Proof of Diagonal Quadratic Profiles on the Noise Wasserstein Space: Moment Bounds, a Lipschitz Estimate, the Gradient Field, and Noise Intrinsic Test Functions

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

Notation. For $z\in X\times X$ we write $x=\pi_{1}(z)$ and $y=\pi_{2}(z)$, and $x_{k}$, $y_{k}$ also denote the functions $z\mapsto\langle\pi_{1}(z),e_{k}\rangle$ and $z\mapsto\langle\pi_{2}(z),e_{k}\rangle$ on $X\times X$; they are Borel, being composites (preimages of preimages, Measurable Function and Real-Valued Measurable Function) of the Borel maps $\pi_{1},\pi_{2}$ of Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §product-sigma with the Borel coordinate functions of Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity. For $\sigma\in\mathcal{P}^{a}_{\rho}$ and $k\in\mathbb{N}$ put $m_{k}(\sigma)=\int_{X}x_{k}^{2}\,\sigma(dx)$, a nonnegative real number. Indeed, $\sigma\in\mathcal{P}_{2}(X)$ 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 §moments, that is, $\int_{X}|x|^{2}\,\sigma(dx)<\infty$ (The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space); the coordinate $x_{k}=\langle x,e_{k}\rangle$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates is a Borel function of $x$ by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, so $x_{k}^{2}$ is Borel by Measure Spaces and the Lebesgue Integral: Standing Notation §measurable; and $|x_{k}|\le|x|\,|e_{k}|=|x|$ by The Cauchy-Schwarz Inequality in a Real Inner Product Space, $e_{k}$ being a unit vector, so $0\le x_{k}^{2}\le|x|^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Hence $\int_{X}x_{k}^{2}\,\sigma(dx)\le\int_{X}|x|^{2}\,\sigma(dx)<\infty$ by the monotonicity in Linearity and Monotonicity of the Lebesgue Integral §nonnegative, so $x_{k}^{2}$ is integrable with respect to $\sigma$ by the criterion of Measure Spaces and the Lebesgue Integral: Standing Notation §integral, with integral in $[0,\infty)$. For a measure space and a $2$-integrable function $h$ on it, the norm of its class in $L^{2}$ satisfies $\lVert h\rVert^{2}=\langle h,h\rangle=\int h^{2}$, by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product and the definition of the norm.

Step 0 (Marginals and pull-backs). Let $\sigma\in\mathcal{P}(X)$, $\pi\in\mathcal{P}(X\times X)$ and $i\in\{1,2\}$ with $(\pi_{i})_{\#}\pi=\sigma$. By claims 1 and 2 of Image Measures, Measures with Densities, and Change of Variables, through Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward: $\pi(\pi_{i}^{-1}(N))=\sigma(N)$ for every Borel $N\subseteq X$; $\int_{X\times X}h\circ\pi_{i}\,d\pi=\int_{X}h\,d\sigma$ for every Borel $h:X\to[0,\infty]$; and a Borel $h:X\to\mathbb{R}$ is integrable with respect to $\sigma$ if and only if $h\circ\pi_{i}$ is integrable with respect to $\pi$, with the same integral. In particular, if two Borel functions on $X$ agree $\sigma$-almost everywhere, their composites with $\pi_{i}$ agree $\pi$-almost everywhere.

(i) Let $\mu,\nu\in\mathcal{P}^{a}_{\rho}$ and $\pi\in\Pi(\mu,\nu)$, so that $\pi$ has marginals $\mu$ and $\nu$ by Couplings of Two Borel Probability Measures on a Hilbert Space and Their Quadratic Cost §coupling. Then $x_{k}^{2}$ and $y_{k}^{2}$ are integrable with respect to $\pi$, with integrals $m_{k}(\mu)$ and $m_{k}(\nu)$. Since $(y_{k}-x_{k})^{2}\le2x_{k}^{2}+2y_{k}^{2}$ pointwise by Products and Sums of Weighted Square-Summable Sequences of Real Numbers §pointwise, Linearity and Monotonicity of the Lebesgue Integral §nonnegative gives $\int(y_{k}-x_{k})^{2}\,d\pi\le2m_{k}(\mu)+2m_{k}(\nu)<\infty$, so $(y_{k}-x_{k})^{2}$ is integrable by the criterion of Measure Spaces and the Lebesgue Integral: Standing Notation §integral. Thus $x_{k}$, $y_{k}$ and $y_{k}-x_{k}$ are $2$-integrable with respect to $\pi$, the product of any two of them is integrable by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product, and we put

