Hilbert space, with inner product $\langle\cdot,\cdot\rangle_{\gamma_{c}}$ and norm $\lVert\cdot\rVert_{\gamma_{c}}$, by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields.

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

2. (The noise gradient)¶ For $F\in\mathbb{D}^{1,2}_{a}$ and an approximating sequence $(F_{j})$, the Cauchy sequence $(\nabla_{a}F_{j})$ converges in the complete space $L^{2}(\gamma_{c};X^{a})$, and its limit, the noise gradient $\nabla_{a}F$ 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 $\lVert F_{j}-F'_{j}\rVert_{2}\le\lVert F_{j}-F\rVert_{2}+\lVert F-F'_{j}\rVert_{2}\to0$, and $\nabla_{a}(F_{j}-F'_{j})=\nabla_{a}F_{j}-\nabla_{a}F'_{j}\to G-G'$. Indeed, 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, and if $F_{j}$, $F'_{j}$ and $F_{j}-F'_{j}$ have representations $(n,\psi)$, $(n',\psi')$ and $(m,\chi)$, all three sums may be taken up to $k\le\max(n,n',m)$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial; and the partial derivatives $\partial_{k}$ are linear: if $\varphi,\varphi':X\to\mathbb{R}$ are differentiable at $x\in X$, then for $z\in X$ the number $(\varphi-\varphi')(x+z)-(\varphi-\varphi')(x)-\langle D\varphi(x)-D\varphi'(x),z\rangle$ is the difference of the two remainders $\varphi(x+z)-\varphi(x)-\langle D\varphi(x),z\rangle$ and $\varphi'(x+z)-\varphi'(x)-\langle D\varphi'(x),z\rangle$ of Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §differentiable, so, given a positive $\varepsilon$, it is at most $\varepsilon|z|$ in absolute value whenever $|z|$ is less than both radii provided there for $\varphi$ and for $\varphi'$ with $\varepsilon/2$ in place of $\varepsilon$; hence $\varphi-\varphi'$ is differentiable at $x$ with gradient $D(\varphi-\varphi')(x)=D\varphi(x)-D\varphi'(x)$ by Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §gradient, and since the partial derivatives of Partial Derivatives along an Orthonormal Basis of a Function Differentiable on a Hilbert Space §partial are the coordinates of the gradient, $\partial_{k}(\varphi-\varphi')=\partial_{k}\varphi-\partial_{k}\varphi'$ for functions differentiable on $X$, as $F_{j}$ and $F'_{j}$ are by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §gradient. So $G=G'$ by Gaussian Integration by Parts for Products of Cylindrical Functions, and Closability of the Noise Gradient in the Gaussian Lebesgue Space §closable. The noise gradient depends only on the class $F$, since approximating sequences are constrained only through $\lVert F_{j}-F\rVert_{2}$. For $F\in\mathcal{F}C^{1}_{b}(X)$ the constant sequence approximates $F$, so $F\in\mathbb{D}^{1,2}_{a}$ and this noise gradient is the class of the noise gradient of The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient.