Differentiability Along Couplings of a Function on the Torus Wasserstein Space, and Its Gradient

In the setting of Optimal Transport on the Flat Torus: Standing Notation, let $\varphi:\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$ and $\mu\in\mathcal{P}(\mathbb{T}^{d})$, let $L^{2}(\mu;\mathbb{R}^{d})$ be as in Optimal Transport on the Flat Torus: Standing Notation §fields, and let $\mathcal{J}_{\mathbb{T}}$ be the torus displacement pairing.

1. (Differentiability along couplings)¶ Let $\eta\in L^{2}(\mu;\mathbb{R}^{d})$. The function $\varphi$ is differentiable along couplings at $\mu$ with gradient $\eta$ if for every real $\varepsilon>0$ there is a real $\theta>0$ such that

$$\bigl|\varphi(\nu)-\varphi(\mu)-\mathcal{J}_{\mathbb{T}}(\eta,\gamma)\bigr|\le\varepsilon\sqrt{I_{\mathbb{T}}(\gamma)}$$

for every $\nu\in\mathcal{P}(\mathbb{T}^{d})$ and every $\gamma\in\Pi(\mu,\nu)$ with $I_{\mathbb{T}}(\gamma)<\theta^{2}$. It is differentiable along couplings at $\mu$ if some $\eta\in L^{2}(\mu;\mathbb{R}^{d})$ has this property.

2. (Gradient)¶ If $\varphi$ is differentiable along couplings at $\mu$, exactly one $\eta$ has the property of clause 1. Indeed, if $\eta$ and $\eta'$ both have it, let $\varepsilon>0$, take $\theta$ and $\theta'$ as in clause 1 for $\eta$ and $\eta'$ with $\varepsilon/2$ in place of $\varepsilon$, and let $\theta''$ be the smaller of the two (Elementary Properties of the Minimum of Two Elements). For $\gamma$ with $I_{\mathbb{T}}(\gamma)<(\theta'')^{2}$ both estimates hold, and by The Torus Displacement Pairing: Linearity, the Cost Bound, and Vanishing of a Field with First-Order Small Pairings §linear and the triangle inequality (claim 5 of Properties of the Absolute Value in an Ordered Field) $|\mathcal{J}_{\mathbb{T}}(\eta-\eta',\gamma)|\le\varepsilon\sqrt{I_{\mathbb{T}}(\gamma)}$. Hence $\eta-\eta'=0$ by The Torus Displacement Pairing: Linearity, the Cost Bound, and Vanishing of a Field with First-Order Small Pairings §vanishing. This $\eta$ is written $\nabla\varphi(\mu)$ and called the gradient along couplings of $\varphi$ at $\mu$.