$$D_{k}(\pi)=\int_{X\times X}(y_{k}-x_{k})^{2}\,\pi(dz)\ge0 .$$

(ii) Let $w:X\to X^{a}$ be measurable and square-integrable with respect to $\sigma$, with coordinates $w_{k}(x)=\langle w(x),f_{k}\rangle_{a}$. Then $w\circ\pi_{i}$ is measurable into $X^{a}$ (again by Measurable Function and Real-Valued Measurable Function), the function $|w|_{a}^{2}$ is Borel 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, and $\int|w\circ\pi_{i}|_{a}^{2}\,d\pi=\int|w|_{a}^{2}\,d\sigma<\infty$; so $w\circ\pi_{i}$ has a class in $L^{2}(\pi;X^{a})$, and the coordinate of a map along $f_{k}$ being taken pointwise, the coordinate of $w\circ\pi_{i}$ along $f_{k}$ is $w_{k}\circ\pi_{i}$. If the class of $w_{k}$ in $L^{2}(\sigma)$ is that of a Borel $g:X\to\mathbb{R}$, then $w_{k}=g$ $\sigma$-almost everywhere (The Lebesgue Space of Power-Integrable Functions §equivalence), hence $w_{k}\circ\pi_{i}=g\circ\pi_{i}$ $\pi$-almost everywhere, and the coordinate along $f_{k}$ of the class of $w\circ\pi_{i}$, in the sense of The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates, is the class of $g\circ\pi_{i}$.

Step 1 (A coordinate bound along couplings). Let $t$ and $M$ be as in claim 1, let $\mu,\nu\in\mathcal{P}^{a}_{\rho}$ and $\pi\in\Pi^{a}(\mu,\nu)$. We show that $\sum_{k=1}^{\infty}t_{k}D_{k}(\pi)$ converges with sum at most $M\,I^{a}(\pi)$. For each $k$, multiplying $t_{k}a_{k}\le M$ by the positive number $a_{k}^{-1}$ gives $t_{k}\le M a_{k}^{-1}$. Let $z\in D_{a}$. Then $y-x\in X^{a}$, its $k$-th coordinate is $y_{k}-x_{k}$ by linearity of the inner product (Real Inner Product Space §inner-product), and by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §noise-space and The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §inner-product

$$c_{a}(z)=|y-x|_{a}^{2}=\sum_{k=1}^{\infty}a_{k}^{-1}(y_{k}-x_{k})^{2},$$

a series of nonnegative terms, so by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates, for every $n\in\mathbb{N}$,

$$\sum_{k=1}^{n}t_{k}(y_{k}-x_{k})^{2}\le M\sum_{k=1}^{n}a_{k}^{-1}(y_{k}-x_{k})^{2}\le M\,c_{a}(z).$$

Since $\pi(D_{a})=1$ by Couplings of Finite Noise Cost and Their Noise Cost §finite, this holds for $\pi$-almost every $z$, both sides being nonnegative Borel functions; by 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,

$$\sum_{k=1}^{n}t_{k}D_{k}(\pi)\le M\int_{X\times X}c_{a}\,d\pi=M\,I^{a}(\pi)\qquad(n\in\mathbb{N}).$$

The terms $t_{k}D_{k}(\pi)$ are nonnegative, so by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §criterion the series converges and its sum, the supremum of these partial sums, is at most $M\,I^{a}(\pi)$.

