Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, let $\mu,\rho,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, let $\pi_{12}\in\Pi(\mu,\rho)$ and $\pi_{23}\in\Pi(\rho,\nu)$ be couplings, with $M_{2}$ the second moment and $I$ the quadratic cost fixed there, and let $\mathrm{q}_{1},\mathrm{q}_{2},\mathrm{q}_{3}:\mathbb{R}^{3d}\to\mathbb{R}^{d}$ be the coordinate maps of the threefold product, which are Borel, as are the pairings $(\mathrm{q}_{1},\mathrm{q}_{2})$, $(\mathrm{q}_{2},\mathrm{q}_{3})$ and $(\mathrm{q}_{1},\mathrm{q}_{3})$. For $\sigma\in\mathcal{P}(\mathbb{R}^{3d})$, $L^{2}(\sigma;\mathbb{R}^{d})$ is the space of square-integrable vector fields against $\sigma$, read with $3d$ in place of $q$ and $d$ in place of $r$, with its inner product $\langle\cdot,\cdot\rangle_{\sigma}$ and norm $\lVert\cdot\rVert_{\sigma}$; the class of a Borel map $\mathbb{R}^{3d}\to\mathbb{R}^{d}$ that is square-integrable against $\sigma$ is denoted by the same symbol as the map. Push-forwards are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv). Then the following hold.

1. (Gluing)(i) There is $\sigma\in\mathcal{P}(\mathbb{R}^{3d})$ with

$$(\mathrm{q}_{1},\mathrm{q}_{2})_{\#}\sigma=\pi_{12},\qquad(\mathrm{q}_{2},\mathrm{q}_{3})_{\#}\sigma=\pi_{23};$$

such a $\sigma$ is called a gluing of $\pi_{12}$ and $\pi_{23}$.

2. (The coordinate fields of a gluing)(ii) Let $\sigma$ be a gluing of $\pi_{12}$ and $\pi_{23}$. Then the classes of $\mathrm{q}_{1}$, $\mathrm{q}_{2}$ and $\mathrm{q}_{3}$ belong to $L^{2}(\sigma;\mathbb{R}^{d})$, with

$$\lVert\mathrm{q}_{1}\rVert_{\sigma}^{2}=M_{2}(\mu),\qquad\lVert\mathrm{q}_{2}\rVert_{\sigma}^{2}=M_{2}(\rho),\qquad\lVert\mathrm{q}_{3}\rVert_{\sigma}^{2}=M_{2}(\nu),$$ $$\lVert\mathrm{q}_{1}-\mathrm{q}_{2}\rVert_{\sigma}^{2}=I(\pi_{12}),\qquad\lVert\mathrm{q}_{2}-\mathrm{q}_{3}\rVert_{\sigma}^{2}=I(\pi_{23}).$$

3. (The composite coupling)(iii) Let $\sigma$ be a gluing of $\pi_{12}$ and $\pi_{23}$. Then $(\mathrm{q}_{1},\mathrm{q}_{3})_{\#}\sigma$ belongs to $\Pi(\mu,\nu)$,

$$\lVert\mathrm{q}_{1}-\mathrm{q}_{3}\rVert_{\sigma}^{2}=I\bigl((\mathrm{q}_{1},\mathrm{q}_{3})_{\#}\sigma\bigr),$$

and

$$\sqrt{I\bigl((\mathrm{q}_{1},\mathrm{q}_{3})_{\#}\sigma\bigr)}\le\sqrt{I(\pi_{12})}+\sqrt{I(\pi_{23})}.$$