Proof of Integrals of Bounded $C^2$ Cylindrical Functions are Noise Intrinsic Test Functions with the Noise Gradient as Gradient

Each result cited is universally quantified over the data in its own statement.

Throughout, $x=\pi_{1}(z)$ and $y=\pi_{2}(z)$ for $z\in X\times X$, as in Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §background, and $\bar{a}$ is the positive bound on the noise weights of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §weights.

Step 0 (Constants and the representation). By Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded, $f$ is of class $C^{2}$ on $\mathbb{R}^{n}$ and the finitely many functions $f$, $\partial_{i}f$ and $\partial_{j}\partial_{i}f$ ($i,j\in[n]$) are bounded; adding the nonnegative bounds of the functions $\partial_{j}\partial_{i}f$ we obtain a real number $\Lambda\ge0$ with $|\partial_{j}\partial_{i}f(u)|\le \Lambda$ for all $u\in\mathbb{R}^{n}$ and $i,j\in[n]$. By clause 2 of C^k Maps on a Euclidean Open Set (read through its clause 3), $f$ is of class $C^{1}$ on $\mathbb{R}^{n}$ and, for each $k\in[n]$, the function $g_{k}=\partial_{k}f$ is of class $C^{1}$ on $\mathbb{R}^{n}$, its partial derivative with respect to the $j$-th variable being $\partial_{j}\partial_{k}f$ by clause 4 there. Hence $f\in C^{1}_{b}(\mathbb{R}^{n})$ by Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded, so $(n,f)$ is a representation of $\varphi$, and Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial gives $\partial_{k}\varphi=(\partial_{k}f)\circ p_{n}$ for $k\le n$. Put

$$K=\tfrac{1}{2}\,n\,\Lambda\,\bar{a},\qquad C=n^{2}\,\Lambda^{2}\,\bar{a}^{2},$$

both nonnegative real numbers.

Step 1 (Pointwise estimates on $D_{a}$). Let $z\in D_{a}$, so that $y-x\in X^{a}$ by the description of $D_{a}$ in The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §pairs, and put $u=p_{n}(x)$ and $w=p_{n}(y)$ in $\mathbb{R}^{n}$. By linearity of the inner product, $(y-x)_{k}=y_{k}-x_{k}=w_{k}-u_{k}$ for $k\le n$.

(1a) Comparison of distances. By Euclidean Distance on $\mathbb{R}^n$, $d_{E}(u,w)^{2}=\sum_{k=1}^{n}(y_{k}-x_{k})^{2}$. For each $k$ we have $0<a_{k}\le\bar{a}$, so $1=a_{k}a_{k}^{-1}\le\bar{a}\,a_{k}^{-1}$, and termwise comparison of finite sums gives $d_{E}(u,w)^{2}\le\bar{a}\,S_{n}(y-x)$, with $S_{n}(v)=\sum_{k=1}^{n}a_{k}^{-1}v_{k}^{2}$ the partial sum of The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability. By The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §partial-sums, $|y-x|_{a}^{2}=\sup_{N}S_{N}(y-x)\ge S_{n}(y-x)$, and $|y-x|_{a}^{2}=c_{a}(z)$ by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §noise-space. Hence

$$d_{E}(u,w)^{2}\le\bar{a}\,c_{a}(z). \tag{1}$$

(1b) Noise pairings with finitely many coordinates. Let $v,v'\in X^{a}$ with $v_{k}=0$ for every $k>n$ (no condition is imposed on $v'$). By The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §inner-product, $\langle v,v'\rangle_{a}$ is the sum of the series $\sum_{k}a_{k}^{-1}v_{k}v'_{k}$, whose terms vanish for $k>n$; its partial sums are therefore constant from the $n$-th on, so by Series of Real Numbers §convergent its sum is $\sum_{k=1}^{n}a_{k}^{-1}v_{k}v'_{k}$. By The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient, $\nabla_{a}\varphi(x)\in X^{a}$ has coordinates $a_{k}\,\partial_{k}\varphi(x)=a_{k}\,\partial_{k}f(u)$ for $k\le n$ (Step 0) and $0$ for $k>n$, and likewise at $y$. Applying this with $v=\nabla_{a}\varphi(x)$ and $v'=y-x$ gives

$$\langle\nabla_{a}\varphi(x),y-x\rangle_{a}=\sum_{k=1}^{n}\partial_{k}f(u)\,(w_{k}-u_{k}), \tag{2}$$

and applying it with $v=v'=\nabla_{a}\varphi(x)-\nabla_{a}\varphi(y)$, which lies in $X^{a}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert and has coordinates $a_{k}(\partial_{k}f(u)-\partial_{k}f(w))$ for $k\le n$ and $0$ for $k>n$, gives

$$|\nabla_{a}\varphi(x)-\nabla_{a}\varphi(y)|_{a}^{2}=\sum_{k=1}^{n}a_{k}\,\bigl(\partial_{k}f(u)-\partial_{k}f(w)\bigr)^{2}. \tag{3}$$

(1c) First-order remainder. The set $\mathbb{R}^{n}$ is open (claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous) and contains the segment from $u$ to $w$, and $f$ is of class $C^{2}$ with $|\partial_{j}\partial_{i}f|\le \Lambda$ there, so part (ii) of Multivariate Taylor Expansion with Uniform Second-Order Remainder, together with (2) and (1), gives

$$\bigl|\varphi(y)-\varphi(x)-\langle\nabla_{a}\varphi(x),y-x\rangle_{a}\bigr|=\Bigl|f(w)-f(u)-\sum_{k=1}^{n}\partial_{k}f(u)(w_{k}-u_{k})\Bigr|\le\tfrac{1}{2}\,n\,\Lambda\,d_{E}(u,w)^{2}\le K\,c_{a}(z). \tag{4}$$

(1d) Lipschitz bound for the gradient. For $k\in[n]$, $g_{k}$ is of class $C^{1}$ on $\mathbb{R}^{n}$ with partial derivatives $\partial_{j}\partial_{k}f$ bounded by $\Lambda$ (Step 0), so part (i) of Multivariate Taylor Expansion with Uniform Second-Order Remainder gives $|\partial_{k}f(w)-\partial_{k}f(u)|\le\sqrt{n}\,\Lambda\,d_{E}(u,w)$; squaring both nonnegative sides and using (1) yields $(\partial_{k}f(u)-\partial_{k}f(w))^{2}\le n\,\Lambda^{2}\,\bar{a}\,c_{a}(z)$. With $a_{k}\le\bar{a}$ and (3),

$$|\nabla_{a}\varphi(x)-\nabla_{a}\varphi(y)|_{a}^{2}\le\sum_{k=1}^{n}\bar{a}\,n\,\Lambda^{2}\,\bar{a}\,c_{a}(z)=C\,c_{a}(z). \tag{5}$$

(1e) The complement of $D_{a}$ is null. If $\sigma,\sigma'\in\mathcal{P}(X)$ and $\pi\in\Pi^{a}(\sigma,\sigma')$, then $\pi(D_{a})=1$ by Couplings of Finite Noise Cost and Their Noise Cost §finite, so the Borel set $(X\times X)\setminus D_{a}$ has $\pi$-measure $\pi(X\times X)-\pi(D_{a})=0$ by additivity of measures (Measure Spaces and the Lebesgue Integral: Standing Notation §space) and is $\pi$-null by Null Set of a Measure. Thus (4) and (5) hold for $\pi$-almost every $z$.

Step 2 (Clause gradient). Let $\mu\in\mathcal{P}^{a}_{\rho}$, $\nu\in\mathcal{P}^{a}_{\rho}$ and $\pi\in\Pi^{a}(\mu,\nu)$. Then $\pi\in\Pi(\mu,\nu)$ by Couplings of Finite Noise Cost and Their Noise Cost §couplings, so $(\pi_{1})_{\#}\pi=\mu$ and $(\pi_{2})_{\#}\pi=\nu$ by Couplings of Two Borel Probability Measures on a Hilbert Space and Their Quadratic Cost §coupling. The function $\varphi$ is 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 integrable with respect to $\mu$ and $\nu$ 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 the change of variables formula (claim 2 of Image Measures, Measures with Densities, and Change of Variables, as recorded in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward) shows that $\varphi\circ\pi_{1}$ and $\varphi\circ\pi_{2}$ are integrable with respect to $\pi$, with $U_{\varphi}(\mu)=\int_{X\times X}\varphi(x)\,\pi(dz)$ and $U_{\varphi}(\nu)=\int_{X\times X}\varphi(y)\,\pi(dz)$. Let $\delta$ be the displacement field of Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §displacement-field for $\pi$. The class of $\nabla_{a}\varphi$ lies in $L^{2}(\mu;X^{a})$ by The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient. By Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §pairing, applied with the representative $\nabla_{a}\varphi$ of its class, the function $z\mapsto\langle\nabla_{a}\varphi(x),\delta(z)\rangle_{a}$ is Borel and $\pi$-integrable with integral $\mathcal{J}^{a}(\nabla_{a}\varphi,\pi)$, and it equals $\langle\nabla_{a}\varphi(x),y-x\rangle_{a}$ on $D_{a}$. Let

$$R(z)=\varphi(y)-\varphi(x)-\langle\nabla_{a}\varphi(x),\delta(z)\rangle_{a}\qquad(z\in X\times X).$$