Claim 1. Let $\mu\in\mathcal{P}^{a}_{\rho}$. 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 §reference and 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, $\rho\in\mathcal{P}^{a}_{\rho}$ and $(\mu,\rho)$ is noise-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 gives a noise-optimal $\pi\in\Pi^{a}(\mu,\rho)$, that is, $I^{a}(\pi)=W_{a}(\mu,\rho)^{2}$ by Noise-Optimal Couplings §optimal. Since $\rho=\gamma_{c}$, Moments of a Diagonal Gaussian Measure on a Hilbert Space: Coordinate Covariances, Finite Second Moment, and Exponential Moments of the Squared Norm §coordinates with $j=k$ gives $m_{k}(\rho)=c_{k}$, which is nonnegative. Pointwise on $X\times X$, by Products and Sums of Weighted Square-Summable Sequences of Real Numbers §pointwise applied with $s=y_{k}$ and $t=x_{k}-y_{k}$,

$$x_{k}^{2}\le2y_{k}^{2}+2(y_{k}-x_{k})^{2},$$

and integrating against $\pi$ (Step 0 (i) and Linearity and Monotonicity of the Lebesgue Integral §integrable) gives $m_{k}(\mu)\le2c_{k}+2D_{k}(\pi)$. Multiplying by $t_{k}\ge0$ and summing over $k\le n$, then using Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates for the convergent series $\sum_{k}t_{k}c_{k}$ of nonnegative terms and Step 1 with $\nu=\rho$,

$$\sum_{k=1}^{n}t_{k}m_{k}(\mu)\le2\sum_{k=1}^{\infty}t_{k}c_{k}+2M\,W_{a}(\mu,\rho)^{2}\qquad(n\in\mathbb{N}).$$

The terms $t_{k}m_{k}(\mu)$ are nonnegative, so by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §criterion the series $S_{t}(\mu)$ converges and its sum, the supremum of the partial sums, satisfies the claimed bound.

Claim 2. The first assertion is Step 1. For the second, write $I=I^{a}(\pi)$ and fix $n\in\mathbb{N}$. Let $\sqrt{t_{k}}$ be the nonnegative square root. In $L^{2}(\pi)=L^{2}(X\times X,\mathcal{B}(X\times X),\pi)$ let $g_{k}$ and $h_{k}$ be the classes of $\sqrt{t_{k}}\,x_{k}$ and of $\sqrt{t_{k}}\,y_{k}$ for $k\le n$, and the zero class for $k>n$; these functions are $2$-integrable by Step 0 (i), and $\lVert g_{k}\rVert^{2}=t_{k}m_{k}(\mu)$ and $\lVert h_{k}\rVert^{2}=t_{k}m_{k}(\nu)$ for $k\le n$ by Step 0. The series $\sum_{k}\lVert g_{k}\rVert^{2}$ and $\sum_{k}\lVert h_{k}\rVert^{2}$ have only finitely many nonzero terms and converge, so The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §synthesis, applied to the measure space $(X\times X,\mathcal{B}(X\times X),\pi)$ with $E=X^{a}$ and the basis $(f_{k})$ as in Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, gives $U,\omega\in L^{2}(\pi;X^{a})$ with $U_{k}=g_{k}$ and $\omega_{k}=h_{k}$ for every $k$. By The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates, $(U-\omega)_{k}=g_{k}-h_{k}$ is the class of $\sqrt{t_{k}}(x_{k}-y_{k})$ for $k\le n$ and zero for $k>n$, and

$$\lVert U\rVert_{\pi}^{2}=\sum_{k=1}^{n}t_{k}m_{k}(\mu),\qquad\lVert\omega\rVert_{\pi}^{2}=\sum_{k=1}^{n}t_{k}m_{k}(\nu),\qquad\lVert U-\omega\rVert_{\pi}^{2}=\sum_{k=1}^{n}t_{k}D_{k}(\pi)\le M\,I,$$

