Squared Noise Wasserstein Distances, Their Convergent Series and Linear Combinations are Noise Intrinsic Test Functions

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, let $Q$ be a subset of the set $\mathcal{P}^{a}_{\rho}$ of The Measures Noise-Connected to the Reference Measure §space that has the noise map property, and let $W_{a}$ be the noise Wasserstein distance. Noise intrinsic test functions on $Q$ are those of that definition, and the gradient along noise couplings $\nabla\varphi(\mu)\in L^{2}(\mu;X^{a})$ of such a function $\varphi$ at $\mu\in Q$ is that of Differentiability of a Function on the Noise-Connected Measures Along Noise Couplings, and Its Gradient §gradient. Noise-optimal maps $S$ from $\mu$ to $\nu$ and their displacements $S-\mathrm{id}\in L^{2}(\mu;X^{a})$ are those of that definition, and $\mathrm{id}-S$ denotes $-(S-\mathrm{id})$; for $\mu\in Q$ and $\nu\in\mathcal{P}^{a}_{\rho}$ such a map exists by the noise map property. For $\mu,\nu\in\mathcal{P}^{a}_{\rho}$ the ordered 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 that $W_{a}(\mu,\nu)$ is defined, and $\rho\in\mathcal{P}^{a}_{\rho}$ 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. For $\mu\in\mathcal{P}^{a}_{\rho}$ the noise tangent space $T^{a}_{\mu}$, in which the gradients of noise intrinsic test functions lie, is a closed linear subspace of $L^{2}(\mu;X^{a})$ by Linearity of the Noise Gradient, and the Noise Tangent Space is a Closed Linear Subspace §subspace. Sequences, convergent series of real numbers and convergent series in the real Hilbert space $L^{2}(\mu;X^{a})$ are those of those items. For $\varphi_{1},\varphi_{2}:\mathcal{P}^{a}_{\rho}\to\mathbb{R}$ and $s,t\in\mathbb{R}$, $s\varphi_{1}+t\varphi_{2}$ is the function on $\mathcal{P}^{a}_{\rho}$ with value $s\,\varphi_{1}(\mu)+t\,\varphi_{2}(\mu)$ at $\mu$. Then the following hold.

1. (The squared distance to a fixed measure)¶ Let $\nu_{0}\in\mathcal{P}^{a}_{\rho}$ and let $\psi:\mathcal{P}^{a}_{\rho}\to\mathbb{R}$ be the function $\psi(\mu)=W_{a}(\mu,\nu_{0})^{2}$. Then $\psi$ is a noise intrinsic test function on $Q$, and for $\mu\in Q$ and any noise-optimal map $S_{\mu}$ from $\mu$ to $\nu_{0}$,

$$\nabla\psi(\mu)=2\,(\mathrm{id}-S_{\mu}).$$

2. (Convergent series of distances)¶ Let $B\in\mathbb{R}$ with $0\le B$, let $(\mu_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathcal{P}^{a}_{\rho}$ with $W_{a}(\mu_{k},\rho)\le B$ for every $k\in\mathbb{N}$, and let $(\beta_{k})_{k\in\mathbb{N}}$ be a sequence of positive real numbers whose series converges. Then for every $\mu\in\mathcal{P}^{a}_{\rho}$ the series $\sum_{k=1}^{\infty}\beta_{k}\,W_{a}(\mu,\mu_{k})^{2}$ and $\sum_{k=1}^{\infty}\beta_{k}\,W_{a}(\mu,\mu_{k})$ converge.

3. (The series is a test function)¶ With $B$, $(\mu_{k})_{k\in\mathbb{N}}$ and $(\beta_{k})_{k\in\mathbb{N}}$ as in claim 2, the function $\psi:\mathcal{P}^{a}_{\rho}\to\mathbb{R}$,

$$\psi(\mu)=\sum_{k=1}^{\infty}\beta_{k}\,W_{a}(\mu,\mu_{k})^{2},$$

whose defining series converges by claim 2, is a noise intrinsic test function on $Q$.

4. (The gradient of the series)¶ With $(\mu_{k})_{k\in\mathbb{N}}$, $(\beta_{k})_{k\in\mathbb{N}}$ and $\psi$ as in claim 3, for $\mu\in Q$ and, for every $k\in\mathbb{N}$, a noise-optimal map $S_{k}$ from $\mu$ to $\mu_{k}$, the series $\sum_{k=1}^{\infty}2\beta_{k}\,(\mathrm{id}-S_{k})$ converges in $L^{2}(\mu;X^{a})$ to $\nabla\psi(\mu)$, and

$$\lVert\nabla\psi(\mu)\rVert_{\mu}\le2\sum_{k=1}^{\infty}\beta_{k}\,W_{a}(\mu,\mu_{k}).$$

5. (Linear combinations)¶ Let $\varphi_{1}$ and $\varphi_{2}$ be noise intrinsic test functions on $Q$ and let $s,t\in\mathbb{R}$. Then $s\varphi_{1}+t\varphi_{2}$ is a noise intrinsic test function on $Q$, and

$$\nabla(s\varphi_{1}+t\varphi_{2})(\mu)=s\,\nabla\varphi_{1}(\mu)+t\,\nabla\varphi_{2}(\mu)\qquad(\mu\in Q).$$