By Linearity and Monotonicity of the Lebesgue Integral §integrable, $R$ is $\pi$-integrable, $\int_{X\times X}R\,d\pi=U_{\varphi}(\nu)-U_{\varphi}(\mu)-\mathcal{J}^{a}(\nabla_{a}\varphi,\pi)$, and $|\int_{X\times X}R\,d\pi|\le\int_{X\times X}|R|\,d\pi$. By (4) and Step (1e), $|R(z)|\le K\,c_{a}(z)$ for $\pi$-almost every $z$, both sides being nonnegative and Borel, so The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison and the homogeneity in Linearity and Monotonicity of the Lebesgue Integral §nonnegative give, with Couplings of Finite Noise Cost and Their Noise Cost §cost,

$$\bigl|U_{\varphi}(\nu)-U_{\varphi}(\mu)-\mathcal{J}^{a}(\nabla_{a}\varphi,\pi)\bigr|\le K\int_{X\times X}c_{a}\,d\pi=K\,I^{a}(\pi). \tag{6}$$

Now let $\varepsilon>0$ be given and choose $\theta=\varepsilon/(K+1)>0$; then $K\theta\le\varepsilon$. Let $\nu\in\mathcal{P}^{a}_{\rho}$ and $\pi\in\Pi^{a}(\mu,\nu)$ satisfy $I^{a}(\pi)<\theta^{2}$. Since $I^{a}(\pi)\ge0$ and $\theta>0$, monotonicity of squares gives $\sqrt{I^{a}(\pi)}<\theta$, so by (6)

$$\bigl|U_{\varphi}(\nu)-U_{\varphi}(\mu)-\mathcal{J}^{a}(\nabla_{a}\varphi,\pi)\bigr|\le K\sqrt{I^{a}(\pi)}\,\sqrt{I^{a}(\pi)}\le K\theta\sqrt{I^{a}(\pi)}\le\varepsilon\sqrt{I^{a}(\pi)}.$$

As noted above, the class of $\nabla_{a}\varphi$ lies in $L^{2}(\mu;X^{a})$, so $U_{\varphi}$ is differentiable along noise couplings at $\mu$ with gradient $\nabla_{a}\varphi$ in the sense of Differentiability of a Function on the Noise-Connected Measures Along Noise Couplings, and Its Gradient §differentiable, and by the uniqueness in Differentiability of a Function on the Noise-Connected Measures Along Noise Couplings, and Its Gradient §gradient, $\nabla U_{\varphi}(\mu)=\nabla_{a}\varphi$ in $L^{2}(\mu;X^{a})$. This proves clause 2 (Gradient). Moreover $\nabla_{a}\varphi\in T^{a}_{\mu}$ by Noise Gradients of Bounded C^2 Cylindrical Functions Are Dense in the Noise Tangent Space §inclusion, since $\varphi\in\mathcal{F}C^{2}_{b}(X)$ by Bounded C^2 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical.

Step 3 (Continuity). Let $\mu\in\mathcal{P}^{a}_{\rho}$, put $B=\lVert\nabla_{a}\varphi\rVert_{\mu}\ge0$, and let $\theta_{1}>0$ be the number of Step 2 for $\varepsilon=1$. Let $\varepsilon'>0$ be given, and choose $\delta=\min\{\theta_{1},\varepsilon'/(B+2)\}$, the minimum, which is positive and satisfies $\delta\le\theta_{1}$ and $\delta\le\varepsilon'/(B+2)$ by claims 1 and 2 of Elementary Properties of the Minimum of Two Elements. Let $\nu\in\mathcal{P}^{a}_{\rho}$ with $W_{a}(\mu,\nu)<\delta$. The pair $(\mu,\nu)$ is noise-connected by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §connected, so The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §optimal provides a noise-optimal $\pi\in\Pi^{a}(\mu,\nu)$, with $I^{a}(\pi)=W_{a}(\mu,\nu)^{2}$ by Noise-Optimal Couplings §optimal; as $W_{a}(\mu,\nu)\ge0$ by The Noise Wasserstein Distance §distance, uniqueness of square roots gives $\sqrt{I^{a}(\pi)}=W_{a}(\mu,\nu)$, and $I^{a}(\pi)<\theta_{1}^{2}$ by monotonicity of squares. By the triangle inequality, Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §bound and the estimate of Step 2 with $\varepsilon=1$,

