Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, with the coordinates $x_{k}$, the maps $Q_{n}$ and the spans $X_{n}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, let $\mu$ belong to the set $\mathcal{P}_{2}(X)$ of The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space and let $n\in\mathbb{N}$. Throughout, the dimension written $d$ or $q$ in the Euclidean items cited below is $n$, and the letter $d$ is kept for the distance of $X$. For $k\in\mathbb{N}$, $a_{k}^{1/2}$ is the positive square root of the noise weight $a_{k}$ given by Existence and Uniqueness of the Nonnegative Square Root, and $a_{k}^{-1/2}$ is its multiplicative inverse. The rescaled head map is $r_{n}:X\to\mathbb{R}^{n}$,

$$r_{n}(x)=\bigl(a_{1}^{-1/2}x_{1},\dots,a_{n}^{-1/2}x_{n}\bigr)\qquad(x\in X),$$

and $\tilde{\mu}=(r_{n})_{\#}\mu$ is its push-forward of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward, defined because $r_{n}$ is Borel by claim 1. For a probability measure $\lambda$ on $\mathbb{R}^{n}$, $L^{2}(\lambda;\mathbb{R}^{n})$ is the space of square-integrable vector fields against $\lambda$, with inner product $\langle\cdot,\cdot\rangle_{\lambda}$; for a test function $\psi\in C_{c}^{\infty}(\mathbb{R}^{n})$ the gradient map $\nabla\psi$ is Borel with $\int_{\mathbb{R}^{n}}\lVert\nabla\psi\rVert^{2}\,d\lambda<\infty$ by Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §test; and for $\lambda$ in the set $\mathcal{P}_{2}(\mathbb{R}^{n})$ of The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space, $T_{\lambda}$ is the tangent space at $\lambda$. $L^{2}(\mu;X^{a})$ is the space of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, with inner product $\langle\cdot,\cdot\rangle_{\mu}$; $\mathcal{F}C^{1}_{b}(X)$ is the set of bounded $C^{1}$ cylindrical functions, $\nabla_{a}\varphi$ is the noise gradient of $\varphi\in\mathcal{F}C^{1}_{b}(X)$, and $T^{a}_{\mu}$ is the noise tangent space at $\mu$. For a Borel map $\eta:\mathbb{R}^{n}\to\mathbb{R}^{n}$ with coordinates $\eta_{1},\dots,\eta_{n}$, its lift is the map $\Lambda_{n}\eta:X\to X$,

$$\Lambda_{n}\eta(x)=\sum_{k=1}^{n}a_{k}^{1/2}\,\eta_{k}\bigl(r_{n}(x)\bigr)\,e_{k}\qquad(x\in X).$$

1. (The rescaled head)¶ $r_{n}$ is continuous from $(X,d)$ to $(\mathbb{R}^{n},d_{E})$, hence Borel, and $\tilde{\mu}\in\mathcal{P}_{2}(\mathbb{R}^{n})$.

2. (The lift)¶ Let $\eta:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be Borel with $\int_{\mathbb{R}^{n}}\lVert\eta\rVert^{2}\,d\tilde{\mu}<\infty$. Then $\Lambda_{n}\eta$ is Borel as a map from $X$ to $X$, takes its values in $X_{n}$ and in $X^{a}$, is measurable as a map into $X^{a}$, and satisfies $|\Lambda_{n}\eta(x)|_{a}^{2}=\lVert\eta(r_{n}(x))\rVert^{2}$ for every $x\in X$; in particular it is square-integrable with respect to $\mu$. Its class in $L^{2}(\mu;X^{a})$ depends only on the class of $\eta$ in $L^{2}(\tilde{\mu};\mathbb{R}^{n})$, and the induced map, again written $\Lambda_{n}:L^{2}(\tilde{\mu};\mathbb{R}^{n})\to L^{2}(\mu;X^{a})$, is linear and satisfies

$$\langle\Lambda_{n}\xi,\Lambda_{n}\eta\rangle_{\mu}=\langle\xi,\eta\rangle_{\tilde{\mu}}\qquad\text{for all }\xi,\eta\in L^{2}(\tilde{\mu};\mathbb{R}^{n}).$$

3. (Shifted heads and tails)¶ Let $\eta:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be Borel. Then for every $x\in X$

$$r_{n}\bigl(x+\Lambda_{n}\eta(x)\bigr)=r_{n}(x)+\eta\bigl(r_{n}(x)\bigr),\qquad Q_{n}\bigl(x+\Lambda_{n}\eta(x)\bigr)=Q_{n}x .$$

4. (Gradients)¶ Let $\psi\in C_{c}^{\infty}(\mathbb{R}^{n})$. Then $\psi\circ r_{n}\in\mathcal{F}C^{1}_{b}(X)$ and $\Lambda_{n}(\nabla\psi)(x)=\nabla_{a}(\psi\circ r_{n})(x)$ for every $x\in X$.

5. (Tangent fields)¶ $\Lambda_{n}$ maps $T_{\tilde{\mu}}$ into $T^{a}_{\mu}$: if $\eta\in T_{\tilde{\mu}}$, then $\Lambda_{n}\eta\in T^{a}_{\mu}$.