the inequality by Step 1 and Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates. Since $\lVert\cdot\rVert_{\pi}$ is the norm of the inner product of the real Hilbert space $L^{2}(\pi;X^{a})$ (The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §hilbert), The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle gives

$$\lVert U\rVert_{\pi}\le\lVert\omega\rVert_{\pi}+\lVert U-\omega\rVert_{\pi},\qquad\lVert\omega\rVert_{\pi}\le\lVert U\rVert_{\pi}+\lVert U-\omega\rVert_{\pi},$$

writing $U=\omega+(U-\omega)$ and $\omega=U-(U-\omega)$. Now $\sum_{k\le n}t_{k}m_{k}(\nu)\le S_{t}(\nu)$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates, $S_{t}(\nu)$ converging by claim 1, and $\sqrt{M I}=\sqrt{M}\sqrt{I}$ by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root, the right side being nonnegative with square $MI$. So claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, applied to nonnegative square roots, gives $\lVert\omega\rVert_{\pi}\le\sqrt{S_{t}(\nu)}$ and $\lVert U-\omega\rVert_{\pi}\le\sqrt{M}\sqrt{I}$, whence $\lVert U\rVert_{\pi}\le R$ with $R=\sqrt{S_{t}(\nu)}+\sqrt{M}\sqrt{I}$, and, by the same claim, $\sum_{k\le n}t_{k}m_{k}(\mu)=\lVert U\rVert_{\pi}^{2}\le R^{2}$. This holds for every $n$, so $S_{t}(\mu)\le R^{2}$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §criterion, and $\sqrt{S_{t}(\mu)}\le R$ by Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field again. The second triangle inequality gives in the same way $\sqrt{S_{t}(\nu)}\le\sqrt{S_{t}(\mu)}+\sqrt{M}\sqrt{I}$. The two bounds together are the claimed estimate.

Claim 3. Let $B$ be a bound for $b$ and put $t_{k}=a_{k}b_{k}^{2}\ge0$. Then $t_{k}a_{k}=(a_{k}|b_{k}|)^{2}\le B^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, since $0\le a_{k}|b_{k}|\le B$; and $0\le t_{k}c_{k}=(a_{k}|b_{k}|)\,|b_{k}|c_{k}\le B\,|b_{k}|c_{k}$, using $c_{k}>0$ (Variance Sequences and Their Truncations §variances, via A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §gaussian), the series $\sum_{k}B|b_{k}|c_{k}$ converging by admissibility and Elementary Properties of Series of Real Numbers §linearity, so $\sum_{k}t_{k}c_{k}$ converges by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison. Thus $t$ and $M=B^{2}$ satisfy the hypotheses of claim 1, and the series $\sum_{k}a_{k}b_{k}^{2}m_{k}(\mu)$ converges. Let $g_{k}\in L^{2}(\mu)$ be the class of $x\mapsto a_{k}^{1/2}b_{k}x_{k}$, a $2$-integrable function since $x_{k}^{2}$ is $\mu$-integrable; then $\lVert g_{k}\rVert_{L^{2}(\mu)}^{2}=a_{k}b_{k}^{2}m_{k}(\mu)$. By The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §synthesis, with $E=X^{a}$ and the basis $(f_{k})$ as in Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, there is exactly one $V_{b}(\mu)\in L^{2}(\mu;X^{a})$ with coordinate $g_{k}$ along $f_{k}$ for every $k$, and The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates gives $\lVert V_{b}(\mu)\rVert_{\mu}^{2}=\sum_{k}\lVert g_{k}\rVert_{L^{2}(\mu)}^{2}=\sum_{k}a_{k}b_{k}^{2}m_{k}(\mu)$.

