Half the Squared Torus Wasserstein Distance to a Fixed Measure is an Intrinsic Test Function on the Absolutely Continuous Measures

In the setting of Optimal Transport on the Flat Torus: Standing Notation, let $\nu\in\mathcal{P}(\mathbb{T}^{d})$, let $\Phi_{\nu}(\mu)=\tfrac12W_{\mathbb{T}}(\mu,\nu)^{2}$ as in Half the Squared Torus Wasserstein Distance to a Fixed Measure: the One-Sided Bound Along Couplings, Differentiability at an Absolutely Continuous Source, and the Tangent Gradient, and let $\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$ denote the set of absolutely continuous members of $\mathcal{P}(\mathbb{T}^{d})$. For $\mu\in\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$ an optimal map from $\mu$ to $\nu$ 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. Then the following hold.

1. (Stability of the displacement in the source)¶ Let $\mu\in\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$ and, for $n\in\mathbb{N}$, $\mu_{n}\in\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$; let $T$ and $T_{n}$ be optimal maps from $\mu$ and from $\mu_{n}$ to $\nu$, and let $\gamma_{n}\in\Pi(\mu_{n},\mu)$ be such that $(I_{\mathbb{T}}(\gamma_{n}))_{n\in\mathbb{N}}$ converges to $0$. Then

$$\Bigl(\int_{\mathbb{R}^{d+d}}\bigl\lVert v_{T_{n}}(\mathrm{pr}_{1}(w))-v_{T}(\mathrm{pr}_{2}(w))\bigr\rVert^{2}\,\gamma_{n}(dw)\Bigr)_{n\in\mathbb{N}}$$

converges to $0$.

2. (Test function)¶ $\Phi_{\nu}$ is an intrinsic test function on $\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$, with $\nabla\Phi_{\nu}(\mu)=-v_{T}$ for $\mu\in\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$ and any optimal map $T$ from $\mu$ to $\nu$.