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

In the setting of Optimal Transport on the Flat Torus: Standing Notation, let $\nu\in\mathcal{P}(\mathbb{T}^{d})$ and let $\Phi_{\nu}:\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$ be $\Phi_{\nu}(\mu)=\tfrac12W_{\mathbb{T}}(\mu,\nu)^{2}$. Let $\mathcal{J}_{\mathbb{T}}$ be the torus displacement pairing, $T_{\mu}$ the tangent space, and differentiability along couplings and the gradient along couplings as defined there. For $\mu\in\mathcal{P}(\mathbb{T}^{d})$ and an optimal map $T$ from $\mu$ to $\nu$, the displacement field $v_{T}$ is Borel with $\lVert v_{T}\rVert\le\sqrt d/2$ by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range, so its class belongs to $L^{2}(\mu;\mathbb{R}^{d})$. Then the following hold.

1. (One-sided bound)¶ Let $\mu\in\mathcal{P}(\mathbb{T}^{d})$ and let $T$ be an optimal map from $\mu$ to $\nu$. Then $\Phi_{\nu}(\mu)=\tfrac12\lVert v_{T}\rVert_{\mu}^{2}$, and for every $\mu'\in\mathcal{P}(\mathbb{T}^{d})$ and every $\gamma\in\Pi(\mu,\mu')$,

$$\Phi_{\nu}(\mu')\le\Phi_{\nu}(\mu)-\mathcal{J}_{\mathbb{T}}(v_{T},\gamma)+\tfrac12I_{\mathbb{T}}(\gamma).$$

2. (Differentiability)¶ Let $\mu\in\mathcal{P}(\mathbb{T}^{d})$ be absolutely continuous and 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. Then $\Phi_{\nu}$ is differentiable along couplings at $\mu$ with $\nabla\Phi_{\nu}(\mu)=-v_{T}$.

3. (Tangent gradient)¶ In the situation of clause 2, the class of $v_{T}$ belongs to $T_{\mu}$.