Claim 4. With $B$ a bound for $b$, the sequence $t=|b|=(|b_{k}|)_{k}$ and $M=B$ satisfy the hypotheses of claim 1, since $|b_{k}|a_{k}\le B$ and $\sum_{k}|b_{k}|c_{k}$ converges. So $\sum_{k}|b_{k}|\,m_{k}(\mu)$ converges, and $|b_{k}m_{k}(\mu)|=|b_{k}|\,m_{k}(\mu)$ as $m_{k}(\mu)\ge0$: the series $\sum_{k}b_{k}m_{k}(\mu)$ converges absolutely, hence converges by An Absolutely Convergent Series of Real Numbers Converges §convergence, and $\Phi_{b}$ is well defined.

Claim 5. Let $\pi\in\Pi^{a}(\mu,\nu)$, 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, and let $v$ be a representative of $V_{b}(\mu)$. By Step 0 (ii) with $\sigma=\mu$ and $i=1$, $v\circ\pi_{1}$ has a class $A\in L^{2}(\pi;X^{a})$ whose coordinate $A_{k}$ along $f_{k}$ is the class of $a_{k}^{1/2}b_{k}x_{k}$. For $z\in D_{a}$, $\delta(z)=y-x$ and $\langle\delta(z),f_{k}\rangle_{a}=a_{k}^{-1/2}(y_{k}-x_{k})$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §basis; as $\pi(D_{a})=1$, the coordinate $h_{k}$ of $\delta$ along $f_{k}$ is the class of $a_{k}^{-1/2}(y_{k}-x_{k})$ (The Lebesgue Space of Power-Integrable Functions §equivalence). By the defining formula of Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §pairing, The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations and The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates, the latter series converging,

$$\mathcal{J}^{a}(V_{b}(\mu),\pi)=\int_{X\times X}\langle v(x),\delta(z)\rangle_{a}\,\pi(dz)=\langle A,\delta\rangle_{\pi}=\sum_{k=1}^{\infty}\langle A_{k},h_{k}\rangle_{L^{2}(\pi)}=\sum_{k=1}^{\infty}b_{k}\int_{X\times X}x_{k}(y_{k}-x_{k})\,\pi(dz),$$

using The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product and $a_{k}^{1/2}a_{k}^{-1/2}=1$ in the last step. Pointwise $y_{k}^{2}-x_{k}^{2}=2x_{k}(y_{k}-x_{k})+(y_{k}-x_{k})^{2}$; all three functions are $\pi$-integrable by Step 0 (i), so Linearity and Monotonicity of the Lebesgue Integral §integrable and Step 0 give

$$\tfrac12 b_{k}\bigl(m_{k}(\nu)-m_{k}(\mu)\bigr)=b_{k}\int_{X\times X}x_{k}(y_{k}-x_{k})\,\pi(dz)+\tfrac12 b_{k}D_{k}(\pi)\qquad(k\in\mathbb{N}).$$

By claim 4 and Elementary Properties of Series of Real Numbers §linearity the series of the left sides converges with sum $\Phi_{b}(\nu)-\Phi_{b}(\mu)$, and the series of the first terms on the right converges with sum $\mathcal{J}^{a}(V_{b}(\mu),\pi)$, as just shown. By Elementary Properties of Series of Real Numbers §linearity again, $\sum_{k}\tfrac12 b_{k}D_{k}(\pi)$ converges with sum $\Phi_{b}(\nu)-\Phi_{b}(\mu)-\mathcal{J}^{a}(V_{b}(\mu),\pi)$. Since $D_{k}(\pi)\ge0$, $|\tfrac12 b_{k}D_{k}(\pi)|=\tfrac12|b_{k}|D_{k}(\pi)$, and by Step 1 with $t=|b|$ and $M=B$ (admissible as in claim 4) and Elementary Properties of Series of Real Numbers §linearity, $\sum_{k}\tfrac12|b_{k}|D_{k}(\pi)$ converges with sum at most $\tfrac{B}{2}I^{a}(\pi)$. Therefore An Absolutely Convergent Series of Real Numbers Converges §dominated gives

$$\bigl|\Phi_{b}(\nu)-\Phi_{b}(\mu)-\mathcal{J}^{a}(V_{b}(\mu),\pi)\bigr|\le\sum_{k=1}^{\infty}\tfrac12|b_{k}|D_{k}(\pi)\le\frac{B}{2}\,I^{a}(\pi).$$

