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

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, with the reference measure $\rho\in\mathcal{P}_{2}(X)$ of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §reference, the sets $\Pi^{a}(\mu,\nu)$ of couplings of finite noise cost and noise-connectedness are those of Couplings of Finite Noise Cost and Their Noise Cost; $W_{a}$ is the noise Wasserstein distance, a nonnegative real-valued function on the noise-connected ordered pairs of members of $\mathcal{P}(X)$; noise-optimal couplings are as defined there; and $\mathcal{P}^{a}_{\rho}$ is the set of measures noise-connected to $\rho$. Then claims 1 to 3 hold, claims 4 to 6 hold for all $\mu,\nu,\lambda\in\mathcal{P}^{a}_{\rho}$, and claims 7 to 9 hold.

1. (The reference measure and finite second moments)¶ $\rho\in\mathcal{P}^{a}_{\rho}$. ¶ $\mathcal{P}^{a}_{\rho}\subseteq\mathcal{P}_{2}(X)$.

2. (Noise-connectedness)¶ For any two members $\mu,\nu$ of $\mathcal{P}^{a}_{\rho}$, distinct or not, the ordered pair $(\mu,\nu)$ is noise-connected, so that $W_{a}(\mu,\nu)$ is defined.

3. (Noise-optimal couplings)¶ For all $\mu,\nu\in\mathcal{P}(X)$ such that $(\mu,\nu)$ is noise-connected there is a noise-optimal coupling $\pi\in\Pi^{a}(\mu,\nu)$.

4. (Symmetry)¶ $W_{a}(\mu,\nu)=W_{a}(\nu,\mu)$.

5. (Separation)¶ $W_{a}(\mu,\nu)=0$ if and only if $\mu=\nu$.

6. (Triangle inequality)¶ $W_{a}(\mu,\lambda)\le W_{a}(\mu,\nu)+W_{a}(\nu,\lambda)$.

7. (The noise Wasserstein space)¶ The restriction of $W_{a}$ to pairs of members of $\mathcal{P}^{a}_{\rho}$ is a metric on $\mathcal{P}^{a}_{\rho}$; the metric space $(\mathcal{P}^{a}_{\rho},W_{a})$ is called the noise Wasserstein space based at $\rho$.

8. (Comparison)¶ For all $\mu,\nu\in\mathcal{P}^{a}_{\rho}$, which belong to $\mathcal{P}_{2}(X)$ by claim 1, the quadratic Wasserstein distance satisfies $W_{2}(\mu,\nu)\le\sqrt{\bar{a}}\,W_{a}(\mu,\nu)$, with $\bar{a}$ the positive bound on the noise weights of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §weights.

9. (Lower semicontinuity)¶ Let $\mu_{j},\nu_{j}\in\mathcal{P}^{a}_{\rho}$ for $j\in\mathbb{N}$ and $\mu,\nu\in\mathcal{P}(X)$ satisfy $\mu_{j}\Rightarrow\mu$ and $\nu_{j}\Rightarrow\nu$, and suppose that the sequence $(W_{a}(\mu_{j},\nu_{j}))_{j\in\mathbb{N}}$ of real numbers is bounded. Then $(\mu,\nu)$ is noise-connected and $W_{a}(\mu,\nu)\le\liminf_{j}W_{a}(\mu_{j},\nu_{j})$, the limit inferior of that bounded sequence. ¶ If moreover $\mu\in\mathcal{P}^{a}_{\rho}$, then $\nu\in\mathcal{P}^{a}_{\rho}$.