Proof of Gaussian Integration by Parts for Products of Cylindrical Functions, and Closability of the Noise Gradient in the Gaussian Lebesgue Space

Each result cited is universally quantified over the data in its own statement. Elementary real arithmetic and order (The Real Numbers: Standing Notation and Background §background) are used without citation. Integrals and integrability are those of Measure Spaces and the Lebesgue Integral: Standing Notation; linearity of the integral for integrable functions is Linearity and Monotonicity of the Lebesgue Integral §integrable, and monotonicity for nonnegative measurable functions is Linearity and Monotonicity of the Lebesgue Integral §nonnegative. The variances $c_{k}$ are positive by Variance Sequences and Their Truncations §variances. Throughout, $\langle\cdot,\cdot\rangle$ and $\lVert\cdot\rVert$ denote the inner product and norm of $L^{2}(\gamma_{c})$; by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product, $\langle f,g\rangle=\int_{X}fg\,d\gamma_{c}$ for $2$-integrable $f,g$ (the product being integrable) and $\lVert f\rVert=\lVert f\rVert_{2}$.

Step 1 (Calculus of profiles). For $N\in\mathbb{N}$ the set $\mathbb{R}^{N}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, and the class $C^{1}$ and the partial derivatives on it are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, that is, of C^k Maps on a Euclidean Open Set, which are the notions of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set and A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$.

(a) Sums and products. Let $N\in\mathbb{N}$, let $f,g:\mathbb{R}^{N}\to\mathbb{R}$ be of class $C^{1}$ on $\mathbb{R}^{N}$, and let $t\in\mathbb{R}$. By claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, applied to $g$ and the scalar $t$, then to $f$ and $tg$, and to $f$ and $g$, the functions $f+tg$ and $fg$ are of class $C^{1}$ on $\mathbb{R}^{N}$. The partial derivatives of $f$ and $g$ exist at every point by clause 1 of C^k Maps on a Euclidean Open Set, read through its clause 3, so claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, applied in the same way, gives for $u\in\mathbb{R}^{N}$ and $i\in[N]$

$$\partial_{i}(f+tg)(u)=\partial_{i}f(u)+t\,\partial_{i}g(u),\qquad\partial_{i}(fg)(u)=\partial_{i}f(u)\,g(u)+f(u)\,\partial_{i}g(u).$$

If moreover $f$, $g$ and all $\partial_{i}f$, $\partial_{i}g$ are bounded, then so are $f+tg$, $fg$ and their partial derivatives, by the displayed formulas; thus the set $C^{1}_{b}(\mathbb{R}^{N})$ of Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded is closed under sums, real multiples and pointwise products.