Claim 6. Let $B$ be a bound for $b$. We verify the three properties of Noise Intrinsic Test Functions on the Noise Wasserstein Space §test, proving (a) and (b) at every $\mu\in\mathcal{P}^{a}_{\rho}$, which covers every $Q$.

(a) Let $\mu\in\mathcal{P}^{a}_{\rho}$ and $0<\varepsilon$. Put $C=\lVert V_{b}(\mu)\rVert_{\mu}+\tfrac{B}{2}+1>0$ and $\theta=\min\{1,\varepsilon C^{-1}\}>0$. Let $\nu\in\mathcal{P}^{a}_{\rho}$ with $W=W_{a}(\mu,\nu)<\theta$. 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 and 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 there is a noise-optimal $\pi\in\Pi^{a}(\mu,\nu)$, so $I^{a}(\pi)=W^{2}$ (Noise-Optimal Couplings §optimal) and $\sqrt{I^{a}(\pi)}=W$ by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root. By claim 5, the triangle inequality for the absolute value and Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §bound,

$$|\Phi_{b}(\nu)-\Phi_{b}(\mu)|\le\lVert V_{b}(\mu)\rVert_{\mu}\,W+\tfrac{B}{2}W^{2}\le\bigl(\lVert V_{b}(\mu)\rVert_{\mu}+\tfrac{B}{2}\bigr)W\le C\,W<\varepsilon,$$

using $W^{2}\le W$ (as $0\le W<1$) and $W<\varepsilon C^{-1}$. So $\Phi_{b}$ is continuous at $\mu$ in the sense of Continuous Map Between Metric Spaces, for the metric $W_{a}$ of 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 the absolute-value metric; this is property (a).

(b) Let $\mu\in\mathcal{P}^{a}_{\rho}$ and $0<\varepsilon$, and put $\theta=2\varepsilon(B+1)^{-1}>0$. Let $\nu\in\mathcal{P}^{a}_{\rho}$ and $\pi\in\Pi^{a}(\mu,\nu)$ with $I^{a}(\pi)<\theta^{2}$; then $\sqrt{I^{a}(\pi)}<\theta$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, and claim 5 gives

$$\bigl|\Phi_{b}(\nu)-\Phi_{b}(\mu)-\mathcal{J}^{a}(V_{b}(\mu),\pi)\bigr|\le\tfrac{B}{2}\sqrt{I^{a}(\pi)}\,\sqrt{I^{a}(\pi)}\le\tfrac{B}{2}\,\theta\,\sqrt{I^{a}(\pi)}=\tfrac{B}{B+1}\,\varepsilon\,\sqrt{I^{a}(\pi)}\le\varepsilon\,\sqrt{I^{a}(\pi)} .$$

So $\Phi_{b}$ is differentiable along noise couplings at $\mu$ with gradient $V_{b}(\mu)$ in the sense of Differentiability of a Function on the Noise-Connected Measures Along Noise Couplings, and Its Gradient §differentiable, and $\nabla\Phi_{b}(\mu)=V_{b}(\mu)$ by the uniqueness in Differentiability of a Function on the Noise-Connected Measures Along Noise Couplings, and Its Gradient §gradient.

