The Noise Wasserstein Distance

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, the sets $\Pi^{a}(\mu,\nu)$ of couplings of finite noise cost, the noise cost $I^{a}$ and noise-connectedness are those of Couplings of Finite Noise Cost and Their Noise Cost.

1. (Noise Wasserstein distance)¶ Let $\mu,\nu\in\mathcal{P}(X)$ be such that $(\mu,\nu)$ is noise-connected. Then $\{I^{a}(\pi):\pi\in\Pi^{a}(\mu,\nu)\}$ is a set of real numbers, nonempty because $\Pi^{a}(\mu,\nu)$ is nonempty by Couplings of Finite Noise Cost and Their Noise Cost §connected, each nonnegative by Couplings of Finite Noise Cost and Their Noise Cost §cost, hence bounded below by $0$; it has a unique greatest lower bound $\inf\{I^{a}(\pi):\pi\in\Pi^{a}(\mu,\nu)\}$ by Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below, which is nonnegative, $0$ being a lower bound. The noise Wasserstein distance between $\mu$ and $\nu$ is

$$W_{a}(\mu,\nu)=\sqrt{\inf\{I^{a}(\pi):\pi\in\Pi^{a}(\mu,\nu)\}}\ ,$$

the nonnegative square root of that greatest lower bound; thus $W_{a}(\mu,\nu)$ is a nonnegative real number with $W_{a}(\mu,\nu)^{2}\le I^{a}(\pi)$ for every $\pi\in\Pi^{a}(\mu,\nu)$.