(b) Lifting. Let $n,N\in\mathbb{N}$ with $n\le N$, let $\pi:\mathbb{R}^{N}\to\mathbb{R}^{n}$, $\pi(u)=(u_{1},\dots,u_{n})$, and let $\psi:\mathbb{R}^{n}\to\mathbb{R}$ be of class $C^{1}$ on $\mathbb{R}^{n}$. Put $\tilde{\psi}=\psi\circ\pi$. For $u\in\mathbb{R}^{N}$ and $i\le n$, the slice function of $\tilde{\psi}$ at $u$ in the $i$-th variable is the slice function of $\psi$ at $\pi(u)$ in the $i$-th variable (both taken on a common interval, any positive radius being admissible in claim 1 of Slice Function and the Partial Derivative, the domains being whole Euclidean spaces), because $\pi$ maps the point $u[s]$ (the $i$-th coordinate of $u$ replaced by $s$) to $\pi(u)[s]$; for $n<i\le N$ it is constant, because $\pi(u[s])=\pi(u)$. By claim 2 of Slice Function and the Partial Derivative and claim 1 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, $\partial_{i}\tilde{\psi}(u)=\partial_{i}\psi(\pi(u))$ for $i\le n$ and $\partial_{i}\tilde{\psi}(u)=0$ for $n<i\le N$. The coordinate functions of $\pi$ are the coordinate functions $u\mapsto u_{l}$ ($l\in[n]$) of $\mathbb{R}^{N}$, which are smooth, hence of class $C^{1}$, on $\mathbb{R}^{N}$ by claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set and Smooth Map on a Euclidean Open Set; so $\pi$ is of class $C^{1}$ on $\mathbb{R}^{N}$ by clauses 1 and 3 of C^k Maps on a Euclidean Open Set, and $\tilde{\psi}$ is of class $C^{1}$ on $\mathbb{R}^{N}$ by claim 2 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$, applied with $F=\pi$, the open set $\mathbb{R}^{n}$ in place of $V$, and $G=\psi$. If $\psi\in C^{1}_{b}(\mathbb{R}^{n})$, then $\tilde{\psi}$ and its partial derivatives are bounded by the formulas just obtained, so $\tilde{\psi}\in C^{1}_{b}(\mathbb{R}^{N})$. Since $p_{n}(x)=(x_{1},\dots,x_{n})=\pi(p_{N}(x))$ by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, $\psi\circ p_{n}=\tilde{\psi}\circ p_{N}$. Consequently, if $(n,\psi)$ is a representation of $\varphi\in\mathcal{F}C^{1}_{b}(X)$ and $N\ge n$, then $(N,\tilde{\psi})$ is a representation of $\varphi$.

Step 2 (Products, and coordinates of noise gradients). (a) Let $F,\varphi\in\mathcal{F}C^{1}_{b}(X)$ have representations $(n_{1},\psi_{1})$ and $(n_{2},\psi_{2})$, and let $N=\max\{n_{1},n_{2}\}$. By Step 1(b) they have representations $(N,\tilde{\psi}_{1})$ and $(N,\tilde{\psi}_{2})$, and by Step 1(a) $\psi=\tilde{\psi}_{1}\tilde{\psi}_{2}\in C^{1}_{b}(\mathbb{R}^{N})$, so $F\varphi=\psi\circ p_{N}\in\mathcal{F}C^{1}_{b}(X)$ with representation $(N,\psi)$ by Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical. 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, applied to the three representations, and Step 1(a), for $k\le N$ and $x\in X$

$$\partial_{k}(F\varphi)(x)=\partial_{k}\psi(p_{N}(x))=\partial_{k}\tilde{\psi}_{1}(p_{N}(x))\,\tilde{\psi}_{2}(p_{N}(x))+\tilde{\psi}_{1}(p_{N}(x))\,\partial_{k}\tilde{\psi}_{2}(p_{N}(x))=\partial_{k}F(x)\,\varphi(x)+F(x)\,\partial_{k}\varphi(x),$$

while for $k>N$ the three functions $\partial_{k}(F\varphi)$, $\partial_{k}F$ and $\partial_{k}\varphi$ vanish identically.

(b) Let $\varphi\in\mathcal{F}C^{1}_{b}(X)$ with representation $(n,\psi)$, let $x\in X$ and $k\in\mathbb{N}$. By The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient the $j$-th coordinate of $\nabla_{a}\varphi(x)$ is $a_{j}\,\partial_{j}\varphi(x)$ for $j\le n$ and $0=a_{j}\,\partial_{j}\varphi(x)$ for $j>n$, the latter 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. By Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §noise-space, $f_{k}=a_{k}^{1/2}e_{k}$, whose $j$-th coordinate is $a_{k}^{1/2}$ for $j=k$ and $0$ otherwise, the basis $(e_{j})$ being orthonormal. Hence, by The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §inner-product, the series defining $\langle\nabla_{a}\varphi(x),f_{k}\rangle_{a}$ has the single nonzero term $a_{k}^{-1}\,a_{k}\,\partial_{k}\varphi(x)\,a_{k}^{1/2}$, so

$$\langle\nabla_{a}\varphi(x),f_{k}\rangle_{a}=a_{k}^{1/2}\,\partial_{k}\varphi(x).$$

Thus the coordinate along $f_{k}$, in the sense of The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates, of the class $\nabla_{a}\varphi\in L^{2}(\gamma_{c};X^{a})$ is the class of $a_{k}^{1/2}\,\partial_{k}\varphi$, which lies 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.