It remains to show $V_{b}(\mu)\in T^{a}_{\mu}$. Fix $n\in\mathbb{N}$, let $M_{n}$ be the diagonal (hence symmetric) real $n\times n$ matrix with diagonal entries $a_{1}b_{1},\dots,a_{n}b_{n}$, and let $F_{n}(u)=\tfrac12u\cdot(M_{n}u)$ on $\mathbb{R}^{n}$. By Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §quadratic (with $q=0$, $c=0$), $F_{n}$ is of class $C^{2}$ on $\mathbb{R}^{n}$ with $DF_{n}(u)=M_{n}u=(a_{1}b_{1}u_{1},\dots,a_{n}b_{n}u_{n})$ and $D^{2}F_{n}(u)=M_{n}$, so by Hessian Matrix of a C^2 Function every iterated second partial derivative of $F_{n}$ is an entry of $M_{n}$ and has absolute value at most $B$, since the diagonal entries satisfy $|a_{k}b_{k}|=a_{k}|b_{k}|\le B$ and the off-diagonal entries are $0$, with $|0|=0\le B$. Since $\mu\in\mathcal{P}_{2}(X)$ 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 §moments, $\tilde{\mu}_{n}=(r_{n})_{\#}\mu\in\mathcal{P}_{2}(\mathbb{R}^{n})$ by Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §head. By Integrals of Functions with a Bounded Hessian are Intrinsic Test Functions on the Wasserstein Space §integrable and Integrals of Functions with a Bounded Hessian are Intrinsic Test Functions on the Wasserstein Space §tangent (with $d=n$, $f=F_{n}$ and constant $B$), $DF_{n}$ is Borel with $\int\lVert DF_{n}\rVert^{2}\,d\tilde{\mu}_{n}<\infty$ and its class lies in $T_{\tilde{\mu}_{n}}$. By Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §lift and Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §tangent, the class $V^{(n)}$ of the lift $\Lambda_{n}(DF_{n})$ lies in $T^{a}_{\mu}$, and

$$\Lambda_{n}(DF_{n})(x)=\sum_{k=1}^{n}a_{k}^{1/2}\,a_{k}b_{k}\,a_{k}^{-1/2}x_{k}\,e_{k}=\sum_{k=1}^{n}a_{k}b_{k}x_{k}\,e_{k}\qquad(x\in X).$$

By orthonormality of $(e_{k})$ the $j$-th coordinate of this vector is $a_{j}b_{j}x_{j}$ for $j\le n$ and $0$ for $j>n$, so by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §basis its coordinate along $f_{j}$ is $a_{j}^{1/2}b_{j}x_{j}$ for $j\le n$ and $0$ for $j>n$. Hence, by The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates and claim 3, $V_{b}(\mu)-V^{(n)}$ has coordinate the zero class along $f_{k}$ for $k\le n$ and the class of $a_{k}^{1/2}b_{k}x_{k}$ for $k>n$, and with $s_{N}=\sum_{k\le N}a_{k}b_{k}^{2}m_{k}(\mu)$ and $s_{\infty}=\lVert V_{b}(\mu)\rVert_{\mu}^{2}$ its squared norm is the sum of the series whose $N$-th partial sum is $s_{N}-s_{n}$ for $N\ge n$:

$$\lVert V_{b}(\mu)-V^{(n)}\rVert_{\mu}^{2}=s_{\infty}-s_{n}.$$

By Series of Real Numbers §convergent, $s_{\infty}-s_{n}\to0$ as $n\to\infty$; given $0<\varepsilon$, for all large $n$ we have $s_{\infty}-s_{n}<\varepsilon^{2}$, hence $\lVert V_{b}(\mu)-V^{(n)}\rVert_{\mu}<\varepsilon$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. So $(V^{(n)})_{n}$ converges to $V_{b}(\mu)$ in the metric of $L^{2}(\mu;X^{a})$ (Real Hilbert Space §topology, The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §metric), and $V_{b}(\mu)$ lies in the closure of $T^{a}_{\mu}$ by Sequential Characterization of the Closure in a Metric Space. As $T^{a}_{\mu}$ is closed (Linearity of the Noise Gradient, and the Noise Tangent Space is a Closed Linear Subspace §subspace), it equals its closure by claim 4 of The Closure is the Smallest Closed Superset, so $V_{b}(\mu)\in T^{a}_{\mu}$. This is property (b), with $\nabla\Phi_{b}(\mu)=V_{b}(\mu)$.