$$|U_{\varphi}(\nu)-U_{\varphi}(\mu)|\le\bigl|\mathcal{J}^{a}(\nabla_{a}\varphi,\pi)\bigr|+\bigl|U_{\varphi}(\nu)-U_{\varphi}(\mu)-\mathcal{J}^{a}(\nabla_{a}\varphi,\pi)\bigr|\le(B+1)\,W_{a}(\mu,\nu)\le(B+2)\,W_{a}(\mu,\nu)<(B+2)\,\delta\le\varepsilon'.$$

Since the distance of $(\mathcal{P}^{a}_{\rho},W_{a})$ is $W_{a}$ by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §metric and that of $\mathbb{R}$ is $|s-t|$ (The Absolute Value Metric on the Real Line), $U_{\varphi}$ is continuous at $\mu$ relative to $\mathcal{P}^{a}_{\rho}$ in the sense of Continuous Map Between Metric Spaces; $\mu$ being arbitrary, $U_{\varphi}$ is continuous on $\mathcal{P}^{a}_{\rho}$.

Step 4 (Gradient continuity). Let $\mu\in\mathcal{P}^{a}_{\rho}$, let $(\mu_{j})_{j\in\mathbb{N}}$ be a sequence in $\mathcal{P}^{a}_{\rho}$ and let $(\pi_{j})_{j\in\mathbb{N}}$ be a sequence of couplings of vanishing noise cost from $(\mu_{j})$ to $\mu$ (the index of Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost is renamed $j$, as $n$ is fixed); thus $\pi_{j}\in\Pi^{a}(\mu_{j},\mu)\subseteq\Pi(\mu_{j},\mu)$ and $\lim_{j\to\infty}I^{a}(\pi_{j})=0$. By Step 2, $\nabla U_{\varphi}(\mu_{j})=\nabla_{a}\varphi$ in $L^{2}(\mu_{j};X^{a})$ and $\nabla U_{\varphi}(\mu)=\nabla_{a}\varphi$ in $L^{2}(\mu;X^{a})$. By Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §discrepancy, applied with $\mu_{j}$ and $\mu$ in place of its $\nu$ and $\mu$, $q=\nabla_{a}\varphi\in L^{2}(\mu_{j};X^{a})$, $\eta=\nabla_{a}\varphi\in L^{2}(\mu;X^{a})$ and the representative $\nabla_{a}\varphi$ for both, the function $z\mapsto|\nabla_{a}\varphi(x)-\nabla_{a}\varphi(y)|_{a}^{2}$ is Borel, nonnegative and $\pi_{j}$-integrable, and its integral $d_{j}$ is the discrepancy of the definition. By (5) and Step (1e) for $\pi_{j}$, this function is at most $C\,c_{a}$ $\pi_{j}$-almost everywhere, so The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison, Linearity and Monotonicity of the Lebesgue Integral §nonnegative and Couplings of Finite Noise Cost and Their Noise Cost §cost give

$$0\le d_{j}\le C\,I^{a}(\pi_{j})\qquad(j\in\mathbb{N}).$$

By Arithmetic of Limits of Real Sequences §scalar, $\lim_{j\to\infty}C\,I^{a}(\pi_{j})=0$, and the constant sequence $0$ converges to $0$, so the squeeze of claim 2 of Order Properties of Limits of Real Sequences gives $\lim_{j\to\infty}d_{j}=0$. By Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §strong, the sequence $(\nabla U_{\varphi}(\mu_{j}))_{j}$ converges strongly to $\nabla U_{\varphi}(\mu)$ along $(\pi_{j})_{j}$.

Step 5 (Clause test). Let $Q\subseteq\mathcal{P}^{a}_{\rho}$. Property (a) of Noise Intrinsic Test Functions on the Noise Wasserstein Space §test is Step 3; property (b) holds at every $\mu\in Q\subseteq\mathcal{P}^{a}_{\rho}$ by Step 2; and property (c) holds for every $\mu\in Q$, sequence $(\mu_{j})$ in $Q$ and couplings of vanishing noise cost from $(\mu_{j})$ to $\mu$ by Step 4, which was proved for arbitrary members of $\mathcal{P}^{a}_{\rho}$. Hence $U_{\varphi}$ is a noise intrinsic test function on $Q$, proving clause 1 (Test function).