Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, let $\mu,\nu\in\mathcal{P}(X)$. 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, the function $n_{a}:X\to\mathbb{R}$ is that of The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §borel, the set $D_{a}$ and the function $c_{a}:X\times X\to\mathbb{R}$ are those of The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §pairs, and $\mathrm{id}_{X}$ is the identity map of $X$. Then the following hold.

1. (Support bound)¶ Every $\pi\in\Pi^{a}(\mu,\nu)$ satisfies $I(\pi)\le\bar{a}\,I^{a}(\pi)$; in particular $I(\pi)<\infty$. Consequently, if $\Pi^{a}(\mu,\nu)$ is nonempty and one of $\mu$ and $\nu$ belongs to $\mathcal{P}_{2}(X)$, then so does the other.

2. (Swap)¶ Let $\sigma:X\times X\to X\times X$ be the swap $\sigma(x,y)=(y,x)$ of Couplings on a Hilbert Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, the Lipschitz Bound and the Moment Bound §swap. For every $\pi\in\Pi^{a}(\mu,\nu)$ one has $\sigma_{\#}\pi\in\Pi^{a}(\nu,\mu)$ and $I^{a}(\sigma_{\#}\pi)=I^{a}(\pi)$. In particular $(\mu,\nu)$ is noise-connected if and only if $(\nu,\mu)$ is noise-connected.

3. (Displacement couplings)¶ Let $S:X\to X$ be Borel with $S(x)-x\in X^{a}$ for every $x\in X$. The map $(\mathrm{id}_{X},S):X\to X\times X$ is Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §pairing, so the function $x\mapsto n_{a}(S(x)-x)$, which is $c_{a}\circ(\mathrm{id}_{X},S)$ with $c_{a}$ the nonnegative Borel function of The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §pairs, is Borel and nonnegative by claim 4 of Borel Measurability and Bounded Integration on a Metric Space, and its integral against $\mu$ is defined in $[0,\infty]$. If $\int_{X}n_{a}(S(x)-x)\,\mu(dx)<\infty$, then $(\mathrm{id}_{X},S)_{\#}\mu\in\Pi^{a}(\mu,S_{\#}\mu)$ and

$$I^{a}\bigl((\mathrm{id}_{X},S)_{\#}\mu\bigr)=\int_{X}n_{a}(S(x)-x)\,\mu(dx).$$

In particular $(\mathrm{id}_{X},\mathrm{id}_{X})_{\#}\mu\in\Pi^{a}(\mu,\mu)$ has noise cost $0$, so that $(\mu,\mu)$ is noise-connected.

4. (Lower semicontinuity of the noise cost)¶ Let $\mu_{j},\nu_{j}\in\mathcal{P}(X)$ for $j\in\mathbb{N}$ with $\mu_{j}\Rightarrow\mu$ and $\nu_{j}\Rightarrow\nu$, let $\pi_{j}\in\Pi^{a}(\mu_{j},\nu_{j})$, and let $\pi\in\mathcal{P}(X\times X)$ with $\pi_{j}\Rightarrow\pi$. If the sequence $(I^{a}(\pi_{j}))_{j\in\mathbb{N}}$ of real numbers is bounded, then $\pi\in\Pi^{a}(\mu,\nu)$ and $I^{a}(\pi)\le\liminf_{j}I^{a}(\pi_{j})$, the limit inferior of that bounded sequence.