The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, let $\mathcal{F}C^{1}_{b}(X)$ be the set of bounded $C^{1}$ cylindrical functions, with their representations. Every $\varphi\in\mathcal{F}C^{1}_{b}(X)$ is differentiable on $X$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §gradient, and for $k\in\mathbb{N}$, $\partial_{k}\varphi$ is its $k$-th partial derivative. For $\mu\in\mathcal{P}(X)$, $L^{2}(\mu;X^{a})$ is the real Hilbert space of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, and measurability of maps into $X^{a}$ is that fixed there.

1. (Noise gradient)¶ Let $\varphi\in\mathcal{F}C^{1}_{b}(X)$ have a representation $(n,\psi)$. The noise gradient of $\varphi$ is the map $\nabla_{a}\varphi:X\to X^{a}$,

$$\nabla_{a}\varphi(x)=\sum_{k=1}^{n}a_{k}\,\partial_{k}\varphi(x)\,e_{k}\qquad(x\in X).$$

It does not depend on the representation: 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, $\partial_{k}\varphi(x)=0$ for $k>n$ and every $x\in X$, so for a second representation $(m,\chi)$ the sums over $k\le n$ and over $k\le m$ coincide. Its values lie in $X^{a}$, which is a linear subspace of $X$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert and contains each $e_{k}=a_{k}^{-1/2}f_{k}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §basis. By the orthonormality of $(e_{k})_{k\in\mathbb{N}}$ the $k$-th coordinate of $\nabla_{a}\varphi(x)$ is $a_{k}\,\partial_{k}\varphi(x)$ for $k\le n$ and $0$ for $k>n$, so that, by The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §inner-product,

$$|\nabla_{a}\varphi(x)|_{a}^{2}=\sum_{k=1}^{n}a_{k}\,\bigl(\partial_{k}\varphi(x)\bigr)^{2}\qquad(x\in X).$$

The coordinate function $x\mapsto a_{k}\,\partial_{k}\varphi(x)$ is Borel for $k\le n$, $\partial_{k}\varphi$ being Borel by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §bounded-borel, and the coordinate function for $k>n$, being constant $0$, is Borel; so $\nabla_{a}\varphi$ is measurable into $X^{a}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §measurable, and the function $|\nabla_{a}\varphi|_{a}^{2}:X\to\mathbb{R}$, the finite sum $\sum_{k=1}^{n}a_{k}(\partial_{k}\varphi)^{2}$ of the display, is Borel and nonnegative by Measurable Maps into a Hilbert Space with an Orthonormal Basis: Norms, Inner Products and Linear Combinations, Synthesis from Coordinates, Square-Integrability, and Almost-Everywhere Equality §operations, applied with $E=X^{a}$ and $v=\nabla_{a}\varphi$. By Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §bounded-borel there are also real numbers $B_{k}\ge0$ with $|\partial_{k}\varphi(x)|\le B_{k}$ for $k\le n$ and $x\in X$, so the display gives $|\nabla_{a}\varphi(x)|_{a}^{2}\le B$ for every $x\in X$, with $B=\sum_{k=1}^{n}a_{k}B_{k}^{2}$. Hence for every $\mu\in\mathcal{P}(X)$ the monotonicity of Linearity and Monotonicity of the Lebesgue Integral §nonnegative, applied with $f=|\nabla_{a}\varphi|_{a}^{2}$ and $g$ the constant function $B$, gives $\int_{X}|\nabla_{a}\varphi|_{a}^{2}\,d\mu\le B\,\mu(X)=B<\infty$, and $\nabla_{a}\varphi$ is square-integrable with respect to $\mu$ in the sense of The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §space.

2. (Noise gradients of cylindrical functions)¶ For $\mu\in\mathcal{P}(X)$, $G^{a}_{\mu}$ denotes the set of the classes in $L^{2}(\mu;X^{a})$ of the noise gradients $\nabla_{a}\varphi$ of the functions $\varphi\in\mathcal{F}C^{1}_{b}(X)$.

3. (Noise tangent space)¶ For $\mu\in\mathcal{P}(X)$, the noise tangent space at $\mu$ is the closure of $G^{a}_{\mu}$ in $L^{2}(\mu;X^{a})$,

$$T^{a}_{\mu}=\overline{G^{a}_{\mu}} .$$