Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings

In the setting of The Flat Torus: Standing Notation, used with $n=d$ for a natural number $d$ with $1\le d$, and of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $Q$ be the half-open unit cell, which belongs to $\mathcal{B}(\mathbb{R}^{d})$ by The Half-Open Unit Cell Tiles Euclidean Space §cell, and let $d_{\mathbb{T}}$ be the flat torus distance. For $\mu,\nu\in\mathcal{P}(\mathbb{R}^{d})$, $\Pi(\mu,\nu)$ is the set of their couplings; it is nonempty by Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §product.

1. (Measures on the torus)¶ $\mathcal{P}(\mathbb{T}^{d})$ denotes the set of those $\mu\in\mathcal{P}(\mathbb{R}^{d})$ with $\mu(Q)=1$.

2. (Torus cost)¶ For $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$ and $\gamma\in\Pi(\mu,\nu)$, the torus cost of $\gamma$ is

$$I_{\mathbb{T}}(\gamma)=\int_{\mathbb{R}^{d+d}}d_{\mathbb{T}}\bigl(\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w)\bigr)^{2}\,\gamma(dw).$$

The integrand is Borel by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §lipschitz and takes values in $[0,d/4]$ by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range, so it is integrable by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures, and $I_{\mathbb{T}}(\gamma)$ is a real number with $0\le I_{\mathbb{T}}(\gamma)\le d/4$ by the monotonicity of the integral, claim 1 of Linearity and Monotonicity of the Lebesgue Integral.

3. (Torus Wasserstein distance)¶ For $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$ the set $\{I_{\mathbb{T}}(\gamma):\gamma\in\Pi(\mu,\nu)\}$ is nonempty and bounded below by $0$, so it has a greatest lower bound by Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below, which is nonnegative. The torus Wasserstein distance is

$$W_{\mathbb{T}}(\mu,\nu)=\sqrt{\inf\{I_{\mathbb{T}}(\gamma):\gamma\in\Pi(\mu,\nu)\}}\ .$$

4. (Optimal coupling)¶ For $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$, a coupling $\gamma\in\Pi(\mu,\nu)$ is optimal if $I_{\mathbb{T}}(\gamma)=W_{\mathbb{T}}(\mu,\nu)^{2}$.