Wasserstein-1 coupling cost on dimensionless phase space

Use the dimensionless one-particle phase space $Z:={\mathbb{T}}^d\times {\mathbb{R}}^d$ from Smooth periodic finite-N and scaled mean-field phase data, with its product topology and Borel sigma algebra. The finite torus is second-countable by The one-dimensional torus and its normalized Haar integral, and Euclidean space is second-countable. Consequently $Z$ and $Z\times Z$ are second-countable, and the Borel sigma algebra of each finite product equals the product of the factor Borel sigma algebras: every open set is a countable union of basic open rectangles. Use the product topology and its Borel sigma algebra on $Z\times Z$ as well.

For $x,y\in {\mathbb{T}}^d$, let

$d_{{\mathbb{T}}^d}(x,y):={\min }_{\mathrm{\ell }\in {\mathbb{Z}}^d}\mathrm{\mid }\tilde{x}-\tilde{y}+\mathrm{\ell }\mathrm{\mid },$

where $\tilde{x},\tilde{y}\in {\mathbb{R}}^d$ are any representatives; the minimum is independent of the representatives. Equip $Z$ with

$d_Z((x,v),(y,w)):=d_{{\mathbb{T}}^d}(x,y)+\mathrm{\mid }v-w\mathrm{\mid }\mathrm{.}$

Let ${\mathcal{P}}_1(Z)$ be the Borel probability measures $\mu$ on $Z$ with ${\int }_Z\mathrm{\mid }v\mathrm{\mid }d\mu (x,v)<\mathrm{\infty }$. For $\mu ,\nu \in {\mathcal{P}}_1(Z)$, a coupling is a Borel probability $\pi$ on $Z\times Z$ satisfying $\pi (A\times Z)=\mu (A)$ and $\pi (Z\times B)=\nu (B)$ for all Borel $A,B\subseteq Z$; write $\mathrm{\Pi }(\mu ,\nu )$ for the set of couplings. Define the phase-space Wasserstein-1 coupling cost by

${\mathcal{W}}_1(\mu ,\nu ):={\inf }_{\pi \in \mathrm{\Pi }(\mu ,\nu )}{\int }_{Z\times Z}{d}_Z(z,z^{\mathrm{\prime }})d\pi (z,z^{\mathrm{\prime }})\mathrm{.}$

By For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique, $\mu \otimes \nu$ is a probability on $\mathcal{B}(Z)\otimes \mathcal{B}(Z)=\mathcal{B}(Z\times Z)$ and its marginals are $\mu ,\nu$; hence $\mathrm{\Pi }(\mu ,\nu )$ is nonempty. The infimum is finite: this product coupling has torus cost at most $\sqrt{d}\mathrm{/}2$, and Tonelli's theorem bounds its velocity cost by $\int \mathrm{\mid }v\mathrm{\mid }d\mu +\int \mathrm{\mid }w\mathrm{\mid }d\nu$. No minimizing coupling is assumed to exist.