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

In the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, so that the reference measure is $\rho=\gamma_{c}$, every $\mu\in\mathcal{P}^{a}_{\rho}$ belongs 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 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, so that for every $k\in\mathbb{N}$ the Borel function $x\mapsto x_{k}^{2}$, which is at most $|x|^{2}$, is integrable with respect to $\mu$. For $\mu,\nu\in\mathcal{P}^{a}_{\rho}$, $\Pi^{a}(\mu,\nu)$ is the set of couplings of finite noise cost and $I^{a}(\pi)$ the noise cost of Couplings of Finite Noise Cost and Their Noise Cost §cost, and $\mathcal{J}^{a}(\eta,\pi)$ is the noise displacement pairing; the coordinate of an element of $L^{2}(\mu;X^{a})$ along $f_{k}$ is that of The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates, and $\langle\cdot,\cdot\rangle_{\mu}$, $\lVert\cdot\rVert_{\mu}$ are the inner product and norm of $L^{2}(\mu;X^{a})$. Noise intrinsic test functions and gradients along noise couplings are those of those definitions, square roots are those of Existence and Uniqueness of the Nonnegative Square Root, and convergent series are those of that definition. A real sequence $b=(b_{k})_{k\in\mathbb{N}}$ is called admissible if the series $\sum_{k=1}^{\infty}|b_{k}|\,c_{k}$ converges and there is $B\in\mathbb{R}$ with $0\le B$ and $|b_{k}|\,a_{k}\le B$ for every $k\in\mathbb{N}$; such a $B$ is called a bound for $b$.¶

1. (Moments)¶ Let $t=(t_{k})_{k\in\mathbb{N}}$ be a sequence of nonnegative real numbers such that $\sum_{k=1}^{\infty}t_{k}c_{k}$ converges, and let $M\in\mathbb{R}$ satisfy $0\le M$ and $t_{k}a_{k}\le M$ for every $k\in\mathbb{N}$. For every $\mu\in\mathcal{P}^{a}_{\rho}$ the series $S_{t}(\mu)=\sum_{k=1}^{\infty}t_{k}\int_{X}x_{k}^{2}\,\mu(dx)$ converges, and

$$S_{t}(\mu)\le2\sum_{k=1}^{\infty}t_{k}c_{k}+2M\,W_{a}(\mu,\rho)^{2}.$$

2. (A Lipschitz estimate)¶ Let $t$ and $M$ be as in claim 1, and let $\mu,\nu\in\mathcal{P}^{a}_{\rho}$ and $\pi\in\Pi^{a}(\mu,\nu)$. Then the series $\sum_{k=1}^{\infty}t_{k}\int_{X\times X}(y_{k}-x_{k})^{2}\,\pi(d(x,y))$ converges with sum at most $M\,I^{a}(\pi)$, and

$$\Bigl|\sqrt{S_{t}(\mu)}-\sqrt{S_{t}(\nu)}\Bigr|\le\sqrt{M}\,\sqrt{I^{a}(\pi)} .$$

3. (The gradient field)¶ Let $b$ be admissible and $\mu\in\mathcal{P}^{a}_{\rho}$. There is exactly one $V_{b}(\mu)\in L^{2}(\mu;X^{a})$ whose coordinate along $f_{k}$ is, for every $k\in\mathbb{N}$, the class of the function $x\mapsto a_{k}^{1/2}\,b_{k}\,x_{k}$, and

$$\lVert V_{b}(\mu)\rVert_{\mu}^{2}=\sum_{k=1}^{\infty}a_{k}\,b_{k}^{2}\int_{X}x_{k}^{2}\,\mu(dx).$$

4. (The profile)¶ Let $b$ be admissible. For every $\mu\in\mathcal{P}^{a}_{\rho}$ the series $\sum_{k=1}^{\infty}b_{k}\int_{X}x_{k}^{2}\,\mu(dx)$ converges absolutely, and the diagonal quadratic profile with coefficients $b$ is the function

$$\Phi_{b}:\mathcal{P}^{a}_{\rho}\to\mathbb{R},\qquad\Phi_{b}(\mu)=\frac{1}{2}\sum_{k=1}^{\infty}b_{k}\int_{X}x_{k}^{2}\,\mu(dx).$$

5. (First-order expansion)¶ Let $b$ be admissible with bound $B$. For $\mu,\nu\in\mathcal{P}^{a}_{\rho}$ and $\pi\in\Pi^{a}(\mu,\nu)$,

$$\bigl|\Phi_{b}(\nu)-\Phi_{b}(\mu)-\mathcal{J}^{a}(V_{b}(\mu),\pi)\bigr|\le\frac{B}{2}\,I^{a}(\pi).$$

6. (Test function)¶ Let $b$ be admissible. For every subset $Q\subseteq\mathcal{P}^{a}_{\rho}$, $\Phi_{b}$ is a noise intrinsic test function on $Q$, and $\nabla\Phi_{b}(\mu)=V_{b}(\mu)$ for every $\mu\in Q$.

7. (Scaling)¶ Let $b$ be admissible and $s\in\mathbb{R}$. Then the sequence $sb=(s\,b_{k})_{k\in\mathbb{N}}$ is admissible, and $\Phi_{sb}(\mu)=s\,\Phi_{b}(\mu)$ and $V_{sb}(\mu)=s\,V_{b}(\mu)$ for every $\mu\in\mathcal{P}^{a}_{\rho}$.