(c) Let $\mu\in Q$, let $(\mu_{n})$ be a sequence in $Q$ and $(\pi_{n})$ a sequence of couplings of vanishing noise cost from $(\mu_{n})$ to $\mu$ (Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §couplings), so $\pi_{n}\in\Pi^{a}(\mu_{n},\mu)$. Let $v_{n}$ and $v$ be representatives of $V_{b}(\mu_{n})=\nabla\Phi_{b}(\mu_{n})$ and $V_{b}(\mu)=\nabla\Phi_{b}(\mu)$. By Step 0 (ii), the classes $A_{n}$ of $v_{n}\circ\pi_{1}$ and $A'_{n}$ of $v\circ\pi_{2}$ in $L^{2}(\pi_{n};X^{a})$ have coordinates along $f_{k}$ the classes of $a_{k}^{1/2}b_{k}x_{k}$ and $a_{k}^{1/2}b_{k}y_{k}$. The map $z\mapsto v_{n}(x)-v(y)$ is $v_{n}\circ\pi_{1}-v\circ\pi_{2}$, whose class is $A_{n}-A'_{n}$ (The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations), so the discrepancy of Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §discrepancy is

$$\Delta_{n}=\int_{X\times X}|v_{n}(x)-v(y)|_{a}^{2}\,\pi_{n}(dz)=\lVert A_{n}-A'_{n}\rVert_{\pi_{n}}^{2}=\sum_{k=1}^{\infty}a_{k}b_{k}^{2}D_{k}(\pi_{n})\le B^{2}\,I^{a}(\pi_{n}),$$

by The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations, The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates (the coordinate of $A_{n}-A'_{n}$ along $f_{k}$ being the class of $a_{k}^{1/2}b_{k}(x_{k}-y_{k})$) and Step 1 with $t_{k}=a_{k}b_{k}^{2}$ and $M=B^{2}$, admissible as in claim 3. Given $0<\varepsilon$, since $I^{a}(\pi_{n})\to0$ there is $N$ with $I^{a}(\pi_{n})<\varepsilon(B^{2}+1)^{-1}$ for $n\ge N$, and then

$$0\le\Delta_{n}\le B^{2}\,I^{a}(\pi_{n})\le B^{2}\varepsilon(B^{2}+1)^{-1}<\varepsilon ,$$

the last inequality because $B^{2}<B^{2}+1$ and $0<\varepsilon$. So $\Delta_{n}\to0$ (Limit of a Sequence of Real Numbers): $(\nabla\Phi_{b}(\mu_{n}))$ converges strongly to $\nabla\Phi_{b}(\mu)$ along $(\pi_{n})$ in the sense of Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §strong, which is property (c).

Claim 7. Let $B$ be a bound for $b$. The series $\sum_{k}|sb_{k}|c_{k}=\sum_{k}|s|\,|b_{k}|c_{k}$ converges by Elementary Properties of Series of Real Numbers §linearity, and $|sb_{k}|a_{k}=|s|\,|b_{k}|a_{k}\le|s|B$ with $0\le|s|B$; so $sb$ is admissible with bound $|s|B$. For $\mu\in\mathcal{P}^{a}_{\rho}$, $\Phi_{sb}(\mu)=\tfrac12\sum_{k}s\,b_{k}m_{k}(\mu)=s\,\Phi_{b}(\mu)$ by Elementary Properties of Series of Real Numbers §linearity. By The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates, the coordinate of $s\,V_{b}(\mu)$ along $f_{k}$ is $s$ times the class of $a_{k}^{1/2}b_{k}x_{k}$, that is the class of $a_{k}^{1/2}(sb_{k})x_{k}$; by the uniqueness in claim 3 applied to $sb$, $V_{sb}(\mu)=s\,V_{b}(\mu)$.