The Torus Wasserstein Distance: Comparison with the Euclidean Distance, Wrapping, and Integrals of Periodic Functions

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 $\pi$ be the wrapping map and $d_{\mathbb{T}}$ the flat torus distance; let $\mathcal{P}(\mathbb{T}^{d})$, the torus cost $I_{\mathbb{T}}$ and the torus Wasserstein distance $W_{\mathbb{T}}$ be as in Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §measures. For $\mu,\nu\in\mathcal{P}(\mathbb{R}^{d})$ let $\Pi(\mu,\nu)$ and the quadratic cost $I$ be as in Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling, let $M_{2}$ be the second moment and $\mathcal{P}_{2}(\mathbb{R}^{d})$ as in The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space, and let $W_{2}$ be the quadratic Wasserstein distance. Then the following hold.

1. (Inclusion)¶ Every $\mu\in\mathcal{P}(\mathbb{T}^{d})$ belongs to $\mathcal{P}_{2}(\mathbb{R}^{d})$, with $M_{2}(\mu)\le d$.

2. (Comparison with the Euclidean cost)¶ For $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$ and $\gamma\in\Pi(\mu,\nu)$ one has $I_{\mathbb{T}}(\gamma)\le I(\gamma)$; hence $W_{\mathbb{T}}(\mu,\nu)\le W_{2}(\mu,\nu)$.

3. (Wrapping)¶ For $\mu'\in\mathcal{P}(\mathbb{R}^{d})$ one has $\pi_{\#}\mu'\in\mathcal{P}(\mathbb{T}^{d})$. For $\mu',\nu'\in\mathcal{P}(\mathbb{R}^{d})$ and $\gamma'\in\Pi(\mu',\nu')$, the map $\pi\times\pi=(\pi\circ\mathrm{pr}_{1},\pi\circ\mathrm{pr}_{2})$ of $\mathbb{R}^{d+d}$ is Borel, $(\pi\times\pi)_{\#}\gamma'\in\Pi(\pi_{\#}\mu',\pi_{\#}\nu')$, and

$$I_{\mathbb{T}}\bigl((\pi\times\pi)_{\#}\gamma'\bigr)=\int_{\mathbb{R}^{d+d}}d_{\mathbb{T}}\bigl(\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w)\bigr)^{2}\,\gamma'(dw)\le I(\gamma').$$

Hence $W_{\mathbb{T}}(\pi_{\#}\mu',\pi_{\#}\nu')\le W_{2}(\mu',\nu')$ whenever $\mu',\nu'\in\mathcal{P}_{2}(\mathbb{R}^{d})$.

4. (Integrals of periodic functions)¶ Let $f\in C_{\mathrm{per}}$, the continuous $\mathbb{Z}^{d}$-periodic functions. Then $f$ is bounded, by Elementary Properties of Lattice-Periodic Functions §bounded, and Borel, being continuous (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps), hence integrable with respect to every $\mu\in\mathcal{P}(\mathbb{R}^{d})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures. If $f$ is Lipschitz with constant $L$, for a real $L\ge0$, then $\bigl|\int f\,d\mu-\int f\,d\nu\bigr|\le L\,W_{\mathbb{T}}(\mu,\nu)$ for all $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$. If $\mu_{j},\mu\in\mathcal{P}(\mathbb{T}^{d})$ for $j\in\mathbb{N}$ and the real sequence $(W_{\mathbb{T}}(\mu_{j},\mu))_{j\in\mathbb{N}}$ converges to $0$, then $(\int f\,d\mu_{j})_{j\in\mathbb{N}}$ converges to $\int f\,d\mu$.