Nearly Optimal Composite Displacements on the Torus Converge to the Optimal Displacement

In the setting of Optimal Transport on the Flat Torus: Standing Notation, let $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$ with $\mu$ absolutely continuous, let $T$ be an optimal map from $\mu$ to $\nu$, which exists by McCann's Theorem on the Flat Torus: Optimal Couplings out of an Absolutely Continuous Measure are Induced by a Unique Map with a Periodic Potential §map, and let $v_{T}$ be its displacement field. Let $q\in\mathbb{N}$ and, for each $n\in\mathbb{N}$, let $\Sigma_{n}\in\mathcal{P}(\mathbb{R}^{q})$ and let $X_{n},Y_{n},Z_{n}:\mathbb{R}^{q}\to\mathbb{R}^{d}$ be Borel maps such that

$$(X_{n})_{\#}\Sigma_{n}=\mu,\qquad(Y_{n})_{\#}\Sigma_{n}=\nu,\qquad Y_{n}(w)-X_{n}(w)-Z_{n}(w)\in\mathbb{Z}^{d}\ \text{ for every }w\in\mathbb{R}^{q},$$

and $\int\lVert Z_{n}\rVert^{2}\,d\Sigma_{n}<\infty$. Suppose that for every real $\varepsilon>0$ there is $N\in\mathbb{N}$ with $\int\lVert Z_{n}\rVert^{2}\,d\Sigma_{n}\le W_{\mathbb{T}}(\mu,\nu)^{2}+\varepsilon$ for every $n\ge N$.

1. (Convergence to the optimal displacement)¶ The sequence of real numbers $\bigl(\int_{\mathbb{R}^{q}}\lVert Z_{n}-v_{T}\circ X_{n}\rVert^{2}\,d\Sigma_{n}\bigr)_{n\in\mathbb{N}}$ converges to $0$.