Step 3 (Claim 1). Let $F,\varphi\in\mathcal{F}C^{1}_{b}(X)$ and $k\in\mathbb{N}$, and let $(N,\psi)$ be the representation of $F\varphi$ of Step 2(a). The finitely many functions $\partial_{i}\psi$ ($i\in[N]$) are bounded by Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded, so there is a nonnegative $M\in\mathbb{R}$ with $|\partial_{i}\psi(y)|\le M\le M(1+\lVert y\rVert)$ for all $y\in\mathbb{R}^{N}$ and $i\in[N]$. Hence Gaussian Integration by Parts for Functions of Finitely Many Coordinates Relative to a Diagonal Gaussian Measure on a Hilbert Space applies with $n=N$ and this $\psi$ and $M$; the functions written $\varphi$ and $\varphi_{i}$ there are $F\varphi$ and $(\partial_{i}\psi)\circ p_{N}=\partial_{i}(F\varphi)$ (Step 2(a)).

Integrability. 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 (with $\mu=\gamma_{c}$), $F$, $\varphi$, $\partial_{k}F$ and $\partial_{k}\varphi$ are $2$-integrable with respect to $\gamma_{c}$, so $\partial_{k}F\,\varphi$ and $F\,\partial_{k}\varphi$ are integrable by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product. The function $x\mapsto x_{k}F(x)\varphi(x)$ is integrable by Gaussian Integration by Parts for Functions of Finitely Many Coordinates Relative to a Diagonal Gaussian Measure on a Hilbert Space §integrable. Hence, by linearity, $x\mapsto F(x)\bigl(x_{k}\varphi(x)/c_{k}-\partial_{k}\varphi(x)\bigr)=c_{k}^{-1}x_{k}F(x)\varphi(x)-F(x)\partial_{k}\varphi(x)$ is integrable.

The identity. If $k\le N$, Gaussian Integration by Parts for Functions of Finitely Many Coordinates Relative to a Diagonal Gaussian Measure on a Hilbert Space §coordinate with $i=k$, Step 2(a) and linearity give

$$\int_{X}\partial_{k}F\,\varphi\,d\gamma_{c}+\int_{X}F\,\partial_{k}\varphi\,d\gamma_{c}=\int_{X}\partial_{k}(F\varphi)\,d\gamma_{c}=\frac{1}{c_{k}}\int_{X}x_{k}F(x)\varphi(x)\,\gamma_{c}(dx),$$

which rearranges, by linearity, to the asserted identity. If $k>N$, then $\partial_{k}F=0$ and $\partial_{k}\varphi=0$ by Step 2(a), so the left-hand side of the asserted identity is $0$ and its right-hand side is $c_{k}^{-1}\int_{X}x_{k}F(x)\varphi(x)\,\gamma_{c}(dx)$, which is $0$ by Gaussian Integration by Parts for Functions of Finitely Many Coordinates Relative to a Diagonal Gaussian Measure on a Hilbert Space §orthogonal. This proves claim 1.

Step 4 (Claim 2). Let $(F_{j})_{j\in\mathbb{N}}$ and $G$ be as in claim 2, fix $k\in\mathbb{N}$, and let $G_{k}\in L^{2}(\gamma_{c})$ be the coordinate of $G$ along $f_{k}$ (The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates).

(i) By The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates (coordinates are additive and homogeneous) and Step 2(b), the coordinate of $\nabla_{a}F_{j}-G$ along $f_{k}$ is $a_{k}^{1/2}\partial_{k}F_{j}-G_{k}$, and the series $\sum_{l=1}^{\infty}\lVert(\nabla_{a}F_{j}-G)_{l}\rVert^{2}$ of nonnegative terms converges with sum $\lVert\nabla_{a}F_{j}-G\rVert_{\gamma_{c}}^{2}$. Its $k$-th term is at most its $k$-th partial sum, which is at most the sum by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates; so

$$\lVert a_{k}^{1/2}\partial_{k}F_{j}-G_{k}\rVert\le\lVert\nabla_{a}F_{j}-G\rVert_{\gamma_{c}}\longrightarrow0\qquad(j\to\infty).$$

(ii) Let $\varphi\in\mathcal{F}C^{1}_{b}(X)$, and let $B\ge0$ with $|\varphi(x)|\le B$ for every $x\in X$ (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). Put $r(x)=x_{k}\varphi(x)/c_{k}-\partial_{k}\varphi(x)$. The coordinate function $x\mapsto x_{k}$ is Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity and $\varphi$, $\partial_{k}\varphi$ are 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, so $r$ is Borel by claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. Since $(x_{k}\varphi(x))^{2}\le B^{2}x_{k}^{2}$ and $x\mapsto x_{k}^{2}$ is integrable with respect to $\gamma_{c}$ by Moments of a Diagonal Gaussian Measure on a Hilbert Space: Coordinate Covariances, Finite Second Moment, and Exponential Moments of the Squared Norm §coordinates (with $j=k$), monotonicity shows that $x\mapsto x_{k}\varphi(x)$ is $2$-integrable; $\partial_{k}\varphi$ is $2$-integrable 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; so $r$ is $2$-integrable by Minkowski's Inequality and the Seminormed Space of Power-Integrable Functions §vector-space, and its class lies in $L^{2}(\gamma_{c})$. By claim 1 (Step 3) applied to $F_{j}$ and $\varphi$,

$$\langle a_{k}^{1/2}\partial_{k}F_{j},\varphi\rangle=a_{k}^{1/2}\int_{X}\partial_{k}F_{j}\,\varphi\,d\gamma_{c}=a_{k}^{1/2}\int_{X}F_{j}\,r\,d\gamma_{c}=a_{k}^{1/2}\langle F_{j},r\rangle .$$

By bilinearity of the inner product (Real Inner Product Space §inner-product),

$$\langle G_{k},\varphi\rangle=a_{k}^{1/2}\langle F_{j},r\rangle-\langle a_{k}^{1/2}\partial_{k}F_{j}-G_{k},\varphi\rangle,$$

so the Cauchy-Schwarz inequality The Cauchy-Schwarz Inequality in a Real Inner Product Space gives, for every $j\in\mathbb{N}$,

$$|\langle G_{k},\varphi\rangle|\le a_{k}^{1/2}\,\lVert F_{j}\rVert_{2}\,\lVert r\rVert+\lVert a_{k}^{1/2}\partial_{k}F_{j}-G_{k}\rVert\,\lVert\varphi\rVert .$$

The right-hand side tends to $0$ as $j\to\infty$, by the hypothesis $\lVert F_{j}\rVert_{2}\to0$ and by (i); since the left-hand side does not depend on $j$, $\langle G_{k},\varphi\rangle=0$.

(iii) As $\varphi\in\mathcal{F}C^{1}_{b}(X)$ was arbitrary, $G_{k}$ is the zero vector of $L^{2}(\gamma_{c})$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §density, applied with $\mu=\gamma_{c}$. This holds for every $k\in\mathbb{N}$, so by The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates

$$\lVert G\rVert_{\gamma_{c}}^{2}=\sum_{k=1}^{\infty}\lVert G_{k}\rVert^{2}=0 .$$

$L^{2}(\gamma_{c};X^{a})$ is a real Hilbert space whose norm is that of its inner product by The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §hilbert, so $\lVert G\rVert_{\gamma_{c}}$, the nonnegative square root of $\lVert G\rVert_{\gamma_{c}}^{2}=0$, is $0$, and $G$ is the zero vector of $L^{2}(\gamma_{c};X^{a})$ by Elementary Identities in a Real Inner Product Space §vanishing. This proves claim 2.